diff --git a/-NE4T4oBgHgl3EQfDgur/content/tmp_files/2301.04870v1.pdf.txt b/-NE4T4oBgHgl3EQfDgur/content/tmp_files/2301.04870v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a362733bb93610121ffdca440f732ae6a9beb1be --- /dev/null +++ b/-NE4T4oBgHgl3EQfDgur/content/tmp_files/2301.04870v1.pdf.txt @@ -0,0 +1,1484 @@ +Semantic Segmentation via Pixel-to-Center Similarity Calculation +Dongyue Wu1, Zilin Guo1, Aoyan Li1, Changqian Yu2, Changxin Gao1, Nong Sang1 +1National Key Laboratory of Science and Technology on Multispectral Information Processing, School of +Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China +{dongyue wu,zilin guo,aoyanli,cgao,nsang}@hust.edu.cn +2Meituan Inc., Beijing, China +changqianyu@meituan.com +Abstract +Since the fully convolutional network has achieved +great success in semantic segmentation, lots of works +have been proposed focusing on extracting discrimina- +tive pixel feature representations. However, we observe +that existing methods still suffer from two typical chal- +lenges, i.e. (i) large intra-class feature variation in dif- +ferent scenes, (ii) small inter-class feature distinction in +the same scene. In this paper, we first rethink semantic +segmentation from a perspective of similarity between +pixels and class centers. Each weight vector of the seg- +mentation head represents its corresponding semantic +class in the whole dataset, which can be regarded as the +embedding of the class center. Thus, the pixel-wise clas- +sification amounts to computing similarity in the final +feature space between pixels and the class centers. Un- +der this novel view, we propose a Class Center Similarity +layer (CCS layer) to address the above-mentioned chal- +lenges by generating adaptive class centers conditioned +on different scenes and supervising the similarities be- +tween class centers. It utilizes a Adaptive Class Center +Module (ACCM) to generate class centers conditioned +on each scene, which adapt the large intra-class vari- +ation between different scenes. Specially designed loss +functions are introduced to control both inter-class and +intra-class distances based on predicted center-to-center +and pixel-to-center similarity, respectively. Finally, the +CCS layer outputs the processed pixel-to-center simi- +larity as the segmentation prediction. Extensive experi- +ments demonstrate that our model performs favourably +against the state-of-the-art CNN-based methods. +Keywords: semantic segmentation, similarity, adaptive +class center, intra-class variation, intra-class distinction. +1. Introduction +Semantic segmentation aims to assign each pixel with a +semantic category, which is a fundamental and challenging +task in the computer vision field. Benefited from the devel- +opment of deep convolutional networks [23, 14, 22, 12], the +fully convolutional network (FCN) [18] has been the domi- +nant solution in the semantic segmentation task. +Due to the simplicity of the architecture of FCN, exist- +ing methods mainly focus on enhancing the feature repre- +sentations to improve visual recognition capability. They +aggregate rich contextual information via large receptive +field [31, 6], multi-scale methods [37, 4], or attention mech- +anisms [25, 9, 38, 8, 26]. However, we observe that these +methods still suffer from two challenges. (i) Large intra- +class feature variation between different scenes. Accord- +ing to Fig. 1, the features of the “trees” (colored in red) near +the wall in Sample A greatly differ from those on the bank +in Sample B, despite that they belong to the same semantic +category. Similarly, plant pixels of Sample A locate signif- +icantly far from those of Sample B, which also suggesting +large intra-class variation between pixels in different scenes. +(ii) Small inter-class distinction inside each scene. Take +Sample A in Fig. 1 for example, it is difficult for the seg- +mentation network to tell pixels belonging to the “tree” and +pixels belonging to the “plant” apart, since pixels belong- +ing to the “tree” are near to pixels of the “plant”, indicating +that the two groups of pixels of different semantic classes +bear similar features. These two challenges make it hard +to category pixels from the same class but different scenes +into the correct class while telling groups of pixels from the +different classes but the same scene apart from each other. +To solve the aforementioned challenges, we rethink pre- +vious methods from a perspective of similarity between pix- +els and class centers. Most previous methods utilize a 1 × 1 +convolutional layer as the segmentation head to get the fi- +nal prediction. The weights of the convolutional layer are +trained through all training samples and conduct a convolu- +tion operation on the feature maps in the inference phase. In +other words, the learned weights perform correlation calcu- +lation with each pixel and output the map of inner produc- +tion which is widely used to measure the similarity between +features. As each vector of the weights can be regarded as +1 +arXiv:2301.04870v1 [cs.CV] 12 Jan 2023 + +Figure 1. Illustration of the overall distribution of “tree” and “plant” pixels on the ADE20K validation set. We show the distribution of the +two classes in feature space. For clarity, we only label the pixels belonging to “tree” and “plant”, which are in red and blue, respectively, +while the pixels that belong to other classes keep their original color. Each dot in our plot represents a randomly sampled pixel in the +feature space. The light-colored dots denote pixels sampled from other samples(scenes) in the whole dataset, while the dark-colored ones +are sampled from Sample A and Sample B. According to the plots, the features of both “tree” and “plant” in Sample A are quite different +from those in Sample B, suggesting large intra-class variation between different scenes. The features of “tree” and “plant” are hard to +distinguish in Sample A, because of the small inter-class distinction. The dimension of features is reduced for illustration using t-SNE [24]. +(a) Images +(b) Baseline +(c) Ours +Figure 2. Comparison of pixel distribution in feature space on the +ADE20K validation set. Baseline: ResNet-101 + DeeplabV3+. +Ours: Baseline + CCS layer. Pixels that randomly sampled form +the “tree” and “plant” are colored in red and blue, respectively. +Our distribution of each class is more compact than the baseline. +There are also clear boundaries between different classes in the +feature space of our method, when the pixels of the two classes +are mixed up in the feature space of the baseline. +a learned representation of the corresponding class, the fi- +nal segmentation prediction map can also be regarded as a +pixel-to-center similarity map. These learned representa- +tions contain the common information of their class on the +whole dataset. Therefore, we call them global class centers. +In this view, the classification can be remodeled as follows: +A pixel will be assigned to the class whose global class cen- +ter is the most similar to the pixel among all classes. In con- +clusion, semantic segmentation can be viewed as a task to +predict the similarity between pixels and class centers. +Motivated by this perspective, we argue that there are +two limitations of previous methods responsible for the +above-mentioned challenges: (i) The global class centers +are incapable of adapting the large intra-class variations be- +tween different scenes, since the global class centers are +learned based on the whole dataset and are kept unchanged +and identical for different scenes during the inference stage. +(ii) Since there is no constraint on the similarities between +global class centers, these centers which represent different +classes may share excessively similar representations, lead- +ing to difficulties in correctly distinguishing pixels. These +limitations correspond to the two challenges of the large +intra-class variation between scenes and the small inter- +class distinction inside each scene, respectively. +To solve the challenges, we propose a novel and flexi- +ble Class Center Similarity (CCS) layer, which replaces +the segmentation heads of networks and transfers the pixel- +wise classification task to a pixel-to-center similarity pre- +diction task. Our CCS layer consists of three parts: Adap- +tive Class Center Module (ACCM), Similarity Calculation +Module (SCM), and Class Distance (CD) Loss. +First, +Adaptive Class Center Module (ACCM) generates adaptive + +Tree +Sample B +Plant囍class centers conditioned on each scene (image), which ac- +commodates the large intra-class variance between scenes. +Thus, the prediction only depends on the similarity between +each pixel and the scene-specific adaptive class centers, in- +stead of the immutable global class centers. Then, the adap- +tive class centers are forward Similarity Calculation Mod- +ule (SCM) to compute the pixels-to-centers similarity be- +tween pixels and class centers and the mutual similarity of +class centers as the inter-class similarity. Finally, our CD +Loss is applied to the inter-class similarity and pixels-to- +centers similarity in each scene to supervise the segmenta- +tion prediction while increasing the lack of inter-class dis- +tinction inside each scene simultaneously. The CCS layer +can be integrated into almost arbitrary semantic segmen- +tation architectures substituting for the final segmentation +head, namely the 1 × 1 convolution layer. To demonstrate +the effectiveness of the proposed Class Center Similarity +layer, we add the CCS layer to existing segmentation net- +works and carry out extensive experiments on ADE20K and +Pascal Context. Fig. 2 and Fig. 3 also show that our method +leads to clear and compact clusters for each semantic class +in each scene. In summary, the following contributions are +made in this paper: +• We rethink the semantic segmentation task from a +pixel-to-center (P-C) similarity perspective. Semantic +segmentation can be viewed as a task with two stages: +compute P-C similarity for each pixel and categorize +pixels to the most similar semantic class. +• We propose a class Center Similarity layer that can +generate conditional class centers for each scene under +the constraint of our Class Distance Loss. Our CCS +layer is easy to plug into almost any FCN-based se- +mantic segmentation network. +• We conduct extensive experiments to analyze the ef- +fectiveness of our approach. The proposed method, +called CCSNet, achieves state-of-the-art performance +on two challenging datasets. +Based on ResNet- +101 [11], our model achieves 47.76% mIoU on +ADE20K, and 54.9% mIoU on PASCAL Context. +2. Related Work +Semantic segmentation. +The development of deep neu- +ral networks dramatically boosts semantic segmentation. +Since FCN [18] replaces the fully connected layer in the +traditional classification network with a convolutional to +get pixel-wise predictions, FCNs achieves great success in +semantic segmentation. +Segnet [1], UNet [21] and Re- +fineNet [16] adopt encoder-decoder structure to recover the +spatial information that lost by downsample operation via +cascading upsampling. In order to capture long-range de- +pendencies, lots of works have been done by introducing +CRF [3, 2] and MRF [17] into segmentation tasks. Dilated +convolution [31] and deformable convolution[6], which in- +crease the resolution of feature maps, is used to enlarge +the receptive field. We revisit these typical networks and +find that semantic segmentation can be regarded as catego- +rizing each pixel to the semantic category with the great- +est pixel-to-center similarity (or the smallest pixel-to-center +distance). +Contextual information. +Context is believed of great +significance in semantic segmentation. +Recently, plenty +of works focusing on mining richer context information. +Multi-scale representations are used to capture more context +in PSPNet [37]. the family of DeepLab [3, 4, 5] also cap- +tures the context information from multi-scales. Based on +these approaches, many extensions have been proposed,e.g. +DenseASPP [27] and APCNet [10]. The Attention mecha- +nism is also adopted to capture global contextual informa- +tion [13, 15]. Other studies pay their attention to the sim- +ilarity of pixels to help aggregate contextual information. +OCNet [33], DANet [9] and CFNet [36] aggregate context +that computed on all pixels and augment context represen- +tation to the representation of each pixel. Based on the self- +attention mechanism, these approaches calculate similarity +(or relation) between pixels and aggregate representations +according to similarity. Although a great number of studies +explore discriminate representations to help segmentation, +the large variance between pixels of the same category from +different scenes and the lack of distinction in each scene still +remains. Our work addresses these two challenges based on +similarity as well, but we aim to get better similarity maps +while those utilize context to get better features. In addition, +thanks to our ACCM which generates the varying adaptive +class centers for classification, our method is more effective +with minor computational cost. +Center-based methods in semantic segmentation. +Re- +cently, several works propose approaches with centers in +semantic segmentation networks. In contrast to previous +works, ACFNet [34] presents the concept of class centers +as the global context from a categorical perspective, which +describes the overall representation of each class in a scene. +Then, different class centers are adaptively concatenated +with features according to each pixel for aggregating class- +wise context. OCRNet [32] presents a simple method char- +acterizing a pixel by exploiting the representation of the +corresponding object class. +Under supervision, OCRNet +learns object regions and generate representations of ob- +ject regions, which can be viewed as the object centers of +each region. Pixel-region relation is computed to aggre- +gate information from object centers. Similar to ACFNet, +the object-contextual representations generated from object +centers and pixel-region relation augment the original pixel +representations via concatenation. +Compared with these +methods that also introduce the concept of center in their + +(a) Images +(b) Baseline +(c) Ours +(d) Baseline distribution +(e) Our distribution +Figure 3. Visualization results on ADE20K validation set. We visualize the distribution results to demonstrate our model learns a better +scene-level feature distribution than previous works. Comparing our method with the Baseline DeeplabV3+, we find that our boundaries +between different clusters are far more clear, with most pixels of the same class located in the same cluster. +works, our concern is how to get conditional centers to ad- +dress the drawback of global centers, while they follow the +thought of exploring context from a different aspect. Be- +sides, we apply our inter CD Loss and intra CD Loss on +class centers to enlarge distinction and optimize the distri- +bution of different classes while they do not have any super- +vision on the generated centers. +3. Method +In this section, we first have a review of typical segmen- +tation models in semantic segmentation. Then, we expli- +cate our rethinking-modeling semantic segmentation as a +task to assign each pixel to a category directly based on the +pixel-to-center (P-C) similarity. After a brief overview of +the proposed prediction pipeline, we introduce the details + +wall +floor +ceiling +windowpane +table +plant +curtain +chair +painting +sofa +rug +lamp +cushion +blind +coffee table +pot +blanket +sconcesky +tree +water +rockCOGNAC +JACOUETwall +floor +ceiling +windowpane +cabinet +door +table +plant +chair +painting +lamp +blind +potcase +basket +food +traywall +floor +ceiling +chair +shelf +rug +box +book +light +playthingbuilding +sky +treeK +H +W +𝑀𝑎𝑠𝑘 +1 × 1 +𝐶𝑜𝑛𝑣 +⊗ +𝑟𝑒𝑠ℎ𝑎𝑝𝑒 +K×HW +𝑟𝑒𝑠ℎ𝑎𝑝𝑒 +& +𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒 +HW×D +𝑨𝒅𝒂𝒑𝒕𝒊𝒗𝒆 𝑪𝒍𝒂𝒔𝒔 𝑪𝒆𝒏𝒕𝒆𝒓 𝑴𝒐𝒅𝒖𝒍𝒆 +𝑪 +K +𝑃 − 𝐶 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑀𝑎𝑝𝑠 +H +W +D +H +W +𝐹𝑒𝑎𝑡𝑢𝑟𝑒𝑠 +K +H +W +𝑖𝑛𝑡𝑒𝑟 𝐶𝐷 𝐿𝑜𝑠𝑠 +𝑟𝑒𝑠ℎ𝑎𝑝𝑒 +D×HW +𝐺𝑇 +… +D +K +𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒 𝐶𝑙𝑎𝑠𝑠 𝐶𝑒𝑛𝑡𝑒𝑟𝑠 +K-1 +K-1 +𝐶 − 𝐶 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑀𝑎𝑡𝑟𝑖𝑥 +𝑺𝒊𝒎𝒊𝒍𝒂𝒓𝒊𝒕𝒚 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑴𝒐𝒅𝒖𝒍𝒆 +𝐶𝑒𝑛𝑡𝑒𝑟 − 𝐶𝑒𝑛𝑡𝑒𝑟 +𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 +𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 +𝑃𝑖𝑥𝑒𝑙 − 𝐶𝑒𝑛𝑡𝑒𝑟 +𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 +𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 +𝑖𝑛𝑡ra 𝐶𝐷 𝐿𝑜𝑠𝑠 +𝑪𝑫 𝑳𝒐𝒔𝒔 +Figure 4. Structure of the proposed CCS layer. The Mask is a spatial weight map used for aggregating features for all the centers of each +semantic class. C ∈ RK×D denote the generated K adaptive class centers, where D is the number of channels and K is the number +of classes. Then, the CCS layer calculates the pixel-to-center (P-C) similarity maps of each semantic class and center-to-center (C-C) +similarity between different classes. Finally, the proposed inter-class distance loss and intra-class distance loss are applied. The final +prediction is P-C similarity Maps normalized by the softmax function. +of the Adaptive Class Center Module, Similarity Calcula- +tion Module and Class Distance Loss. +3.1. Revisiting prediction of semantic segmentation +Before we go to our proposed approach, let’s revisit +the architecture of typical segmentation networks. A typ- +ical segmentation network consists of two parts: feature +extractor and classifier. +The feature extractor is usually +a deep convolutional network, which takes images as in- +puts, then extracts high-dimensional representations of each +pixel. The classifier takes the output features of the ex- +tractor as inputs and computes the score maps indicating +the probabilities that every element belongs to each class. +The output of the feature extractor is noted as F ∈ RD×N, +where N is the total number of elements, D is the number +of output channels in the extractor. The representation of +the i-th pixel in the input image is noted as fi ∈ RD×1. +The weights of the classifier are noted as Wc ∈ RK×D, +and wj ∈ R1×D (j = 1, 2, ..., K) denotes the kernel +weight which performs correlation on feature maps to get +the prediction of j-th class. Following the work of FCN, +most segmentation networks replace the FC classifier with +a 1 × 1 convolutional layer, which computes the score map +S ∈ RK×N of input feature F. The computation can be +formulated as: +S = W ⊛ F, +(1) +where ⊛ is the convolution operation. S is then normalized +by soft-max to get a soft-margin prediction of the whole +image. +As classifier employs a 1×1 conv layer, the computation +of the j-th class’ score sij of i-th pixel can be formulated as: +sij = w⊤ +j · fi, +(2) +where · denotes the inner product, sij is the score of i- +th pixel at j-th class, and w⊤ +j is the transposed wj, i = +1, 2, ..., N. The prediction pi assigned to i-th pixel is made +by performing a argmax operation to each location in the +image: +pi = arg max +j +(sij) = arg max +j +(w⊤ +j · fi). +(3) +Hence, for a single pixel, the score sij is actually the in- +ner production similarity, or so-called correlation, between +its feature fi and the transposed weight vector w⊤ +j of class +j. According to the pixel-to-weight similarity map S, the +final prediction is made by categorizing pixels to the se- +mantic class which has the greatest similarity value in fea- +ture space. +Thus, the weight vectors can be deemed as +learned class centers, and the category assignment is equiv- +alent to finding the nearest class center for each pixel. Con- +sequently, the overall inference procedure can be summa- +rized as Alg. 1, where I ∈ R3×N is the input RGB features +of an image, θ is the parameter of network, Wc ∈ RK×D is +the weights of the classifier, namely the learned global class +centers, i = 1, 2, ..., N, j = 1, 2, ..., K. +Algorithm 1 Inference with Global Class Centers +Input: I, network Net(, ) and parameters θ, weights of +classifier Wc. +Extract feature map: F ← Net(I, θ). +Calculate P-C similarity: S ← Simi(F, Wc) +Get prediction: pi ← arg maxj(sij) +Output: Final prediction P. +If pixel i belongs to the j-th class according to ground +truth, we hope that we can get greater relative sij at the j-th +channel compared with other channels and smaller scores at + +Algorithm 2 Inference with Adaptive Class Centers +Input: I, network Net(, ), the parameters θ, and the +ACCM G(·). +Extract feature map: F ← Net(I, θ). +Gnerate Adaptive Class Centers: C ← G(F). +Calculate P-C similarity: S ← Simi(F, C) +Get prediction: pi ← arg maxj(sij) +Output: Final prediction P. +irrelevant channels, which correspond to irrelevant classes. +So, the network is trained under the supervision of Cross- +entropy (CE) Loss that applied on the normalized scores +maps: +LCE += +� +i +−log +� exp(si · yi) +� +j exp(sij) +� += +� +i +−(si · yi) + log +� � +j +exp(sij) +� +, +(4) +where yi is the one-hot label of element i. +In our understanding, CE Loss mainly focuses on enlarg- +ing the relative inner production similarity between the cor- +responding weight vector and pixels of each class by push- +ing pixels and corresponding weight vectors closer in fea- +ture space. Accordingly, we can regard the weight vectors +as a set of global class centers learned by the network on +the whole dataset. The training stage is a process that the +network learns to find the global class centers of each class, +best fitting the distribution of all the pixels which belong to +the same class on the whole training set. Moreover, these +global class centers are identical for all input scenes. +However, semantic segmentation networks are still fac- +ing two challenges: (i) pixels of the same category yet dif- +ferent scenes are significantly different in feature space, +leading to difficulty for networks in categorizing all these +pixels to the same semantic class. (ii) pixels from different +categories but the same scene lack enough distinction, caus- +ing trouble finding distinctive separating plane to tell pixels +belonging to different classes apart. +To demonstrate the challenges, we conduct experiments +and visualize the feature representations, which are the in- +puts of the final convolution, in Fig. 1 of randomly sampled +pixels from “tree” and “plant” classes on ADE20K valida- +tion set, with sampling ratio set as 1%. We adopt t-SNE [24] +for a clear visualization. As shown in Fig. 1, the pixels of +“trees” and “plants” in Sample A are far from those in Sam- +ple B in feature space despite the fact that they belong to +the same class. This phenomenon indicates the large vari- +ance of features between different scenes, making it hard +to assign pixels of the same class yet different scenes with +the same class based on the immutable global class centers. +Moreover, for those pixels in the same scene, they are much +closer to each other despite that they belong to different cat- +egories. These short distances in feature space between pix- +els from the same scene but different classes corroborate the +scarcity of enough distinction to categorize them correctly. +3.2. Overview of our proposed method +As illustrated above, considering the learned class cen- +ters Wc are unchangeable for different input images I, the +first challenges are unavoidable. To solve it, we propose a +new pipeline for semantic segmentation prediction pipeline +as shown in Alg. 2. The pixel-to-center (P-C) similarity +is calculated between pixels in each scene and the adap- +tive class centers which are generated based on the feature +map of the scene. Compared with Wc which is immutable +during inference for every input image, the adaptive class +center should be capable of varying between scenes to ac- +commodate the large intra-class feature variation in differ- +ent scenes. +On the other hand, there is no constraint on the similar- +ity between different class centers inside each scene. So, +the second challenge is aggravated, even though the adap- +tive class centers can mitigate the problem. Therefore, we +propose the Class Distance (CD) Loss to enlarge the inter- +class feature variation in the same scene. CD Loss directly +requires large similarities between pixels and the adaptive +class center of the ground truth class and small mutual sim- +ilarities among adaptive class centers. +Extensive experi- +ments and ablation studies in Sec. 4 corroborate the effec- +tiveness of our proposed prediction pipeline. +3.3. Adaptive Class Center Module +Previous approaches take the transposed learned weight +Wc of the final classifier as global class centers, which are +the representations of each class at the dataset-level. On +account of the feature variances on the dataset from scene +to scene, these approaches are impeded by their global +class centers which are not able to vary between scenes. +Therefore, we propose the Adaptive Class Center Module +(ACCM) to generate unique class centers for each scene +as class representations at the scene-level instead of the +dataset-level. +ACCM performs matrix multiplication of pixel features +and a learned mask to generate coarse class centers for each +input image. This module refines the coarse class centers +and outputs a set of adaptive centers as scene-level repre- +sentations for all classes. The whole computation process +of ACCM can be formulated as: +C = A(M ⊗ F⊤), +(5) +where C ∈ RK×D is the matrix of generated conditional +adaptive class centers, ⊗ is matrix multiplication, A(·) is +the adaptive module consisting of several convolutional lay- +ers, and M ∈ RK×N is a learned weight map, based + +on which ACCM aggregates information and generates the +coarse class centers. In practice, we find that for the adap- +tive module, a 1 × 1 convolution works well to generate +the adaptive class centers. We also conduct experiments to +study the impact of different M and eventually choose the +one supervised by Dice loss [19] for better performance. +The Dice loss is defined as: +LDice = 1 − +2 �N +i mi · yi +�N +i ||mi||2 + �N +i ||yi||2 + ϵ +, +(6) +where mi ∈ RK×1 is the i-th column vector in M corre- +sponding to i-th pixel, ||a|| is the second norm of vector a, +and ϵ is set as 1e−3 to prevent division by zero. +(a) traditional centers +(b) our centers +Figure 5. Illustration of our adaptive class centers. During infer- +ence, traditional approaches calculate pixel-to-centers similarity +maps with fixed centers, while our approach generates adaptive +centers for each input image, respectively. +Based on the adaptive class centers, we perform pixel-to- +center similarity calculation on feature maps to get absolute +P-C similarity maps, followed by soft-max at the dimension +of different classes to generate the relative P-C similarity +maps as the soft-margin prediction. The structure of the +model is shown in Fig. 4. +Moreover, to address the problem that features of differ- +ent classes but the same scene lack enough distinction, pre- +vious methods try to aggregate more context information +into pixel representations, but there is no supervision im- +posed on the representations of class centers. We propose +the Class Distance Loss composed of inter-class distance +loss and intra-class distance loss imposed on class centers +to make features in the same scene more discriminative. We +will introduce it in the next section. +3.4. Similarity Calculation Module +We introduce of similarity function to measure the prox- +imity of two embedding a ∈ RD×1 and b ∈ RD×1 in the +feature space. we define the inner production similarity as: +Simi(a, b)inn = a · b. +(7) +To encourage class centers to be linearly independent +and obviate the influence of the embedding’s norm, we also +introduce the cosine similarity as : +Simi(a, b)cos = abs(a · b) +||a|| ||b|| , +(8) +where the abs(·) denotes the absolute value. +To calculate the C-C similarity, we employ consine sim- +ilarity (Eq. 8) to get rid of the influence of the norms of +class centers. We simply replace a and b in Eq. 8 with +adaptive class centers c⊤ +p and c⊤ +p as Simi(c⊤ +p , c⊤ +q ), where +cq ∈ R1×D is the q-th row in C, namely the adaptive class +center of the q-th class. +For P-C similarity calculation, we employ inner pro- +duction simialrity (Eq. 7). As the prediction is made by +the argmax operation on P-C similarity maps, the absolute +value of similarity between a pixel with a class center alone +is meaningless unless compared with similarities between +the pixel and other class centers. Hence, we introduce rel- +ative similarity as the probability that i-th pixel belongs to +q-th class: +RSimi(fi, c⊤ +q ) = +exp +� +Simi(fi, c⊤ +q ) +� +� +j exp +� +Simi(fi, c⊤ +j ) +�, +(9) +where j = 1, 2, ..., K. +3.5. Class Distance Loss +Since there is no constraint on the similarity between +different class centers inside each scene, the challenge that +different-classes pixels in the same scene lack enough dis- +tinction is aggravated consequently. We believe that it is +easy for pixel-wise classification if pixel representations in +the same scene have large inter-class distinction and small +intra-class distinction. Motivated by this notion, we propose +the Class Distance (CD) Loss. We introduce the inter-class +distance and intra-class distance at first, then explicate the +details of the Class Distance Loss. +Definition of inter-class and intra-class distance. +The +inter-class distance between the p-th and the q-th class is +defined as: +d(p,q) +inter = D(c⊤ +p , c⊤ +q ), +(10) +where p and q are the numbers of two different classes, c⊤ +p +and c⊤ +q are the class center vectors of p-th and q-th class +respectively, D(·, ·) is the distance function. The intra-class +distance of the q-th class is defined as: +dq +intra = +N +� +i=1 +1[yiq = 1]D(fi, c⊤ +q ), +(11) +1[condition] = +� +1 +condition is True +0 +condition is False , +(12) + +Conditional +CentersWe introduce our distance function based on the relative +similarity. The distance should be negative correlated with +relative similarity. Therefore, we define our distance as: +D(fi, c⊤ +q ) = −log +� +RSimi(fi, c⊤ +q ) +� +. +(13) +We apply inner production similarity and cosine similarity +for dintra and dinter, respectively. +Inter-class and intra-class Distance Loss. +As illustrated +above, we propose inter-class distance loss and intra-class +distance loss to enlarge inter-class distances and diminish +intra-class distance. Our loss functions are as: +Lintra = +K +� +q=1 +dq +intra, +(14) +Linter = +K +� +p=1 +K +� +q=1 +1[q ̸= p]exp(−dp,q +inter), +(15) +where Lintra aims at diminishing intra-class distinction un- +der the supervision of GT, and Linter focuses on enlarging +the differences between class centers that can operate with- +out GT. +The weighted summation of the Lintra and Linter is +called Class Distance (CD) Loss for convenience, which +can be formulated as: +LCD = Lintra + αLinter, +(16) +where α is the a hyper-parameter that set empirically. +For dataset-level CD Loss, we regard each w⊤ +j +as +dataset-level class center just as our rethinking of traditional +segmentation networks. So, the dataset-level intra-class dis- +tance loss and inter-class distance loss are calculated re- +spectively as: +Ldataset +intra += +K +� +q=1 +N +� +i=1 +1[yiq = 1]D(fi, w⊤ +q ), +(17) +Ldataset +inter += +K +� +p=1 +K +� +q=1 +1[q ̸= p]exp +� +− D(w⊤ +p , w⊤ +q ) +� +.(18) +Please notice that when combining Eq. 13 with Eq. 11, Eq. 9 +and Eq. 7, the Lintra is in the same format with CE Loss +shown in Eq. 4: +Ldataset +intra += +K +� +q=1 +N +� +i=1 +1[yiq = 1] +� +log +� � +j +exp(fi·c⊤ +j ) +� +−fi·c⊤ +q +� +. +(19) +This shows that the CE Loss is a special case of our pro- +posed intra-class distance loss. +For scene-level CD Loss, we use the proposed ACCM to +calculate the class centers conditioned on each scene. So, +the scene-level CD Loss can be formulated as follows: +Lscene +intra = +K +� +q=1 +N +� +i=1 +1[yiq = 1]D(fi, c⊤ +q ), +(20) +Lscene +inter = +K +� +p=1 +K +� +q=1 +1[q ̸= p]exp +� +− D(c⊤ +p , c⊤ +q ) +� +.(21) +Overall loss function. +The over-all loss function of our +model are as follows: +L += +LCD + βLDice += +Lintra + αLinter + βLDice, +(22) +where LDice is the Dice loss imposed on the mask M and +β is the weight of LDice. Since the scene-level CD Loss is +much more powerful, we employ the Lscene +intra and Lscene +inter in- +stead of Ldataset +intra +and Ldataset +inter +. We conduct experiments to +show the effectiveness of Lscene +intra and Lscene +inter over Ldataset +intra +and Ldataset +inter +in Tab. 5. +4. Experiment +We evaluate our approach on two challenging semantic +segmentation datasets: ADE20K and Pascal Context. We +perform a comprehensive ablation study on the ADE20K +dataset and report the comparison with other methods on the +ADE20K validation set and the Pascal Context validation +set. +4.1. Settings +Dataset. +ADE20K [40] dataset is a large-scale scene pars- +ing benchmark with 150 fine-grained objects and stuff cate- +gories, containing 20,210 images for training, 2,000 images +for validation, and 3352 images for testing. +Pascal Context dataset [20] is a scene parsing dataset that +provides semantic labels for whole scene(both “things” and +“stuff” classes), which augments 10,103 images from PAS- +CAL VOC 2010 [7]. It has 4,998 training and 5,105 vali- +dation images. We use the 59 most common categories for +evaluation. +Training. +We +conduct +our +experiments +using +four +NVIDIA GTX 2080 ti GPUs with four images per GPU. All +of our models are optimized by SGD optimizer with 0.9 mo- +mentum. The initial learning rate is set 1e−2 for ADE20K +and 4e−3 for PASCAL Context. We adopt the polynomial +learning rate decay strategy in training following previous +works [30, 29, 5]. The initial learning rate is multiplied by +(1− +iter +maxiter)0.9. We apply random resizing with a ratio be- +tween 0.5 and 2 in training, random cropping input images + +into (512,512), and random horizontal flipping during train- +ing for all the experiments. We set the total iterations on +ADE20K to 80K and 160K for our ResNet-50 and ResNet- +101 models, respectively. For Pascal Context, models run +80k iterations during training. +Auxiliary loss. +Follow previous work [39], we adopt aux- +iliary segmentation loss to help train our model. We add +an auxiliary FCN head, which outputs prediction under the +supervision of CE Loss multiplied by 0.4. +Evaluation. +During the evaluation, we average the pre- +dictions of multiple scaled following the previous work [37, +29, 5]. Each image is then flipped horizontally, then scaled +to a uniform size with scaling factor (0.5, 0.75, 1.0, 1.25, +1.5, 1.75) for better performance. Besides, we use Synchro- +nized BN in our models. Additionally, we report mean In- +tersection over Union (mIoU) and pixel accuracy (Acc) for +ADE20K and mIoU for PASCAL context. +4.2. Ablation Study On ADE20K +We conduct ablation studies on ADE20K to demonstrate +the effectiveness of our approach. Models are trained on +ADE20K train set and evaluated on val set. All the models +are pretrained on Image-Net without extra data. +Upper-bound verification. +To verify the feasibility of +our proposed methods, we first carry out a simple upper- +bound verification experiment. We directly replace the pre- +dicted Mask, which is supervised by CE loss, of our trained +model with processed ground truth during inference. Ac- +cording to the results shown in Tab. 1, our method pro- +vides a huge improvement compared with the baseline, +while the upper bound method Ours-GT greatly outper- +forms our method when the predicted Mask is replaced by +GT. Since the GT provides better weight maps than the pre- +dicted Mask to generate the adaptive class centers, the per- +formance is dramatically improved, indicating that there is +still a lot of room for improvement. +Method +CCS +mIoU(%) +Acc(%) +Baseline +37.94 +77.98 +Ours +✓ +43.57 +81.06 +Ours-GT +✓ +47.21 +84.12 +Table 1. Upper-bound verification. +Baseline: ResNet-50 FCN. +Ours: Baseline + CCS layer. Ours-GT: Baseline + CCS layer +whose Mask is supplanted by GT to verify the upper bound of +CCS layer. +Ablation study of class centers. +To demonstrate the +effectiveness of the adaptive class centers, we compare +Figure 6. P-C distance comparison between MC & AC. Left: +inter-class P-C distance. Right: intra-class P-C distance. We de- +note pixel-to-mean centers and pixel-to-adaptive centers as MC +and AC, respectively. +the P-C similarity, depending on which we generated the +probability maps. +Taking “tree” and “plant” as exam- +ples, we show the frequency histograms of intra-class +pixel-to-adaptive centers distance and pixel-to-mean cen- +ters (namely the global centers that learned from the whole +dataset) distance in Fig. 6. As we introduced before, the +results demonstrate that the adaptive class centers make +prominent success in increasing the inter-class distance be- +tween “tree” pixels and “plant” class center, while the intra- +class distances of “plant” pixels and adaptive class center of +“plant” are dramatically smaller than the distance between +pixels and global class center. +Ablation study of ACCM. +We conduct ablation studies +of ACCM to explore the best performance. We compared +different loss functions on Mask in order to generate better +adaptive class centers. We conduct experiments on FCN +based on ResNet-50 and report the results in Tab. 2. +Res50 FCN CCSNet +w/o loss +w/ CE loss +w/ Dice loss +mIoU(%) +43.01 +41.56 +43.57 +Acc(%) +80.69 +80.45 +81.06 +Table 2. Ablation study of defferent loss fucntion on Mask. Our +baseline in this experiment is ResNet-50 FCN, with our CS replace +the final convolution of FCN under supervision of scene-level CD +Loss. +Ablation study of hyper-parameter and architecture. +We first conduct experiments with different segmentation +heads. The multi-scale mIoU results of different segmen- +tation heads based on ResNet-50 are reported in Tab. 3. +Among three different segmentation heads, deeplabv3+ has +the best performance with 44.25% ms mIoU. We also ex- +plore proper hyper-parameters for CCSNet. We take FCN +Methods +FCN +PSP +Deeplabv3+ +baseline +37.94 +41.94 +43.57 ++CCS +43.57 +44.07 +44.25 +Table 3. Ablation study of segmentation heads. +CCS layer is +equipped on different segmentation heads based on ResNet-50 are +trained on ADE20K training set for 80k iterations. The mIoU re- +sults on ADE20K validati set of each model are reported. + +0.20 +inter P-MC +Frequency +inter P-AC +0.15 +0.10 +0.05 +0.00 +2 +4 +6 +8 +10 +12 +14 +Inter Distance (tree-to-plant)0.15 +intra P-MC +intra P-AC +Frequ +0.10 +aA +lati +0.05 +Rel +0.00 +2 +4 +6 +8 +10 +12 +14 +Intra Distance (plant-to-plant)mIoU +β = 0.1 +β = 0.5 +β = 1.0 +α = 0.1 +42.79 +42.86 +42.85 +α = 0.5 +43.01 +43.05 +43.57 +α = 1.0 +43.14 +43.50 +43.47 +Table 4. Alblation Study of weight of Lscene +inter and weight of dice +loss. We vary the value of α and β and find when α = 0.5, β = +1.0, model has the best performance. +based on ResNet-50 as a baseline to exploit the best weight +of dinter and the weight of Dice loss. +As shown in +Tab. 4, our approach achieves the best performance when +the weight of dinter is empirically set as 0.5. The best FCN +based model equipped with our approach improves the pix- +Acc and mIoU by 2.59% and 5.63%. +Ablation study of CCS layer. +We break down the +improvements of our work over ResNet-101 based on +DeeplabV3+, which has the best performance. +We add +the proposed components in our approaches step by step +to the DeeplabV3+ baseline. Experiments are conducted +on ADE20K and run 160k iterations. By simply replacing +the global class centers Wc with our adaptive class cen- +ters conditionally generated by ACCM, our network im- +proves the performance by 0.86% in mIoU. This provides +strong support for our assertion that global class centers are +inferior compared to conditional class centers. Moreover, +together with the image-level inter-class distance loss, our +network achieves 47.76% mIoU on the ADE20K validation +set, which demonstrates the effectiveness of our inter-class +distance loss. +4.3. Results on ADE20K +We train the ResNet-101 on the ADE20K training set +and report the mIoU results on the validation set (results are +shown in Tab. 6). Our CCSNet uses a pre-trained backbone +network on ImageNet and achieves 47.76% mIoU, which +outperforms DeeplabV3+ by 1.41% mIoU using the same +backbone network. Visual comparison examples are shown +in Fig. 7. +4.4. Results on PASCAL Context +Tab. 7 reports the comparison results of our net- +work and other state-of-the-the-art approaches. Based on +ResNet-101, our method makes favourable performance +and achieves 54.9% mIoU. CCSNet outperforms previous +methods using the same ResNet-101 backbone. +5. Conclusion +In this paper, we provide a novel perspective to view typ- +ical semantic segmentation models, and re-model the prob- +lem as a task that models compute the similarity maps be- +tween pixels and class centers on each scene, then assign +Method +centers +Linter +Lintra +mIoU +Deeplabv3+ +Wc +Wc +46.35 ++ Ldataset +inter +Wc +Wc +Wc +46.45 ++ACCM&Lscene +inter +Wc +C +Wc +46.94 ++CCS w/o Linter +C +C +47.21 +CCSNet (Ours) +C +C +C +47.76 +Table 5. Ablation study of CCS layer and CD Loss. The “centers” +are those used to calculate the P-C similarity for prediction. For +example, Wc denotes the learned weights of classifier, namely +global class centers, used for final prediction, while the C indicate +that the prediction is based on the P-C similarity using adaptive +class centers. “Linter” and “Linter” shows whether the loss is +applied on Wc (Ldataset +inter +and Ldataset +intra +) or C (Lscene +inter and Lscene +intra). +Method +Baseline +mIoU(%) +CascadeNet [40] +VGG-16 +34.90 +RefineNet [16] +ResNet-152 +40.7 +UperNet [16] +ResNet-101 +42.66 +PSPNet [37] +ResNet-101 +43.51 +PSPNet [37] +ResNet-269 +44.94 +PSANet [38] +ResNet-269 +43.77 +EncNet [35] +ResNet-101 +43.77 +CFNet [36] +ResNet-101 +44.65 +ANL [41] +ResNet-101 +45.24 +OCRNet [32] +ResNet-101 +45.28 +APCNet [10] +ResNet-101 +45.38 +RGNet [28] +ResNet-101 +45.80 +CPNet [29] +ResNet-101 +46.27 +DeeplabV3+ [5] +ResNet-101 +46.35 +CCSNet +ResNet-101 +47.76 +Table 6. Results on the ADE20K validation set. Our model based +on ResNet-101 achieves 47.76% in mIoU and outperforms all pre- +vious methods using the same backbone network. +Method +Baseline +mIoU(%) +RefineNet [16] +ResNet-152 +47.3 +PSPNet [37] +ResNet-101 +47.8 +DeeplabV3+ [5] +ResNet-101 +48.3 +EncNet [35] +ResNet-101 +51.7 +DANet[9] +ResNet-101 +52.6 +ANL [41] +ResNet-101 +52.8 +CFNet [36] +ResNet-101 +54.0 +APCNet [10] +ResNet-101 +54.7 +RGNet [28] +ResNet-101 +53.9 +CPNet [29] +ResNet-101 +53.9 +OCRNet [32] +ResNet-101 +54.8 +CCSNet +ResNet-101 +54.9 +Table 7. Results on the PASCAL Context validation set. We report +our result evaluated on 59 class without background. Our model +based on ResNet-101 achieves 54.9% in mIoU. +pixels to the semantic category with the highest P-C sim- +ilarity. Based on this perspective, we provide solutions to +address the two typical issues, i.e. same category yet dif- +ferent scenes features could be of large variance, while fea- + +(a) Images +(b) GT +(c) FCN +(d) DeeplabV3+ +(e) CCSNet (Ours) +Figure 7. Qualitative results on ADE20K validation set. +tures of different categories but the same scene may be quite +similar to each other. Based on P-C similarity maps, we +propose ACCM to generate adaptive class centers condi- +tioned on each scene to deal with the feature variances of +different scenes and design inter-class and intra-class dis- +tance loss at scene-level for more inter-class distinction in- +side each scene. Our approach is easy yet effective and can +be plugged into most FCN-based architectures. Finally, CC- +SNet achieves state-of-the-the-art performance on two chal- +lenging semantic segmentation datasets. +References +[1] V. Badrinarayanan, A. Kendall, and R. Cipolla. 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Asym- +metric non-local neural networks for semantic seg- +mentation. +In Proceedings of the IEEE/CVF Inter- +national Conference on Computer Vision, pages 593– +602, 2019. 10 + diff --git a/-NE4T4oBgHgl3EQfDgur/content/tmp_files/load_file.txt b/-NE4T4oBgHgl3EQfDgur/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..efa517cff7c1cec43ed97df363cb8f19501eae5f --- /dev/null +++ b/-NE4T4oBgHgl3EQfDgur/content/tmp_files/load_file.txt @@ -0,0 +1,921 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf,len=920 +page_content='Semantic Segmentation via Pixel-to-Center Similarity Calculation Dongyue Wu1, Zilin Guo1, Aoyan Li1, Changqian Yu2, Changxin Gao1, Nong Sang1 1National Key Laboratory of Science and Technology on Multispectral Information Processing, School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China {dongyue wu,zilin guo,aoyanli,cgao,nsang}@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='cn 2Meituan Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=', Beijing, China changqianyu@meituan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='com Abstract Since the fully convolutional network has achieved great success in semantic segmentation, lots of works have been proposed focusing on extracting discrimina- tive pixel feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' However, we observe that existing methods still suffer from two typical chal- lenges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (i) large intra-class feature variation in dif- ferent scenes, (ii) small inter-class feature distinction in the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In this paper, we first rethink semantic segmentation from a perspective of similarity between pixels and class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Each weight vector of the seg- mentation head represents its corresponding semantic class in the whole dataset, which can be regarded as the embedding of the class center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Thus, the pixel-wise clas- sification amounts to computing similarity in the final feature space between pixels and the class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Un- der this novel view, we propose a Class Center Similarity layer (CCS layer) to address the above-mentioned chal- lenges by generating adaptive class centers conditioned on different scenes and supervising the similarities be- tween class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' It utilizes a Adaptive Class Center Module (ACCM) to generate class centers conditioned on each scene, which adapt the large intra-class vari- ation between different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Specially designed loss functions are introduced to control both inter-class and intra-class distances based on predicted center-to-center and pixel-to-center similarity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Finally, the CCS layer outputs the processed pixel-to-center simi- larity as the segmentation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Extensive experi- ments demonstrate that our model performs favourably against the state-of-the-art CNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Keywords: semantic segmentation, similarity, adaptive class center, intra-class variation, intra-class distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Introduction Semantic segmentation aims to assign each pixel with a semantic category, which is a fundamental and challenging task in the computer vision field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Benefited from the devel- opment of deep convolutional networks [23, 14, 22, 12], the fully convolutional network (FCN) [18] has been the domi- nant solution in the semantic segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Due to the simplicity of the architecture of FCN, exist- ing methods mainly focus on enhancing the feature repre- sentations to improve visual recognition capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' They aggregate rich contextual information via large receptive field [31, 6], multi-scale methods [37, 4], or attention mech- anisms [25, 9, 38, 8, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' However, we observe that these methods still suffer from two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (i) Large intra- class feature variation between different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Accord- ing to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 1, the features of the “trees” (colored in red) near the wall in Sample A greatly differ from those on the bank in Sample B, despite that they belong to the same semantic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Similarly, plant pixels of Sample A locate signif- icantly far from those of Sample B, which also suggesting large intra-class variation between pixels in different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (ii) Small inter-class distinction inside each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Take Sample A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 1 for example, it is difficult for the seg- mentation network to tell pixels belonging to the “tree” and pixels belonging to the “plant” apart, since pixels belong- ing to the “tree” are near to pixels of the “plant”, indicating that the two groups of pixels of different semantic classes bear similar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' These two challenges make it hard to category pixels from the same class but different scenes into the correct class while telling groups of pixels from the different classes but the same scene apart from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' To solve the aforementioned challenges, we rethink pre- vious methods from a perspective of similarity between pix- els and class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Most previous methods utilize a 1 × 1 convolutional layer as the segmentation head to get the fi- nal prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The weights of the convolutional layer are trained through all training samples and conduct a convolu- tion operation on the feature maps in the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In other words, the learned weights perform correlation calcu- lation with each pixel and output the map of inner produc- tion which is widely used to measure the similarity between features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' As each vector of the weights can be regarded as 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='04870v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='CV] 12 Jan 2023 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Illustration of the overall distribution of “tree” and “plant” pixels on the ADE20K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We show the distribution of the two classes in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' For clarity, we only label the pixels belonging to “tree” and “plant”, which are in red and blue, respectively, while the pixels that belong to other classes keep their original color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Each dot in our plot represents a randomly sampled pixel in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The light-colored dots denote pixels sampled from other samples(scenes) in the whole dataset, while the dark-colored ones are sampled from Sample A and Sample B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' According to the plots, the features of both “tree” and “plant” in Sample A are quite different from those in Sample B, suggesting large intra-class variation between different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The features of “tree” and “plant” are hard to distinguish in Sample A, because of the small inter-class distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The dimension of features is reduced for illustration using t-SNE [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (a) Images (b) Baseline (c) Ours Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Comparison of pixel distribution in feature space on the ADE20K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Baseline: ResNet-101 + DeeplabV3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ours: Baseline + CCS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Pixels that randomly sampled form the “tree” and “plant” are colored in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our distribution of each class is more compact than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' There are also clear boundaries between different classes in the feature space of our method, when the pixels of the two classes are mixed up in the feature space of the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' a learned representation of the corresponding class, the fi- nal segmentation prediction map can also be regarded as a pixel-to-center similarity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' These learned representa- tions contain the common information of their class on the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Therefore, we call them global class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In this view, the classification can be remodeled as follows: A pixel will be assigned to the class whose global class cen- ter is the most similar to the pixel among all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In con- clusion, semantic segmentation can be viewed as a task to predict the similarity between pixels and class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Motivated by this perspective, we argue that there are two limitations of previous methods responsible for the above-mentioned challenges: (i) The global class centers are incapable of adapting the large intra-class variations be- tween different scenes, since the global class centers are learned based on the whole dataset and are kept unchanged and identical for different scenes during the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (ii) Since there is no constraint on the similarities between global class centers, these centers which represent different classes may share excessively similar representations, lead- ing to difficulties in correctly distinguishing pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' These limitations correspond to the two challenges of the large intra-class variation between scenes and the small inter- class distinction inside each scene, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' To solve the challenges, we propose a novel and flexi- ble Class Center Similarity (CCS) layer, which replaces the segmentation heads of networks and transfers the pixel- wise classification task to a pixel-to-center similarity pre- diction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our CCS layer consists of three parts: Adap- tive Class Center Module (ACCM), Similarity Calculation Module (SCM), and Class Distance (CD) Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' First, Adaptive Class Center Module (ACCM) generates adaptive Tree Sample B Plant囍class centers conditioned on each scene (image), which ac- commodates the large intra-class variance between scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Thus, the prediction only depends on the similarity between each pixel and the scene-specific adaptive class centers, in- stead of the immutable global class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Then, the adap- tive class centers are forward Similarity Calculation Mod- ule (SCM) to compute the pixels-to-centers similarity be- tween pixels and class centers and the mutual similarity of class centers as the inter-class similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Finally, our CD Loss is applied to the inter-class similarity and pixels-to- centers similarity in each scene to supervise the segmenta- tion prediction while increasing the lack of inter-class dis- tinction inside each scene simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The CCS layer can be integrated into almost arbitrary semantic segmen- tation architectures substituting for the final segmentation head, namely the 1 × 1 convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' To demonstrate the effectiveness of the proposed Class Center Similarity layer, we add the CCS layer to existing segmentation net- works and carry out extensive experiments on ADE20K and Pascal Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 3 also show that our method leads to clear and compact clusters for each semantic class in each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In summary, the following contributions are made in this paper: We rethink the semantic segmentation task from a pixel-to-center (P-C) similarity perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Semantic segmentation can be viewed as a task with two stages: compute P-C similarity for each pixel and categorize pixels to the most similar semantic class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We propose a class Center Similarity layer that can generate conditional class centers for each scene under the constraint of our Class Distance Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our CCS layer is easy to plug into almost any FCN-based se- mantic segmentation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We conduct extensive experiments to analyze the ef- fectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The proposed method, called CCSNet, achieves state-of-the-art performance on two challenging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Based on ResNet- 101 [11], our model achieves 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='76% mIoU on ADE20K, and 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='9% mIoU on PASCAL Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Related Work Semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The development of deep neu- ral networks dramatically boosts semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Since FCN [18] replaces the fully connected layer in the traditional classification network with a convolutional to get pixel-wise predictions, FCNs achieves great success in semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Segnet [1], UNet [21] and Re- fineNet [16] adopt encoder-decoder structure to recover the spatial information that lost by downsample operation via cascading upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In order to capture long-range de- pendencies, lots of works have been done by introducing CRF [3, 2] and MRF [17] into segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Dilated convolution [31] and deformable convolution[6], which in- crease the resolution of feature maps, is used to enlarge the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We revisit these typical networks and find that semantic segmentation can be regarded as catego- rizing each pixel to the semantic category with the great- est pixel-to-center similarity (or the smallest pixel-to-center distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Context is believed of great significance in semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Recently, plenty of works focusing on mining richer context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Multi-scale representations are used to capture more context in PSPNet [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' the family of DeepLab [3, 4, 5] also cap- tures the context information from multi-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Based on these approaches, many extensions have been proposed,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' DenseASPP [27] and APCNet [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The Attention mecha- nism is also adopted to capture global contextual informa- tion [13, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Other studies pay their attention to the sim- ilarity of pixels to help aggregate contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' OCNet [33], DANet [9] and CFNet [36] aggregate context that computed on all pixels and augment context represen- tation to the representation of each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Based on the self- attention mechanism, these approaches calculate similarity (or relation) between pixels and aggregate representations according to similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Although a great number of studies explore discriminate representations to help segmentation, the large variance between pixels of the same category from different scenes and the lack of distinction in each scene still remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our work addresses these two challenges based on similarity as well, but we aim to get better similarity maps while those utilize context to get better features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In addition, thanks to our ACCM which generates the varying adaptive class centers for classification, our method is more effective with minor computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Center-based methods in semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Re- cently, several works propose approaches with centers in semantic segmentation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In contrast to previous works, ACFNet [34] presents the concept of class centers as the global context from a categorical perspective, which describes the overall representation of each class in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Then, different class centers are adaptively concatenated with features according to each pixel for aggregating class- wise context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' OCRNet [32] presents a simple method char- acterizing a pixel by exploiting the representation of the corresponding object class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Under supervision, OCRNet learns object regions and generate representations of ob- ject regions, which can be viewed as the object centers of each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Pixel-region relation is computed to aggre- gate information from object centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Similar to ACFNet, the object-contextual representations generated from object centers and pixel-region relation augment the original pixel representations via concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Compared with these methods that also introduce the concept of center in their (a) Images (b) Baseline (c) Ours (d) Baseline distribution (e) Our distribution Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Visualization results on ADE20K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We visualize the distribution results to demonstrate our model learns a better scene-level feature distribution than previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Comparing our method with the Baseline DeeplabV3+, we find that our boundaries between different clusters are far more clear, with most pixels of the same class located in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' works, our concern is how to get conditional centers to ad- dress the drawback of global centers, while they follow the thought of exploring context from a different aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Be- sides, we apply our inter CD Loss and intra CD Loss on class centers to enlarge distinction and optimize the distri- bution of different classes while they do not have any super- vision on the generated centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Method In this section, we first have a review of typical segmen- tation models in semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Then, we expli- cate our rethinking-modeling semantic segmentation as a task to assign each pixel to a category directly based on the pixel-to-center (P-C) similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' After a brief overview of the proposed prediction pipeline,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' we introduce the details ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='wall ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='& ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='HW×D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑨𝒅𝒂𝒑𝒕𝒊𝒗𝒆 𝑪𝒍𝒂𝒔𝒔 𝑪𝒆𝒏𝒕𝒆𝒓 𝑴𝒐𝒅𝒖𝒍𝒆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑪 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑃 − 𝐶 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑀𝑎𝑝𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝐹𝑒𝑎𝑡𝑢𝑟𝑒𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑖𝑛𝑡𝑒𝑟 𝐶𝐷 𝐿𝑜𝑠𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑟𝑒𝑠ℎ𝑎𝑝𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='D×HW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝐺𝑇 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒 𝐶𝑙𝑎𝑠𝑠 𝐶𝑒𝑛𝑡𝑒𝑟𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='K-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='K-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝐶 − 𝐶 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑀𝑎𝑡𝑟𝑖𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑺𝒊𝒎𝒊𝒍𝒂𝒓𝒊𝒕𝒚 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑴𝒐𝒅𝒖𝒍𝒆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝐶𝑒𝑛𝑡𝑒𝑟 − 𝐶𝑒𝑛𝑡𝑒𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑃𝑖𝑥𝑒𝑙 − 𝐶𝑒𝑛𝑡𝑒𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑖𝑛𝑡ra 𝐶𝐷 𝐿𝑜𝑠𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='𝑪𝑫 𝑳𝒐𝒔𝒔 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Structure of the proposed CCS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The Mask is a spatial weight map used for aggregating features for all the centers of each semantic class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' C ∈ RK×D denote the generated K adaptive class centers, where D is the number of channels and K is the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Then, the CCS layer calculates the pixel-to-center (P-C) similarity maps of each semantic class and center-to-center (C-C) similarity between different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Finally, the proposed inter-class distance loss and intra-class distance loss are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The final prediction is P-C similarity Maps normalized by the softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' of the Adaptive Class Center Module, Similarity Calcula- tion Module and Class Distance Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Revisiting prediction of semantic segmentation Before we go to our proposed approach, let’s revisit the architecture of typical segmentation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' A typ- ical segmentation network consists of two parts: feature extractor and classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The feature extractor is usually a deep convolutional network, which takes images as in- puts, then extracts high-dimensional representations of each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The classifier takes the output features of the ex- tractor as inputs and computes the score maps indicating the probabilities that every element belongs to each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The output of the feature extractor is noted as F ∈ RD×N, where N is the total number of elements, D is the number of output channels in the extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The representation of the i-th pixel in the input image is noted as fi ∈ RD×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The weights of the classifier are noted as Wc ∈ RK×D, and wj ∈ R1×D (j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=', K) denotes the kernel weight which performs correlation on feature maps to get the prediction of j-th class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Following the work of FCN, most segmentation networks replace the FC classifier with a 1 × 1 convolutional layer, which computes the score map S ∈ RK×N of input feature F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The computation can be formulated as: S = W ⊛ F, (1) where ⊛ is the convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' S is then normalized by soft-max to get a soft-margin prediction of the whole image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' As classifier employs a 1×1 conv layer, the computation of the j-th class’ score sij of i-th pixel can be formulated as: sij = w⊤ j · fi, (2) where · denotes the inner product, sij is the score of i- th pixel at j-th class, and w⊤ j is the transposed wj, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The prediction pi assigned to i-th pixel is made by performing a argmax operation to each location in the image: pi = arg max j (sij) = arg max j (w⊤ j · fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (3) Hence, for a single pixel, the score sij is actually the in- ner production similarity, or so-called correlation, between its feature fi and the transposed weight vector w⊤ j of class j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' According to the pixel-to-weight similarity map S, the final prediction is made by categorizing pixels to the se- mantic class which has the greatest similarity value in fea- ture space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Thus, the weight vectors can be deemed as learned class centers, and the category assignment is equiv- alent to finding the nearest class center for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Con- sequently, the overall inference procedure can be summa- rized as Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 1, where I ∈ R3×N is the input RGB features of an image, θ is the parameter of network, Wc ∈ RK×D is the weights of the classifier, namely the learned global class centers, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=', N, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Algorithm 1 Inference with Global Class Centers Input: I, network Net(, ) and parameters θ, weights of classifier Wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Extract feature map: F ← Net(I, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Calculate P-C similarity: S ← Simi(F, Wc) Get prediction: pi ← arg maxj(sij) Output: Final prediction P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' If pixel i belongs to the j-th class according to ground truth, we hope that we can get greater relative sij at the j-th channel compared with other channels and smaller scores at Algorithm 2 Inference with Adaptive Class Centers Input: I, network Net(, ), the parameters θ, and the ACCM G(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Extract feature map: F ← Net(I, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Gnerate Adaptive Class Centers: C ← G(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Calculate P-C similarity: S ← Simi(F, C) Get prediction: pi ← arg maxj(sij) Output: Final prediction P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' irrelevant channels, which correspond to irrelevant classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' So, the network is trained under the supervision of Cross- entropy (CE) Loss that applied on the normalized scores maps: LCE = � i −log � exp(si · yi) � j exp(sij) � = � i −(si · yi) + log � � j exp(sij) � , (4) where yi is the one-hot label of element i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In our understanding, CE Loss mainly focuses on enlarg- ing the relative inner production similarity between the cor- responding weight vector and pixels of each class by push- ing pixels and corresponding weight vectors closer in fea- ture space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Accordingly, we can regard the weight vectors as a set of global class centers learned by the network on the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The training stage is a process that the network learns to find the global class centers of each class, best fitting the distribution of all the pixels which belong to the same class on the whole training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Moreover, these global class centers are identical for all input scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' However, semantic segmentation networks are still fac- ing two challenges: (i) pixels of the same category yet dif- ferent scenes are significantly different in feature space, leading to difficulty for networks in categorizing all these pixels to the same semantic class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (ii) pixels from different categories but the same scene lack enough distinction, caus- ing trouble finding distinctive separating plane to tell pixels belonging to different classes apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' To demonstrate the challenges, we conduct experiments and visualize the feature representations, which are the in- puts of the final convolution, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 1 of randomly sampled pixels from “tree” and “plant” classes on ADE20K valida- tion set, with sampling ratio set as 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We adopt t-SNE [24] for a clear visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 1, the pixels of “trees” and “plants” in Sample A are far from those in Sam- ple B in feature space despite the fact that they belong to the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' This phenomenon indicates the large vari- ance of features between different scenes, making it hard to assign pixels of the same class yet different scenes with the same class based on the immutable global class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Moreover, for those pixels in the same scene, they are much closer to each other despite that they belong to different cat- egories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' These short distances in feature space between pix- els from the same scene but different classes corroborate the scarcity of enough distinction to categorize them correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Overview of our proposed method As illustrated above, considering the learned class cen- ters Wc are unchangeable for different input images I, the first challenges are unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' To solve it, we propose a new pipeline for semantic segmentation prediction pipeline as shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The pixel-to-center (P-C) similarity is calculated between pixels in each scene and the adap- tive class centers which are generated based on the feature map of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Compared with Wc which is immutable during inference for every input image, the adaptive class center should be capable of varying between scenes to ac- commodate the large intra-class feature variation in differ- ent scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' On the other hand, there is no constraint on the similar- ity between different class centers inside each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' So, the second challenge is aggravated, even though the adap- tive class centers can mitigate the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Therefore, we propose the Class Distance (CD) Loss to enlarge the inter- class feature variation in the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' CD Loss directly requires large similarities between pixels and the adaptive class center of the ground truth class and small mutual sim- ilarities among adaptive class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Extensive experi- ments and ablation studies in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4 corroborate the effec- tiveness of our proposed prediction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Adaptive Class Center Module Previous approaches take the transposed learned weight Wc of the final classifier as global class centers, which are the representations of each class at the dataset-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' On account of the feature variances on the dataset from scene to scene, these approaches are impeded by their global class centers which are not able to vary between scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Therefore, we propose the Adaptive Class Center Module (ACCM) to generate unique class centers for each scene as class representations at the scene-level instead of the dataset-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' ACCM performs matrix multiplication of pixel features and a learned mask to generate coarse class centers for each input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' This module refines the coarse class centers and outputs a set of adaptive centers as scene-level repre- sentations for all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The whole computation process of ACCM can be formulated as: C = A(M ⊗ F⊤), (5) where C ∈ RK×D is the matrix of generated conditional adaptive class centers, ⊗ is matrix multiplication, A(·) is the adaptive module consisting of several convolutional lay- ers, and M ∈ RK×N is a learned weight map, based on which ACCM aggregates information and generates the coarse class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In practice, we find that for the adap- tive module, a 1 × 1 convolution works well to generate the adaptive class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We also conduct experiments to study the impact of different M and eventually choose the one supervised by Dice loss [19] for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The Dice loss is defined as: LDice = 1 − 2 �N i mi · yi �N i ||mi||2 + �N i ||yi||2 + ϵ , (6) where mi ∈ RK×1 is the i-th column vector in M corre- sponding to i-th pixel, ||a|| is the second norm of vector a, and ϵ is set as 1e−3 to prevent division by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (a) traditional centers (b) our centers Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Illustration of our adaptive class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' During infer- ence, traditional approaches calculate pixel-to-centers similarity maps with fixed centers, while our approach generates adaptive centers for each input image, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Based on the adaptive class centers, we perform pixel-to- center similarity calculation on feature maps to get absolute P-C similarity maps, followed by soft-max at the dimension of different classes to generate the relative P-C similarity maps as the soft-margin prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The structure of the model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Moreover, to address the problem that features of differ- ent classes but the same scene lack enough distinction, pre- vious methods try to aggregate more context information into pixel representations, but there is no supervision im- posed on the representations of class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We propose the Class Distance Loss composed of inter-class distance loss and intra-class distance loss imposed on class centers to make features in the same scene more discriminative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We will introduce it in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Similarity Calculation Module We introduce of similarity function to measure the prox- imity of two embedding a ∈ RD×1 and b ∈ RD×1 in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' we define the inner production similarity as: Simi(a, b)inn = a · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (7) To encourage class centers to be linearly independent and obviate the influence of the embedding’s norm, we also introduce the cosine similarity as : Simi(a, b)cos = abs(a · b) ||a|| ||b|| , (8) where the abs(·) denotes the absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' To calculate the C-C similarity, we employ consine sim- ilarity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 8) to get rid of the influence of the norms of class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We simply replace a and b in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 8 with adaptive class centers c⊤ p and c⊤ p as Simi(c⊤ p , c⊤ q ), where cq ∈ R1×D is the q-th row in C, namely the adaptive class center of the q-th class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' For P-C similarity calculation, we employ inner pro- duction simialrity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' As the prediction is made by the argmax operation on P-C similarity maps, the absolute value of similarity between a pixel with a class center alone is meaningless unless compared with similarities between the pixel and other class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Hence, we introduce rel- ative similarity as the probability that i-th pixel belongs to q-th class: RSimi(fi, c⊤ q ) = exp � Simi(fi, c⊤ q ) � � j exp � Simi(fi, c⊤ j ) �, (9) where j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Class Distance Loss Since there is no constraint on the similarity between different class centers inside each scene, the challenge that different-classes pixels in the same scene lack enough dis- tinction is aggravated consequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We believe that it is easy for pixel-wise classification if pixel representations in the same scene have large inter-class distinction and small intra-class distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Motivated by this notion, we propose the Class Distance (CD) Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We introduce the inter-class distance and intra-class distance at first, then explicate the details of the Class Distance Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Definition of inter-class and intra-class distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The inter-class distance between the p-th and the q-th class is defined as: d(p,q) inter = D(c⊤ p , c⊤ q ), (10) where p and q are the numbers of two different classes, c⊤ p and c⊤ q are the class center vectors of p-th and q-th class respectively, D(·, ·) is the distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The intra-class distance of the q-th class is defined as: dq intra = N � i=1 1[yiq = 1]D(fi, c⊤ q ), (11) 1[condition] = � 1 condition is True 0 condition is False , (12) Conditional CentersWe introduce our distance function based on the relative similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The distance should be negative correlated with relative similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Therefore, we define our distance as: D(fi, c⊤ q ) = −log � RSimi(fi, c⊤ q ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (13) We apply inner production similarity and cosine similarity for dintra and dinter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Inter-class and intra-class Distance Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' As illustrated above, we propose inter-class distance loss and intra-class distance loss to enlarge inter-class distances and diminish intra-class distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our loss functions are as: Lintra = K � q=1 dq intra, (14) Linter = K � p=1 K � q=1 1[q ̸= p]exp(−dp,q inter), (15) where Lintra aims at diminishing intra-class distinction un- der the supervision of GT, and Linter focuses on enlarging the differences between class centers that can operate with- out GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The weighted summation of the Lintra and Linter is called Class Distance (CD) Loss for convenience, which can be formulated as: LCD = Lintra + αLinter, (16) where α is the a hyper-parameter that set empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' For dataset-level CD Loss, we regard each w⊤ j as dataset-level class center just as our rethinking of traditional segmentation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' So, the dataset-level intra-class dis- tance loss and inter-class distance loss are calculated re- spectively as: Ldataset intra = K � q=1 N � i=1 1[yiq = 1]D(fi, w⊤ q ), (17) Ldataset inter = K � p=1 K � q=1 1[q ̸= p]exp � − D(w⊤ p , w⊤ q ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (18) Please notice that when combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 13 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 11, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 9 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 7, the Lintra is in the same format with CE Loss shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4: Ldataset intra = K � q=1 N � i=1 1[yiq = 1] � log � � j exp(fi·c⊤ j ) � −fi·c⊤ q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (19) This shows that the CE Loss is a special case of our pro- posed intra-class distance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' For scene-level CD Loss, we use the proposed ACCM to calculate the class centers conditioned on each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' So, the scene-level CD Loss can be formulated as follows: Lscene intra = K � q=1 N � i=1 1[yiq = 1]D(fi, c⊤ q ), (20) Lscene inter = K � p=1 K � q=1 1[q ̸= p]exp � − D(c⊤ p , c⊤ q ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' (21) Overall loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The over-all loss function of our model are as follows: L = LCD + βLDice = Lintra + αLinter + βLDice, (22) where LDice is the Dice loss imposed on the mask M and β is the weight of LDice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Since the scene-level CD Loss is much more powerful, we employ the Lscene intra and Lscene inter in- stead of Ldataset intra and Ldataset inter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We conduct experiments to show the effectiveness of Lscene intra and Lscene inter over Ldataset intra and Ldataset inter in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Experiment We evaluate our approach on two challenging semantic segmentation datasets: ADE20K and Pascal Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We perform a comprehensive ablation study on the ADE20K dataset and report the comparison with other methods on the ADE20K validation set and the Pascal Context validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Settings Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' ADE20K [40] dataset is a large-scale scene pars- ing benchmark with 150 fine-grained objects and stuff cate- gories, containing 20,210 images for training, 2,000 images for validation, and 3352 images for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Pascal Context dataset [20] is a scene parsing dataset that provides semantic labels for whole scene(both “things” and “stuff” classes), which augments 10,103 images from PAS- CAL VOC 2010 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' It has 4,998 training and 5,105 vali- dation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We use the 59 most common categories for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We conduct our experiments using four NVIDIA GTX 2080 ti GPUs with four images per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' All of our models are optimized by SGD optimizer with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='9 mo- mentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The initial learning rate is set 1e−2 for ADE20K and 4e−3 for PASCAL Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We adopt the polynomial learning rate decay strategy in training following previous works [30, 29, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The initial learning rate is multiplied by (1− iter maxiter)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We apply random resizing with a ratio be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='5 and 2 in training, random cropping input images into (512,512), and random horizontal flipping during train- ing for all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We set the total iterations on ADE20K to 80K and 160K for our ResNet-50 and ResNet- 101 models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' For Pascal Context, models run 80k iterations during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Auxiliary loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Follow previous work [39], we adopt aux- iliary segmentation loss to help train our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We add an auxiliary FCN head, which outputs prediction under the supervision of CE Loss multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' During the evaluation, we average the pre- dictions of multiple scaled following the previous work [37, 29, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Each image is then flipped horizontally, then scaled to a uniform size with scaling factor (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='75) for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Besides, we use Synchro- nized BN in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Additionally, we report mean In- tersection over Union (mIoU) and pixel accuracy (Acc) for ADE20K and mIoU for PASCAL context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ablation Study On ADE20K We conduct ablation studies on ADE20K to demonstrate the effectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Models are trained on ADE20K train set and evaluated on val set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' All the models are pretrained on Image-Net without extra data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Upper-bound verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' To verify the feasibility of our proposed methods, we first carry out a simple upper- bound verification experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We directly replace the pre- dicted Mask, which is supervised by CE loss, of our trained model with processed ground truth during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ac- cording to the results shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 1, our method pro- vides a huge improvement compared with the baseline, while the upper bound method Ours-GT greatly outper- forms our method when the predicted Mask is replaced by GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Since the GT provides better weight maps than the pre- dicted Mask to generate the adaptive class centers, the per- formance is dramatically improved, indicating that there is still a lot of room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Method CCS mIoU(%) Acc(%) Baseline 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='94 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='98 Ours ✓ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='57 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='06 Ours-GT ✓ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='21 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='12 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Upper-bound verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Baseline: ResNet-50 FCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ours: Baseline + CCS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ours-GT: Baseline + CCS layer whose Mask is supplanted by GT to verify the upper bound of CCS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ablation study of class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' To demonstrate the effectiveness of the adaptive class centers, we compare Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' P-C distance comparison between MC & AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Left: inter-class P-C distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Right: intra-class P-C distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We de- note pixel-to-mean centers and pixel-to-adaptive centers as MC and AC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' the P-C similarity, depending on which we generated the probability maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Taking “tree” and “plant” as exam- ples, we show the frequency histograms of intra-class pixel-to-adaptive centers distance and pixel-to-mean cen- ters (namely the global centers that learned from the whole dataset) distance in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' As we introduced before, the results demonstrate that the adaptive class centers make prominent success in increasing the inter-class distance be- tween “tree” pixels and “plant” class center, while the intra- class distances of “plant” pixels and adaptive class center of “plant” are dramatically smaller than the distance between pixels and global class center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ablation study of ACCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We conduct ablation studies of ACCM to explore the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We compared different loss functions on Mask in order to generate better adaptive class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We conduct experiments on FCN based on ResNet-50 and report the results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Res50 FCN CCSNet w/o loss w/ CE loss w/ Dice loss mIoU(%) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='01 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='56 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='57 Acc(%) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='69 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='45 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='06 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ablation study of defferent loss fucntion on Mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our baseline in this experiment is ResNet-50 FCN, with our CS replace the final convolution of FCN under supervision of scene-level CD Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ablation study of hyper-parameter and architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We first conduct experiments with different segmentation heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The multi-scale mIoU results of different segmen- tation heads based on ResNet-50 are reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Among three different segmentation heads, deeplabv3+ has the best performance with 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='25% ms mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We also ex- plore proper hyper-parameters for CCSNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We take FCN Methods FCN PSP Deeplabv3+ baseline 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='94 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='94 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='57 +CCS 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='57 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='07 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='25 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ablation study of segmentation heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' CCS layer is equipped on different segmentation heads based on ResNet-50 are trained on ADE20K training set for 80k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The mIoU re- sults on ADE20K validati set of each model are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='20 inter P-MC Frequency inter P-AC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='00 2 4 6 8 10 12 14 Inter Distance (tree-to-plant)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='15 intra P-MC intra P-AC Frequ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='10 aA lati 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='05 Rel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='00 2 4 6 8 10 12 14 Intra Distance (plant-to-plant)mIoU β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='1 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='5 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='79 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='86 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='85 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='01 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='05 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='57 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='14 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='50 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='47 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Alblation Study of weight of Lscene inter and weight of dice loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We vary the value of α and β and find when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='5, β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='0, model has the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' based on ResNet-50 as a baseline to exploit the best weight of dinter and the weight of Dice loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4, our approach achieves the best performance when the weight of dinter is empirically set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The best FCN based model equipped with our approach improves the pix- Acc and mIoU by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='59% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='63%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ablation study of CCS layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We break down the improvements of our work over ResNet-101 based on DeeplabV3+, which has the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We add the proposed components in our approaches step by step to the DeeplabV3+ baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Experiments are conducted on ADE20K and run 160k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' By simply replacing the global class centers Wc with our adaptive class cen- ters conditionally generated by ACCM, our network im- proves the performance by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='86% in mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' This provides strong support for our assertion that global class centers are inferior compared to conditional class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Moreover, together with the image-level inter-class distance loss, our network achieves 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='76% mIoU on the ADE20K validation set, which demonstrates the effectiveness of our inter-class distance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Results on ADE20K We train the ResNet-101 on the ADE20K training set and report the mIoU results on the validation set (results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our CCSNet uses a pre-trained backbone network on ImageNet and achieves 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='76% mIoU, which outperforms DeeplabV3+ by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='41% mIoU using the same backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Visual comparison examples are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Results on PASCAL Context Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 7 reports the comparison results of our net- work and other state-of-the-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Based on ResNet-101, our method makes favourable performance and achieves 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='9% mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' CCSNet outperforms previous methods using the same ResNet-101 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Conclusion In this paper, we provide a novel perspective to view typ- ical semantic segmentation models, and re-model the prob- lem as a task that models compute the similarity maps be- tween pixels and class centers on each scene, then assign Method centers Linter Lintra mIoU Deeplabv3+ Wc Wc 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='35 + Ldataset inter Wc Wc Wc 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='45 +ACCM&Lscene inter Wc C Wc 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='94 +CCS w/o Linter C C 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='21 CCSNet (Ours) C C C 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='76 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Ablation study of CCS layer and CD Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' The “centers” are those used to calculate the P-C similarity for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' For example, Wc denotes the learned weights of classifier, namely global class centers, used for final prediction, while the C indicate that the prediction is based on the P-C similarity using adaptive class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' “Linter” and “Linter” shows whether the loss is applied on Wc (Ldataset inter and Ldataset intra ) or C (Lscene inter and Lscene intra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Method Baseline mIoU(%) CascadeNet [40] VGG-16 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='90 RefineNet [16] ResNet-152 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='7 UperNet [16] ResNet-101 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='66 PSPNet [37] ResNet-101 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='51 PSPNet [37] ResNet-269 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='94 PSANet [38] ResNet-269 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='77 EncNet [35] ResNet-101 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='77 CFNet [36] ResNet-101 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='65 ANL [41] ResNet-101 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='24 OCRNet [32] ResNet-101 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='28 APCNet [10] ResNet-101 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='38 RGNet [28] ResNet-101 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='80 CPNet [29] ResNet-101 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='27 DeeplabV3+ [5] ResNet-101 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='35 CCSNet ResNet-101 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='76 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Results on the ADE20K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our model based on ResNet-101 achieves 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='76% in mIoU and outperforms all pre- vious methods using the same backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Method Baseline mIoU(%) RefineNet [16] ResNet-152 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='3 PSPNet [37] ResNet-101 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='8 DeeplabV3+ [5] ResNet-101 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='3 EncNet [35] ResNet-101 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='7 DANet[9] ResNet-101 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='6 ANL [41] ResNet-101 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='8 CFNet [36] ResNet-101 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='0 APCNet [10] ResNet-101 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='7 RGNet [28] ResNet-101 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='9 CPNet [29] ResNet-101 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='9 OCRNet [32] ResNet-101 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='8 CCSNet ResNet-101 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='9 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Results on the PASCAL Context validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' We report our result evaluated on 59 class without background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our model based on ResNet-101 achieves 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='9% in mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' pixels to the semantic category with the highest P-C sim- ilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Based on this perspective, we provide solutions to address the two typical issues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' same category yet dif- ferent scenes features could be of large variance, while fea- (a) Images (b) GT (c) FCN (d) DeeplabV3+ (e) CCSNet (Ours) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Qualitative results on ADE20K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' tures of different categories but the same scene may be quite similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Based on P-C similarity maps, we propose ACCM to generate adaptive class centers condi- tioned on each scene to deal with the feature variances of different scenes and design inter-class and intra-class dis- tance loss at scene-level for more inter-class distinction in- side each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Our approach is easy yet effective and can be plugged into most FCN-based architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' Finally, CC- SNet achieves state-of-the-the-art performance on two chal- lenging semantic segmentation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 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Asym- metric non-local neural networks for semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Inter- national Conference on Computer Vision, pages 593– 602, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE4T4oBgHgl3EQfDgur/content/2301.04870v1.pdf'} diff --git a/-tFRT4oBgHgl3EQfrzf-/content/tmp_files/2301.13622v1.pdf.txt b/-tFRT4oBgHgl3EQfrzf-/content/tmp_files/2301.13622v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3ab3c4896fddccb19e5e9f2f8647f196a83bce1 --- /dev/null +++ b/-tFRT4oBgHgl3EQfrzf-/content/tmp_files/2301.13622v1.pdf.txt @@ -0,0 +1,1338 @@ +Learning Data Representations with Joint Diffusion Models +Kamil Deja 1 Tomasz Trzcinski 1 2 3 Jakub M. Tomczak 4 † +Abstract +We introduce a joint diffusion model that simul- +taneously learns meaningful internal representa- +tions fit for both generative and predictive tasks. +Joint machine learning models that allow synthe- +sizing and classifying data often offer uneven per- +formance between those tasks or are unstable to +train. In this work, we depart from a set of em- +pirical observations that indicate the usefulness +of internal representations built by contemporary +deep diffusion-based generative models in both +generative and predictive settings. We then intro- +duce an extension of the vanilla diffusion model +with a classifier that allows for stable joint train- +ing with shared parametrization between those +objectives. The resulting joint diffusion model +offers superior performance across various tasks, +including generative modeling, semi-supervised +classification, and domain adaptation. +1. Introduction +Training a single machine learning model that can jointly +synthesize new data as well as to make predictions about +input samples remains a long-standing goal of machine +learning (Jebara, 2012; Lasserre et al., 2006). Shared rep- +resentations created with a combination of those two ob- +jectives promise benefits on many downstream problems +such as calibration of model uncertainty (Chapelle et al., +2009), semi-supervised learning (Kingma et al., 2014), un- +supervised domain adaptation (Ilse et al., 2020) or continual +learning (Masarczyk et al., 2021). +Since the introduction of deep generative models such +as Variational Autoencoders (VAEs) (Kingma & Welling, +2014), a growing body of work takes advantage of shared +deep neural network-based parameterization and latent vari- +ables to build joint models. For instance, Ilse et al. (2020); +† Work done at Vrije Universiteit Amsterdam 1Warsaw Univer- +sity of Technology, Poland 2IDEAS NCBR, Poland 3Tooploox, +Poland 4Eindhoven University of Technology, The Netherlands. +Correspondence to: Kamil Deja , +Jakub M. Tomczak . +Preprint. +Tulyakov et al. (2017); Knop et al. (2020); Yang et al. (2022) +stack a classifier on top of latent variables sampled from a +shared encoder. Similarly, (Nalisnick et al., 2019; Perugachi- +Diaz et al., 2021) use normalizing flows to obtain an invert- +ible representation that is further fed to a classifier. However, +these approaches require modifying the log-likelihood func- +tion by scaling either the conditional log-likelihood or the +marginal log-likelihood. This idea, known as hybrid mod- +eling (Lasserre et al., 2006), leads to the situation where +models concentrate either on synthesizing data or predicting +but not on both of those tasks simultaneously. +We address existing joint models’ limitations and lever- +age the recently introduced diffusion-based deep generative +models (DDGM) (Sohl-Dickstein et al., 2015; Dhariwal & +Nichol, 2021; Kingma et al., 2021). This new family of +methods has become popular because of the unprecedented +quality of the samples they generate. However, relatively +little attention was paid to their inner workings, especially +to the internal representations built by the DDGMs. In this +work, we fill this gap and empirically analyze those repre- +sentations, validating their usefulness for predictive tasks +and beyond. Then, we introduce a joint diffusion model, +where a classifier shares the parametrization with the UNet +encoder by operating on the extracted latent features. This +results in meaningful data representations shared across +discriminative and generative objectives. +We validate our approach in several use cases where we +show how one part of our model can benefit from the other. +First, we investigate how DDGMs benefit from the addi- +tional classifier to conditionally generate new samples or +alter original images. Next, we show the performance im- +provement our method brings in the classification task. Fi- +nally, we extend the evaluation of our joint diffusion model +to semi-supervised learning, domain adaptation, and coun- +terfactual explanations. For all of those tasks, our method +does not require any problem-specific adjustments, which +confirms the flexibility of our approach. +We can summarize the contributions of our work as follows: +• We provide empirical observations with insight into +representations built internally by diffusion models, +on top of which we introduce a joint classifier and +diffusion model with shared parametrization. +arXiv:2301.13622v1 [cs.LG] 31 Jan 2023 + +Diffusion Models Learn Data Representations +• We introduce a conditional sampling algorithm where +we optimize internal diffusion representations with a +classifier. +• We prove that our solution work with several use cases +including the semi-supervised learning, domain adap- +tion and counterfactual explanations. +2. Background +Joint models +Let us consider two random variables: x ∈ +X and y ∈ Y. For instance, in the classification problem we +can have X = RD and Y = {0, 1, . . . , K − 1}. The joint +distribution over these random variables could be factorized +in one of the following two manners: +p(x, y) = p(x|y)p(y) +(1) += p(y|x) p(x). +(2) +In Eq. (2), we get the conditional distribution p(y|x) (e.g., a +classifier) and the marginal distribution p(x). For prediction, +it is enough to learn the conditional distribution, which is +typically parameterized with neural networks. However, +training the joint model with shared parametrization has +many advantages since one part of the model can positively +influence the other. +Diffusion-based Deep Generative Models +In this work, +we follow the formulation of Diffusion-based deep gen- +erative models as presented in (Ho et al., 2020; Sohl- +Dickstein et al., 2015). Given a data distribution x0 ∼ +q(x0), we define a forward noising process q that pro- +duces a sequence of latent variables x1 through xT by +adding Gaussian noise at each time step t, with a vari- +ance of βt ∈ (0, 1), defined by a schedule β1, ..., βT , +namely, q(x1, . . . , xT |x0) = �T +t=1 q(xt|xt−1), where +q(xt|xt−1) = N(xt; √1 − βtxt−1, βtI). +Following (Huang et al., 2021; Kingma et al., 2021; +Tomczak, 2022; Tzen & Raginsky, 2019), we consider +DDGMs as infinitely deep hierarchical VAEs with +a specific family of variational posteriors; +namely, +Gaussian diffusion processes (Sohl-Dickstein et al., +2015). +Therefore, for data point x0, and latent vari- +ables x1, . . . , xT , we want to optimize the marginal +likelihood +pθ(x0) += +� +pθ(x0, . . . , xT )dx1, . . . , xT . +We +define +the +backward +diffusion +process +as +pθ(x0, . . . , xT ) += +p(xT ) +�T +t=0 pθ(xt−1|xt), where +pθ(xt−1|xt) = N(xt−1; µθ(xt, t), Σθ(xt, t)). +We can define the variational lower bound as follows: +ln pθ(x0) ≥ Lvlb(θ) := +Eq(x1|x0)[ln pθ(x0|x1)] +� +�� +� +−L0 +− DKL [q(xT |x0)∥p(xT )] +� +�� +� +LT +− +− +T +� +t=2 +Eq(xt|x0)DKL [q(xt−1|xt, x0)∥pθ(xt−1|xt)] +� +�� +� +Lt−1 +. (3) +that we further optimize with respect to the parameters of +the backward diffusion. +Training objective +In (Ho et al., 2020), authors notice +that instead of estimating the probability of previous la- +tent variable p(xt−1|xt), we can predict the added noise ϵ. +Therefore, a single part of the variational lower bound is +equal to: +Lt(θ) =Ex0,ϵ +� +β2 +t +2σ2 +t αt (1 − αt)× +��ϵ − ϵθ +�√αtx0 + +√ +1 − αtϵ, t +���2� +, +(4) +where ϵ ∼ N(0, I) and ϵθ(·, ·) is a neural network predict- +ing the noise ϵ from xt. +In (Ho et al., 2020), it is also suggested to train the model +with a simplified objective that is a modified version of +equation (4) without scaling, namely: +Lt,simple(θ) = Ex0,ϵ +���ϵ − ϵθ +�√αtx0 + √1 − αtϵ, t +���2� +. +(5) +In practice, a single shared neural network is used for mod- +eling ϵθ. For that end, most of the works (Ho et al., 2020; +Kingma et al., 2021; Nichol & Dhariwal, 2021) use UNet +architecture (Ronneberger et al., 2015) that can be seen as a +specific type of an autoencoder. This is particularly relevant +for this work since we benefit from the Encoder – Decoder +structure of the denoising DDGM model. +3. Related Work +Diffusion models There are several extensions to the base- +line DDGM setup that aim to improve the quality of sampled +generations (Ho et al., 2020; Huang et al., 2021; Kingma +et al., 2021; Song & Ermon, 2019; Song et al., 2020). This +includes noising and sampling schedulers and modified +training objectives. Several works propose to improve the +quality of samples from DDGMs by conditioning the gen- +erations with class identities (Tashiro et al., 2021; Ho & +Salimans, 2022; Huang et al.). Among those works, (Dhari- +wal & Nichol, 2021) introduces a classifier-guided genera- +tion, where a gradient from an externally and independently +trained classifier is added in the process of backward diffu- +sion to guide the generation towards a target class. On top + +Diffusion Models Learn Data Representations +of this approach, (Augustin et al., 2022) present a tool for +investigating the decision of a classifier by generating visual +counterfactual explanations with a diffusion model. In this +work, we simplify both of those methods benefiting from +training a joint model with representations shared between +a diffusion model and a classifier. +Diffusion models and UNet representations Some works +tackle the problem of data representation with diffusion +models. (Abstreiter et al., 2021) introduce additional en- +coded information to the score estimator, which allows them +to use the score matching loss function for learning data rep- +resentations. (Baranchuk et al., 2021) use activations from +the pre-trained diffusion UNet model for the image segmen- +tation task. Other works consider data representations from +the UNet model within other generative models. (Esser +et al., 2018) introduce a conditional UNet-based variational +autoencoder, while (Falck et al., 2022) show the connection +between the UNet architecture and wavelet transformation, +applying it to the hierarchical VAEs. We show that indeed +diffusion models learn useful representations. We further +take advantage of that in utilizing a shared parameterization +between a diffusion model and a classifier in a joint model. +Joint training Apart from latent variable joint models, +(Grathwohl et al., 2019b) show that it is possible to use a +shared parameterization (a neural network-based classifier) +to formulate an energy-based model. This Joint Energy- +based Model (JEM) could be seen as a classifier if a softmax +function is applied to logits or a generator if a Markov-chain +Monte Carlo method is used to sample from the model. +Although it obtains strong empirical results, gradient estima- +tors used to train JEM are unstable and prone to diverging +when optimization parameters are not perfectly tuned, which +limits the robustness and applicability of this method. Alter- +natively, Introspective Neural Networks could be used for +generative modeling and classification by applying a single +parameterization (Jin et al., 2017; Lazarow et al., 2017; Lee +et al., 2018). The idea behind this class of models relies +on utilizing a training procedure that combines adversarial +learning and contrastive learning. Similarly to JEMs, sam- +pling is carried out by running an MCMC method. Here, +we propose to combine diffusion models with a classifier +by sharing parameterization. Thus, our training is entirely +based on the log-likelihood function, and sampling is carried +out by backward diffusion instead of any MCMC algorithm. +4. Diffusion models learn data representations +As mentioned earlier, learning useful data representations is +important for having a good generator or classifier. Ideally, +we would like to train a joint model that allows us to obtain +proper representations for both p(y|x) and p(x) simulta- +neously. In this work, we investigate parameterizations +of DDGMs and, in particular, the use of an autoencoder +𝐱𝐭−𝟏 +𝐱𝐭 +Encoder +Decoder +𝐳𝐭 +𝟏 +𝐳𝐭 +𝟐 +𝐳𝐭 +𝟑 +𝐳𝐭 +Pooling +Figure 1. Data representation zt in a UNet-based diffusion model. +as a denoising decoder pθ(xt−1|xt). Within this architec- +ture, the denoising function can be decomposed into two +parts: encoding of the image at the current timestep into a +set of features Zt = e(xt) and then decoding it to obtain +xt−1 = d(Zt). In particular, for the UNet architecture, a set +of features obtained from an input is a structure composed +of several tensors with image features encoded to different +levels, Zt = {z1 +t, z2 +t . . . zn +t }. For simplification, for all fur- +ther experiments, we propose to pool features encoded by +the same filter and concatenate the averaged representations +into a single vector zt, as presented in Fig. 1 for n = 3. +In particular, we can use average pooling to select average +convolutional filter activations to the whole input. Details +of this procedure are described in Appendix A.1. +4.1. UNet representations are useful for prediction +First, we would like to verify whether averaged representa- +tions z0 extracted from an original image x0 by the UNet +contain information that is in some sense predictive. For +that, we measure the performance of an MLP-based classi- +fier fed with z0. +As presented in Fig 2, representations encoded in z0 are +indeed very informative and, in some cases (e.g., CIFAR- +10), could lead to performance comparable to a stand-alone +classifier with the same architecture as the combination of +the UNet encoder and MLP but trained with the standard +cross-entropy loss function. This observation is in line +with (Baranchuk et al., 2021), where the same activations +from the pre-trained diffusion model were used for semantic +image segmentation. +4.2. Diffusion models learn features of increasing +granularity +The next question is how the data representations zt differ +with diffusion timesteps t. To investigate this issue, we train +an unsupervised DDGM on the CelebA dataset, which we +then use to extract the features zt at different timesteps. On +top of those representations, we fit a binary logistic regres- +sion classifier for each of the 40 attributes in the dataset. In +Fig. 3, we show the performance of those regression models + +Diffusion Models Learn Data Representations +Figure 2. The test-set accuracy of a stand-alone classifier compared +to a classifier trained on top of data representations from a pre- +trained diffusion model extracted from original images x0. +for 6 different attributes when calculated on top of represen- +tations from ten different diffusion timesteps. We observe +that the model learns different data features depending on +the amount of noise added to the original data sample. As +presented in Fig. 3, high-grained data features such as hair +color start to emerge at late diffusion steps (closer to the +noise), while low-grained features (e.g., necklace or glasses) +are not present until the early steps. This observation is in +line with the works on denoising autoencoders where au- +thors observe similar behavior for denoising with different +amounts of added noise (Chandra & Sharma, 2014; Geras +& Sutton, 2014; Zhang & Zhang, 2018). +200 +400 +600 +800 +1000 +Timestep +0.5 +0.6 +0.7 +0.8 +0.9 +AUC +Blond hair +Black hair +Necklace +Mouth slightly open +Pointy nose +Eyeglasses +Figure 3. The area under the ROC curve (AUC) for logistic regres- +sion models fit on data representations extracted with a pre-trained +diffusion model at ten different diffusion timesteps. High-grained +features are already distinguishable at late diffusion steps (closer +to random noise), while low-grained features are only represented +at the earlier stage of the forward diffusion. +5. Method +Taking into account the observations described in Section +4, we propose to train a joint model that is composed of a +classifier and a DDGM. Specifically, we propose to use a +shared parameterization, namely, a shared encoder of the +UNet architecture that serves as the generative part and for +calculating pooled features for the classifier. +5.1. Joint Diffusion Models: DDGMs with classifiers +Following the procedure introduced in Sec. 4, we pool the +latent representations of the data from different levels of the +UNet architecture into one vector z. On top of this vector, +we build a classifier model trained to assign a label to the +data example represented by the vector z. +In particular, we consider the following parameterization +of a denoising diffusion model within a single diffusion +timestep t, pθ(zt−1|zt). We distinguish the encoder eν +with parameters ν that maps input xt into a set of vectors +Zt = eν(xt), where Zt = {z1 +t, z2 +t . . . zn +t }, i.e., a set of +representation vectors derived from each depth level of the +UNet architecture. The second component of the denoising +diffusion model is the decoder dψ with parameters ψ that +reconstructs feature vectors into a denoised sample, xt−1 = +dψ(Zt). Together the encoder and the decoder form the +denoising model pθ with parameters θ = {ν, ψ}. Next, we +introduce a third part of our model, which is the classifier +gω with parameters ω that predicts target class ˆy = gω(Zt). +The first layer of the classifier is the average pooling that +results in a single representation zt. +ො𝐲 +𝐱𝟎 +𝐱𝐓 +𝐱𝐭−𝟏 +𝐱𝐭 +(a) The parameterization of our joint diffusion +𝐱𝟎 +𝐱𝐓 +𝐱𝐭−𝟏 +𝐱𝐭 +ො𝐲𝐓 +ො𝐲𝐭 +ො𝐲𝐭−𝟏 +(b) Additional noisy classifiers +Figure 4. The parameterization of our joint diffusion model. +(a) Each step in the backward diffusion is parameterized by a +shared UNet. The classifier uses the encoder of the UNet together +with the average pooling (green) and additional layers (yellow). +(b) An alternative training that additionally uses the classifier for +noisy images xt (t > 0). +In our approach, we consider a classifier that takes the origi- +nal image x0 for which a vector of probabilities is returned ϕ +and eventually the final prediction is calculated, ˆy = gω(x0). +The visualization of our shared parameterization is pre- +sented in Figure 4(a). As a result, our model could be writ- +ten as follows pν,ψ,ω(x0:T , y) = pν,ω(y|x0) pν,ψ(x0:T ), +and applying the logarithm yields: +ln pν,ψ,ω(x0:T , y) = ln pν,ω(y|x0) + ln pν,ψ(x0:T ). +(6) +The logarithm of the joint distribution (6) could serve as +the training objective in which ln pθ(x0:T ) could be either + +100% +75% +Accuracy +50% +25% +0% +FashionMNIsT +SVHN +CIFAR-10 +CIFAR-100 +I Standard classfier +: Classifier on pre-trained DDGMDiffusion Models Learn Data Representations +approximated by the ELBO for the diffusion-based model +in (3) or the simplified objective with (5)). In this paper, we +follow the simplified objective: +Lt,diff(ν, ψ) = Ex0,ϵ +� +∥ϵ − ˆϵ∥2� +, +(7) +where ˆϵ is a prediction from the decoder: +{z1 +t, z2 +t . . . zn +t } =eν +�√αtx0 + +√ +1 − αtϵ, t +� +(8) +ˆϵ =dψ({z1 +t, z2 +t . . . zn +t }). +(9) +For the classifier, we use the logarithm of the categorical +distribution, i.e., the cross-entropy loss: +Lclass(ν, ω) = −Ex0,y +��K−1 +k=0 1[y = k] log +exp(ϕk) +�K−1 +c=0 exp(ϕc) +� +, +(10) +where y is the target class, ϕ is a vector of probabilities +returned by the classifier gω(eν(x0)), and 1[y = k] is the +indicator function that is 1 if y equals k, and 0 otherwise. +The final loss function in our approach is then the following: +L(ν, ψ, ω) = Lclass(ν, ω)+ +(11) +− L0(ν, ψ) − +T +� +t=2 +Lt,diff(ν, ψ) − LT (ν, ψ). +We optimize the objective in (11) jointly with a single opti- +mizer over parameters {ν, ψ, ω}. +5.2. An alternative training of joint diffusion models +The training of the proposed approach over a batch of data is +straightforward. For a sampled pair (x0, y), the example x0 +is first noised with a forward diffusion to a random timestep, +xt so that the training loss for the denoising model is a +Monte-Carlo approximation of the sum over all timesteps. +Then x0 is fed to a classifier that returns probabilities ϕ, and +the cross-entropy loss is calculated for given y. +However, as discussed in Section 4.2, the diffusion model +trained even in a fully unsupervised manner provides data +representations related to the different granularity of input +features at various diffusion timesteps. Considering this, we +can improve the robustness of our method by applying the +same classifier to intermediate noisy images xt (0 < t < +T), which by reason adds the cross-entropy losses for xt, +namely: +Lt +class(ν, ω) = −Ex0,y +��K−1 +k=0 1[y = k] log +exp(ϕt +k) +�K−1 +c=0 exp(ϕtc) +� +, +(12) +where ϕt +k is a vector of probabilities given by gω(eν(xt)). +Then the extended objective (11) is the following: +LT (ν, ψ, ω) = L(ν, ψ, ω) + +� +t∈T +Lt +class(ν, ω), +(13) +where T ⊆ {1, 2, . . . , T} is the set of timesteps. These +additional noisy classifiers are schematically depicted in +Figure 4(b) in which we highlight that the model is reused +across various noisy images. It is important to mention that +the noisy classifiers serve only for training purposes; they +are not used for prediction. This procedure is similar to the +data augmentation technique, where random noise is added +to the input (Sietsma & Dow, 1991). +5.3. Conditional sampling in joint diffusion models +To improve the quality of samples generated by DDGM, +(Dhariwal & Nichol, 2021) propose a classifier guidance +approach, where an externally trained classifier can be used +to guide the generation of the DDGM trained in an unsu- +pervised way towards the desired class. In the standard +DDGM, at each backward diffusion step, an image is sam- +pled from the output of the diffusion model pθ according to +the following formula: +µ, Σ ←µθ (xt) , Σθ (xt) +xt−1 ← sample from N (µ, Σ) +(14) +(Dhariwal & Nichol, 2021) proposed to change the second +line of this equation and add a scaled gradient with respect +to the target class from an externally trained classifier c(·) +directly to the output of the denoising model: +xt−1 ← sample from N (µ + sΣ∇xtc(xt), Σ) , +(15) +where s is a gradient scale. +With the joint training of a classifier and diffusion model +introduced in this work, we propose to simplify the clas- +sifier guidance technique. Using the alternative training +introduced in the previous section, Section 5.2, we can use +noisy classifiers to formulate conditional sampling. The +encoder model eν encodes input data xt into the represen- +tation vectors Zt that are used to both denoise an example +into the previous diffusion timestep xt−1 ∼ dψ (Zt) as well +as to predict the target label with a classifier ˆy = gω (Zt). +Therefore, to guide the model towards a target label during +sampling, we propose optimizing the representations Zt ac- +cording to the gradient calculated through the classifier with +respect to the desired class. The overview of this procedure +is presented in the algorithm Algorithm 1. +For the reformulation of the diffusion model proposed +by (Ho et al., 2020) where instead of predicting the previous +timestep xt−1 denoising model is optimized to predict noise +ϵ that is subtracted from the image at the current timestep +xt, we adequately change the optimization objective. In- +stead of optimizing the noise to be specific to the target +class, we optimize it to be anything except for the target +class, which we implement by changing the optimization +direction: Z′ +t ← Zt + α∇Zt log gω(y|Zt). + +Diffusion Models Learn Data Representations +Table 1. An evaluation of generative capabilities by measuring the FID score, Precision and Recall of generations from various diffusion- +based models, including our joint diffusion model. +Model +FashionMNIST +CIFAR-10 +CIFAR-100 +CelebA +FID ↓ +Prec ↑ +Rec ↑ +FID ↓ +Prec ↑ +Rec ↑ +FID ↓ +Prec ↑ +Rec ↑ +FID ↓ +Prec ↑ +Rec ↑ +DDGM +7.8 +71.5 +65.3 +7.2 +64.8 +61.2 +29.7 +70.0 +47.8 +5.6 +66.5 +58.7 +DDGM (classifier guidance) +7.9 +66.6 +59.5 +8.1 +63.2 +63.3 +22.1 +69.3 +46.9 +4.9 +66.0 +57.8 +Ours +8.7 +71.1 +61.1 +10 +66.4 +56.3 +17.4 +63.2 +54 +7.5 +66.7 +50.7 +Ours (conditional sampling) +5.9 +63.1 +63.2 +9.4 +66.8 +54.7 +16.8 +63.5 +54.1 +6.6 +64.2 +54.5 +Algorithm 1 Sampling with optimized representations +given a diffusion model (an encoder eν(Zt|xt), a decoder +dφ(xt−1|Zt)), a classifier gω(y|Zt), and a step size α. +Input: class label y, step size α +xT ← sample from N(0, I) +for all t from T to 1 do +Zt ← eν(xt) +Z′ +t ← Zt − α∇Zt log gω(y|Zt) +µ, Σ ← dψ(Z′ +t) +xt−1 ← sample from N(µ, Σ) +end for +return x0 +6. Experiments +In the experiments, we aim for observing the benefits of the +proposed joint diffusion model over a stand-alone classifier +or a marginal diffusion model. To that end, we run a series +of experiments to verify various properties, namely: +• We measure the quality of a classifier to evaluate +whether training together with a diffusion model im- +proves the robustness of the classifier. +• We measure the generative capability of our model to +check if representations optimized by the classifier can +lead to more accurate conditional generations. +• We train our model in a semi-supervised setup to see +if shared representations between the classifier and the +diffusion model can positively influence the classifica- +tion accuracy for a limited number of labeled data. +• We use a domain-adaptation task to check if optimizing +the representations using our approach helps to adapt +to new data compared to a stand-alone classifier. +• We show that our joint model learns abstract features +that can be used for the counterfactual explanation. +We use a UNet-based model with a depth level of three in +all experiments. We pool its latent features with average +pooling into a single vector, on top of which we add a clas- +sifier with two linear layers and the LeakyReLU activation. +All metrics are reported for the standard training with the +objective in (11), except for the conditional sampling where +we additionally train the classifier on noisy samples, i.e., +additional losses as in (13). Hyperparameters and training +details are included in the appendix and code repository1. +6.1. Predictive performance of joint diffusion models +In the first experiment, we evaluate the predictive perfor- +mance of our method. To that end, we report the accu- +racy of our model on four datasets: FashionMNIST, SVHN, +CIFAR-10, and CIFAR-100. We compare our method with +a baseline classifier trained with a standard cross-entropy +loss and the MLP classifier trained on top of representations +extracted from the pre-trained DDGM as in Section 4. The +results of this experiment are presented in Table 2. +As noticed before, a classifier trained on features extracted +from the UNet of a DDGM pre-trained in an unsupervised +manner achieves reasonable performance. However, it is +always outperformed by a stand-alone classifier. Interest- +ingly, the proposed joint diffusion model achieves the best +performance on all four datasets. The reason for that could +be two-fold. First, training a partially shared neural network +(i.e., the encoder in the UNet architecture) benefits from the +unsupervised training, similarly to how the pre-training us- +ing Boltzmann machines benefited finetuning of deep neural +networks (Hinton et al., 2006). Second, the shared encoder +part is more robust since it is used in the backward diffusion +for images with various levels of noise. +Table 2. The classification accuracy calculated on the test sets. For +each training, we used exactly the same architecture. +Model +F-MNIST +SVHN +CIFAR-10 +CIFAR-100 +Classifier +92.0% +95.1% +81% +60.8% +Ours (pre-trained DDGM) +60.6% +79.6% +80.9% +43.8% +Ours +93.3% +95.4% +89.9% +63.6% +6.2. Generative performance of joint diffusion models +In the second experiment, we check how adding a classifier +in our joint diffusion models influences the generative per- +formance. We use the FID score to quantify the quality of +data synthesis. Additionally, we use distributed Precision +1Anonymized due to the double-blind policy. + +Diffusion Models Learn Data Representations +(Prec), and Recall (Rec) for assessing the exactness and +diversity of generated samples (Sajjadi et al., 2018). For our +joint diffusion model, we consider samples from the prior +let through the backward diffusion. We also use the second +sampling scheme in which we use conditional sampling, +namely, the optimization procedure as described in Section +5.3. We compare our approach with a vanilla DDGM, and a +DDGM with classifier guidance (Dhariwal & Nichol, 2021). +Overall, all methods performed similarly. In some cases, the +vanilla DDGM and the DDGM with the classifier guidance +obtain better results in terms of the FID score (CIFAR-10, +CelebA) and Precision (FashionMNIST, CIFAR-100). How- +ever, our joint diffusion model with conditional sampling +outperforms the DDGMs in terms of the FID score (Fash- +ionMNIST, CIFAR-100), Precision (CIFAR-10), and Recall +(FashionMNIST, CIFAR-100). Hence, we conclude that +all models perform well on the data synthesis task. Con- +ditional sampling is beneficial in the joint diffusion model +and yields better-quality generations. This could result from +the fact that the optimization procedure drives Zt to a mode. +Eventually, the backward diffusion generates better samples. +0 +500 +1000 +Step size α +40 +60 +Precision +CIFAR10 +FashionMNIST +0 +500 +1000 +Step size α +40 +50 +60 +70 +Recall +CIFAR10 +FashionMNIST +Precision +Recall +Figure 5. The dependency between the value of the step size α +and the value of Precision and Recall for the joint diffusion with +conditional sampling. +To get further insight into the role of conditional sampling, +we carried out an additional study for the varying value of +α (the step size in Algorithm 1). In Figure 5, we present +how Precision and Recall change for different values of +this parameter. Apparently, increasing the step size value α +leads to more precise but less diverse samples. This is rather +intuitive behavior because larger steps result in features Zt +closer to modes. There seems to be a sweet spot around +α ∈ [100, 250] for which both measures are high. +𝛼 = 0 +𝛼 = 100 +𝛼 = 500 +𝛼 = 1000 +Figure 6. Samples from our joint diffusion model optimized to- +wards a specific class (here: plane) with different step size α. +We visualize this effect in Figure 6. For a chosen class, e.g., +plane, we observe that the larger α, the more precise the +samples are but with limited diversity (i.e., the background +is almost the same). For more samples, see Appendix B. +6.3. A comparison to state-of-the-art approaches +To get a better overview of the performance of our joint +diffusion model, we present a comparison with other joint +models and SOTA discriminative and generative models in +Table 3. Importantly, we present the discriminative model +and the generative model as the bounds of the performance. +Importantly, within the class of the joint models, our joint +diffusion performs on par with Joint Energy-based Model +(JEM) (Grathwohl et al., 2019a) in terms of classification +accuracy, but it visibly outperforms JEM in terms of the FID +score. Moreover, our approach performs significantly better +than flow-based methods (Residual Flows, Glow). +Table 3. A comparison of our joint diffusion model with other +joint models, and the SOTA discriminative model, and the SOTA +generative model on the CIFAR-10 test set. +Class +Model +Accuracy% ↑ +FID↓ +Joint +Residual Flows (Chen et al., 2019) +70.3 +46.4 +Glow (Kingma & Dhariwal, 2018) +67.6 +48.9 +JEM (Grathwohl et al., 2019a) +92.9 +38.4 +Ours +89.9 +9.4 +Disc. +VIT-H (Dosovitskiy et al., 2020) +99.5 +N/A +Gen. +DDGM (our implementation) +N/A +7.2 +LSGM (Vahdat et al., 2021) +N/A +2.1 +6.4. Semi-supervised learning of joint diffusion models +With satisfactory performance, we further evaluate other +setups where one part of the model can benefit from another. +In particular, we propose to assess our approach in the semi- +supervised setup, where we artificially limit the amount of +labeled data to 10%, 5% or 1% in three datasets SVHN, +CIFAR-10, and CIFAR-100. We compare joint diffusion +models to a deep neural network-based classifier and a deep +neural network-based classifier on top of the pre-trained +UNet encoder. The results are presented in Table 4. +In the case of the stand-alone classifier, we observe that +classification accuracy drastically drops with the number of +labeled data. However, in our joint diffusion model, we can +train the classifier on the smaller dataset while still optimiz- +ing the generator part in an unsupervised manner, with all +available unlabelled data. This approach significantly im- +proves the classifier’s performance thanks to the improved +quality of data representations. For CIFAR-10, we observe +that the joint diffusion model with only 5% of labeled data +(250 examples per class) performs almost as well as the +stand-alone classifier trained with the fully labeled train- +ing dataset. In more extreme scenarios, e.g., labeled data +limited to 50, 25, or 5 examples per class, it seems to be +slightly more beneficial to first learn the data representation +in an unsupervised way and then add the classifier on top +of them. However, overall, the joint diffusion model per- +forms extremely well and greatly benefits from available +unlabeled data in terms of classification accuracy. Our ex- + +Diffusion Models Learn Data Representations +Table 4. The accuracy of the classifier trained in the semi-supervised setup, for each dataset we train the classifier with the fully labeled +data or a limited amount of labeled examples and the remaining unlabelled examples. We compare standard classifier with classifier +trained on a pre-trained DDGM as presented in Sec 4 and our joint diffusion method. +SVHN +CIFAR-10 +CIFAR-100 +Labelled data +100% +5% +1% +100% +5% +1% +100% +10% +5% +1% +Images per class +10000 +500 +100 +5000 +250 +50 +500 +50 +25 +5 +Classifier +95.1 +87.8 +75.15 +81 +46.4 +31.5 +60.8 +22.2 +16.6 +6.9 +Ours (pre-trained DDGM) +79.6 +51.7 +66.0 +80.9 +75.1 +65.3 +43.8 +33.9 +28.8 +15.4 +Ours +95.4 +90.2 +76.7 +89.9 +78.2 +64.7 +63.6 +38.6 +21.5 +11.5 +periments align with the observation by (Baranchuk et al., +2021), where DDGMs were used to improve the perfor- +mance in semi-supervised image segmentation. +6.5. Domain adaptation with diffusion-based +fine-tuning +In the previous section, we evaluate whether the classifier +can benefit from the generative part of our model when +trained with limited access to labeled data. Now, we further +extend those experiments and check if joint diffusion can +adapt to the new data domain using only the generative part +– in a fully unsupervised way. For this purpose, we run an +experiment in which we first train the model on the source +labeled data to retrain it on the target dataset without access +to the labels. We compare our approach to a standalone +deep neural network-based classifier, see Table 5. +Table 5. The classification accuracy of the classifier trained in a +domain adaption task. We first train the joint model on the source +dataset, which we adapt to the target domain by retraining it using +only the diffusion loss for the examples in the target one. +SVHN → MNIST +USPS → MNIST +MNIST → USPS +Classifier +78.8 +54.7 +72.2 +Ours +85.5 +90.5 +92.7 +Classifier on target +(upper bound) +96.1 +96.8 +99.4 +As expected, in all three scenarios, the classification accu- +racy of the stand-alone classifier degrades on a target do- +main.2 However, having access to unlabeled data from the +target domain allows our joint diffusion model to adapt sur- +prisingly well. Our approach outperforms the stand-alone +classifier in all three cases by a significant margin. This +result indicates that learning low-level features is essential +for obtaining good predictive power while it is enough to +transfer the classification head unchanged. +6.6. Visual Counterfactual Explanations +In the last experiment, we apply our joint diffusion model to +real-world medical data, the MALARIA dataset (Rajaraman +et al., 2018), that includes 27,558 cell images that are either +2The classification accuracy does not drop to a random level +because all datasets share the same task, i.e., digits classification. +Negative examples +Positive examples +Figure 7. Data samples from the Malaria dataset classified as nega- +tive examples (left) or parasitized cells (right). (top row) original +data examples, (2nd row) data noised with 20% of forward diffu- +sion steps, (3rd row) denoised images with conditional sampling, +(bottom row) the difference between the 3rd and 4th rows. +infected by the malaria parasite or not (a classification task). +The cells have various shapes and different staining (i.e., +colors) and contain or not the parasite (visually apparent as +a purple dot). +After training our joint diffusion model, we obtain high +classification accuracy (98%) on the test set. On top of +this, we introduce an adaptation of visual counterfactual +explanations (VCE) method (Augustin et al., 2022) that +provides an answer to the question: What is the minimal +change to the input image x0 to change the decision of the +classifier. In our setup, we answer this question with a +conditional sampling algorithm that we use to generate the +counterfactual explanations. In Figure 7, we show a few +examples from the negative (left) or positive (right) classes. +We add 20% of noise to these images and run conditional +sampling with the opposite class (i.e., changing negative +examples to positive ones and vice versa). In both cases, +the joint diffusion model with conditional sampling can +either remove the parasite from the image (for the positive +examples) or add the parasite to the image (for the negative +ones). All presented images are not cherry-picked. +This experiment shows that not only we can use our pro- +posed approach to obtain a powerful classifier but also to +visualize some regions of interest. In the considered case, +calculating the difference between the original example and +the image with a changed class label indicates the malaria +plasmodium (see the last row in Figure 7). We provide more +examples from the CelebA data in the Appendix C. + +Diffusion Models Learn Data Representations +7. 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G. +Chroma-vae: Mitigating shortcut learning with generative +classifiers. arXiv preprint arXiv:2211.15231, 2022. + +Diffusion Models Learn Data Representations +Zhang, Q. and Zhang, L. Convolutional adaptive denoising +autoencoders for hierarchical feature extraction. Frontiers +of Computer Science, 12(6):1140–1148, 2018. + +Diffusion Models Learn Data Representations +Appendix +A. Training details and hyperparameters +A.1. Pooling of the UNet features +As discussed in Section 4, we pool the UNet features encoded to different UNet levels with the average pooling function. +Precisely speaking, we take an average convolutional filter activation for a given filter across the whole image. This approach +seems to result in the loss of information, such as the location of particular features extracted by the convolutional filter, but +it allows us to create image representation with reasonable dimensionality. Depending on the dimensionality of input, with +our method, we extract 1856 features for 28 × 28 Grayscale images (e.g. MNIST), 3712 features for 32 × 32 images with 3 +color channels (e.g. CIFAR), and 5248 features for 64 × 64 images with 3 color channels (CelebA). +In all of our experiments, we use average pooling. Although other options such as max or min pooling might be used, our +approach ensures that all of the features across the whole image are shared between the classifier and the generative models. +A.2. Semi-supervised learning +In our semi-supervised learning, we train our joint diffusion model on datasets with limited access to labeled samples. The +simplest approach for this problem is to calculate the loss function on the diffusion using the whole batch of data while +using only the labeled examples for the classifier loss. However, in some scenarios, we artificially omit up to 99% of labeled +data. In practice, this would lead to a situation where for batch size equal to 128 or 254 examples, the classifier loss would +be practically calculated on 1 or 2 samples. Therefore, to stabilize the training we propose to create a buffer where we put +labeled examples from each batch. When the buffer reaches its capacity equal to the batch size, we calculate the classifier +loss using the examples from the buffer and add it to the generative loss according to Equation 11. +A.3. Domain adaptation +In the experiments on the domain adaptation task, we propose the simplest setup, where we first train the joint model on the +source task using the joint loss function (Eq. 11), and then we retrain the model on the target domain using only the DDGM +loss in Equation 7. We show that without any alteration to our basic setup, we can observe a significant performance boost +compared to the baseline classifier. We believe that we can further improve those results if we focus directly on the domain +adaptation task and take advantage of the recent advantages in this field. Further experiments in this direction should for +example include simultaneous training on examples from two domains. To improve the alignment, we can also benefit from +adversarial training as introduced by (Ganin et al., 2016) in DANN. + +Diffusion Models Learn Data Representations +B. Additional results: Conditional generations with optimized representations +In Figure 8 we present how the decision of the classifier changes for sampling with the optimized generations. With a higher +α step size value, optimization converges faster towards target classes. Interestingly, for the CIFAR10 dataset, there are +certain classes (e.g., class 3) that converge later in the backward diffusion process than the others. In Figure 8 we also +present associated samples from our model. Once more they depict that the higher value of the α parameter leads to more +precise but less diverse samples. +0 +200 +400 +600 +800 +1000 +Timestep +0 +2 +4 +6 +8 +Avg. prediction +9 +7 +6 +4 +3 +1 +0 +(a) α = 200 +0 +200 +400 +600 +800 +1000 +Timestep +0 +2 +4 +6 +8 +Avg. prediction +9 +7 +6 +4 +3 +1 +0 +(b) α = 1000 +Figure 8. CIFAR10: Classifier decisions at different diffusion steps, for conditional sampling with different values of step size α and +associated conditional samples + +Diffusion Models Learn Data Representations +Fashion MNIST +CIFAR-100 +Figure 9. Conditional samples from our joint diffusion model for Fashion MNIST dataset (left) and first 10 classes of CIFAR100 dataset +(right). Each row represents samples from one class. +Fashion MNIST +CIFAR-100 +CelebA +Figure 10. Generated examples from our joint diffusion model without conditional sampling for CIFAR-10, CIFAR-100, and CelebA +dataset. + +oDiffusion Models Learn Data Representations +C. Additional results: Counterfactual image generation +In the experiment described in Section 6.6, we presented how we can use our joint diffusion model to generate the +counterfactual explanations to the classifier using the medical dataset. In Figure 11, we present more examples of this +approach by perturbing original examples from the CelebA dataset. We select 3 attributes from the CelebA dataset namely: +young, smiling, and moustache. For each attribute, we select 5 positive examples and 5 negative examples which we alter +using our conditional sampling procedure with the classifier-based optimization. We present original examples (first row) +noised with 20% of noise (second row) and generated towards counterfactual class (third row). In the last row, we show the +differences between the original and modified examples. +(a) Young to old (left), old to young (right) +(b) Smiling to no-smiling (left), no-smiling to smiling (right) +(c) Moustache to no-moustache (left), no-moustache to moustache (right) +Figure 11. Counterfactual image generation for the CelebA dataset using three different attributes on random original examples. For each +attribute, we select 5 positive examples that we change to negative ones and 5 negative ones that we change to positive ones. + diff --git a/-tFRT4oBgHgl3EQfrzf-/content/tmp_files/load_file.txt b/-tFRT4oBgHgl3EQfrzf-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..078f39695d0c0f498b61a082ef2b7f29543cc92a --- /dev/null +++ b/-tFRT4oBgHgl3EQfrzf-/content/tmp_files/load_file.txt @@ -0,0 +1,975 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf,len=974 +page_content='Learning Data Representations with Joint Diffusion Models Kamil Deja 1 Tomasz Trzcinski 1 2 3 Jakub M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' Tomczak 4 † Abstract We introduce a joint diffusion model that simul- taneously learns meaningful internal representa- tions fit for both generative and predictive tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' Joint machine learning models that allow synthe- sizing and classifying data often offer uneven per- formance between those tasks or are unstable to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' In this work, we depart from a set of em- pirical observations that indicate the usefulness of internal representations built by contemporary deep diffusion-based generative models in both generative and predictive settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' We then intro- duce an extension of the vanilla diffusion model with a classifier that allows for stable joint train- ing with shared parametrization between those objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' The resulting joint diffusion model offers superior performance across various tasks, including generative modeling, semi-supervised classification, and domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' Introduction Training a single machine learning model that can jointly synthesize new data as well as to make predictions about input samples remains a long-standing goal of machine learning (Jebara, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' Lasserre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' Shared rep- resentations created with a combination of those two ob- jectives promise benefits on many downstream problems such as calibration of model uncertainty (Chapelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=', 2009), semi-supervised learning (Kingma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=', 2014), un- supervised domain adaptation (Ilse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=', 2020) or continual learning (Masarczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' Since the introduction of deep generative models such as Variational Autoencoders (VAEs) (Kingma & Welling, 2014), a growing body of work takes advantage of shared deep neural network-based parameterization and latent vari- ables to build joint models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' For instance, Ilse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' † Work done at Vrije Universiteit Amsterdam 1Warsaw Univer- sity of Technology, Poland 2IDEAS NCBR, Poland 3Tooploox, Poland 4Eindhoven University of Technology, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFRT4oBgHgl3EQfrzf-/content/2301.13622v1.pdf'} +page_content=' Correspondence to: Kamil Deja ncrit(γ) for a +given value of γ; the border of this section has the equation ncrit(γ) as shown in Figure 9. The side ratio is extended to +the excluded regime of γ < 1. + + +Vertex Angle (deg.) +0 +60 +-1 +50 +Pythagorean Exponent +40 +30 +20 +8 +10 +-9 +-10 +1.6 +1.2 +1.4 +1.8 +2 +Side Ratio12 + + +Figure 11: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio. The gaps between +the plots and the horizontal axis are due to n > ncrit(γ) at those values of γ. The side ratio is extended to the excluded +regime of γ < 1. + +As was the case with eq. (5), the solution to eq. (12) for ( ) +n + + is not analytic. However, it is clear that +for all values of n, +1 + = is once again a solution. By substituting +1 + = into eq. (9), the vertex angle can +be found (eq. (13)) which expectedly has the same mathematical form as eq. (7)). + +( +)( +) +2 +2 +0 +1 +1 +1 +0 +n +n +n +n +d +d + + + + + +− +− += + − ++ +− ++ += + +(12) + +2 +1 +max +cos +1 +2 +n +n + +− +− +−  + += +− + + + + +, +(13) +where +max + +− denotes the maximum value of  +− for a given degree n. + +The second derivative of eq. (9) is the same as the second derivative of eq. (4), and therefore, the +mathematical confirmation that +max + +− is a maximum angle was also performed numerically with Maple®. +A plot of eq. (13) is shown in Figure 12. + + +60 +n=-1 +n=-2 +50 +n=-3 +n = -100 +40 +30 +20 +10 +0 +0 +0.5 +1.5 +2 +2.5 +Side Ratio13 + + +Figure 12: Maximum vertex angle versus Pythagorean exponent. + +In the limit that the Pythagorean exponent becomes infinitely negative, + +( +) +2 +2 +1 +1 +1 +lim +lim +cos +2 +n +n +n +n + + + + + +− +− +→− +→− + + + + ++ − ++ + + + + += += + + + + + + + + + + + + +, + (14) +which is like the case with positive exponents. Even with the imposed condition that 1 +2 + + + +, eq. (14) +must be evaluated as a piecewise function. + + + +Case 1 ( +) += +γ +1 +( +) +1 +60o + + +− += += + + + +Case 2 ( +) +1 < γ < 2 +( +) +1 +1 +2 +cos +2 + + + +− +− + + + + += + + + + + + +Case 3 ( +) +− +→ +γ +2 + +( +) +2 +0 + + +− +− +→ += + + + + + +60 +50 + (deg.) +Maximum Vertex Angle +40 +30 +20 +10 +0 +-15 +-12 +6- +-6 +-3 +0 +Pythagorean Exponent14 + +Case 1 produces the same result as did case 2 for positive exponents ( +) +i.e., +1 + = + – a maximum +vertex angle of 60o and thus, an equilateral triangle. Also, like the positive exponents case, the equilateral +triangle is never actually realized because although γ can equal 1, n cannot be infinite. As illustrated in +Figure 13, the negative infinite degree Pythagorean Theorem can produce real triangles, and the +requirement that 1 +2 + + + + is retained. Furthermore, a straight line will result when + +( ) +( ) +( +) +crit +crit +2 +2 +1 +1 +1 +lim +lim +cos +0 +2 +n +n +n +n +n +n + + + + + + +− +→ +→ + + + + ++ − ++ + + + + += += + + + + + + + + + + + + +. + + + +Figure 13: Plot of the negative infinite degree Pythagorean Theorem. + +4. The Areas of the Associated Triangles + +The area of the scalene triangle in Figure 1, with a real vertex angle, is +1 +sin +2 +A +ab + += +. Its side +lengths are a, b +a + += +, and from eq. (1), +( +) +1 +1 +n +n +c +a  += ++ +. Therefore, its area is + +2 +1 +sin +2 +A +a  + += +. + (15) + +Applying +2 +2 +cos +1 sin + + += − + and eq. (4) or eq. (9) depending on whether n ++ + + or n +− + +, +respectively, eq. (15) can be written in terms of n and γ. + + +15 + +( +) +2 +2 +2 +2 +2 +4 +1 +1 +2 +n +n +a +A + + + + + + + += +− ++ − ++ + + + + + + + + +. + (16) + +The restrictions on γ remain: +1 +  for n ++ + +, and 1 +2 + + + + for n +− + +. However, there are no +restrictions on a (other than +0 +a  +). + +4.1 The Maximum Areas of Triangles for the n-Degree Pythagorean Theorem with Positive +Real Exponents + +A plot of eq. (16) is shown in Figure 14. It is clear as if a is constant and γ increases, the area of the +associated triangle increases without bound. Also, if a is constant, increasing n increases the area, but this +is a negligible effect compared to increasing γ. However, the effect that n has on the area increases as γ +increases. + + +Figure 14: Triangle area (in dimensionless units) with unit side length (a = 1). In the left-side figure, the effect of n +on the area is difficult to distinguish. However, in the right-side figure, the effect of n on the area is clearer. +Determining the conditions that maximize the area of a triangle can be done with an “angular” +approach. From eq. (15), the maximum area of a triangle with given values of a and γ clearly occurs when +90 + = + (corresponding to +2 +n = +) and is +2 +1 +2 +A +a  += +. +In summary, +• +If γ and n are fixed, increasing a will increase the triangle’s area without bound. +• +If a and n are fixed, increasing γ will increase the triangle’s area without bound. + +Therefore, no absolute maximum area triangle exists because the triangle will experience infinite +dilation. The only minimum area is the trivial solution ( +) +0 +A = +. Most significant from this analysis is +that a right triangle, with fixed values of a and γ, has the maximum area. + +Triangle Area +Triangle Area +10 +5 +3 +4.5 +2.8 +0.9 +8 +4 +2.6 +0.8 +Pythagorean Exponent +7 +3.5 +Pythag orean Exponent +2.4 +0.7 +3 +2.2 +0.6 +2.5 +2 +0.5 +5 +2 +0.4 +1.8 +1.5 +0.3 +1.6 +3 +1 +0.2 +1.4 +2 +0.5 +0.1 +1.2 +0 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +Side Ratio +Side Ratio16 + +The above result can, of course, be determined from the dependence of  ++ on n and γ. From eq. (4), +( +) +2 +2 +1 +1 +1 +cos +90 +2 +n +n + + + + ++ +− + + ++ − ++ + + += += + + + + + + + which gives +( +) +2 +2 +1 +1 +n +n + + ++ += ++ +(17) +because +( +) +2 +2 +1 +1 +0 +2 +n +n + + + ++ − ++ += +. The only real solution to eq. (17) for n is +2 +n = +. + +An alternative and more rigorous approach to finding the degree which gives rise to the maximum +area case for a given value of γ is from eq. (16); +0 +A +n + += + + gives + +( +) +( +) +( +) +2 +2 +2 +1 +1 +1 +1 +ln +ln +1 +0 +1 +n +n +n +n +n +n +n +n + + + + + + + + + + +  + ++ ++ − ++ +− ++ += + + + + + + ++ + + + +  + +. + (18) + +But ( +) +2 +1 +0 +n +n + + + + for any value of γ or n, and therefore, + +( +) +( +) +2 +2 +1 +1 +1 +ln +ln +1 +0 +1 +n +n +n +n +n +n + + + + + + + + + +  + ++ − ++ +− ++ += + + + + + + ++ + + + +  + +. + (19) + +( +) +2 +2 +1 +1 +0 +n +n + + + + ++ − ++ += + + + + + iff +2 +n = + (as shown with eq. (17)). +( +) +1 +ln +ln +1 +0 +1 +n +n +n +n + + + + + + + + +− ++ + + + + + ++ + + + + +, +regardless of the value of γ, although it asymptotically approaches zero for increasing n. As expected, this +somewhat more complex approach gives the same result as the “angular approach” used above – the +maximum area triangle for positive real exponents occurs when +2 +n = +. + +Thus, the n-degree Pythagorean Theorem that produces triangles with the largest area for a given a +and γ is the standard Pythagorean Theorem, and the triangles it produces are right angled. Even though +the triangles that result from 1 +2 +n + + + have vertex angles which are larger than 90o, they have smaller +areas than +2 +1 +2 +A +a  += + because sin +1 +  for obtuse  . + + + + + +17 + +For a given a and a given n (not necessarily +2 +n = +), the value of γ that produces the triangle with the +maximum area would be calculated from the derivative of eq. (16) i.e., +0 +A + + + + += + + + + + + which yields + +( +) +( +) +2 +2 1 +2 +2 +1 +1 +1 +1 +4 +n +n +n +n +n + + + + + + +− + + +− + + + + + + ++ − ++ +− ++ += + + + + + +  + +. + (20) + +However, equation (20) has no solution for +0 +n  +. This is physically reasonable because increasing γ +for a given value of a continuously dilates the triangle; therefore, there is no finite maximum area. The +smallest of all of the areas of these continuously enlarging triangles (with a fixed side length a) occurs +when +1 + = and is + +3 +2 +1 +2 +2 +min +2 +1 4 +n +n +n +n +A +a +− +− + + + + + + + + + + + + + + += +− + + + + + + +. + (21) + +Equation (21) (a plot of which is shown in Figure 15) represents the areas of the members of an +infinite set of isosceles triangles. Each of these triangles has the minimum possible area among all of the +various triangles of the same degree. The triangle in this infinite set with the largest area occurs from the +infinite degree case, is denoted by +max +min +A + , and is given by the limit of eq. (21). + +max +2 +min +min +lim +6 +n +A +A +a + +→ += += +, + (22) + +Figure 15: The minimum area (in dimensionless units) of an isosceles triangle with unit side length (a = 1) for the +n-degree Pythagorean Theorem with positive real exponents. The red line indicates +max +min +A + . + + + + +18 + +4.1.1 +The Area of a Triangle with a Fixed Perimeter (n > 0) + +To further examine these ideas, it is sensible to consider the area of a triangle with a fixed perimeter +P. Then, determine the values of γ and n that produce the maximum area for the n ++ + + case (and then in +Section 4.2, for the n +− + + case). + +The triangle perimeter is trivially + +( +) +1 +1 +1 +n +n +P +a +b +c +a  + + + += ++ ++ += ++ + ++ + + + + +. + (23) + +The area, in terms of the fixed perimeter, is denoted +P +A , and it results from combining eqs. (16) and (23). + +( +) +( +) +2 +2 +2 +2 +2 +1 +4 +1 +1 +2 +1 +1 +P +n +n +n +n +P +A + + + + + + + + + + + + + += +− ++ − ++ + + + + + + + + + + ++ + ++ + + + + + + + + + + (24) + +From eq. (24): for a fixed P, and as a function of n, the area of the triangle is a maximum when +1 + = . This area is denoted by +max +P +A + and is given by + +1 +1 +2 +max +2 +1 +4 +16 +16 +1 2 +n +n +n +P +n +n +P +A ++ + + + + + + + + + + + + +− + + + + + + + + + + + + +− += + + +  + + ++ + + + + + + + + + + + + +, + (25) + +Equation (25) represents the areas of an infinite set of maximum area isosceles triangles with a fixed +perimeter. The triangle in that set with the largest area is the infinite degree member which is denoted by +max +P +A + , and its area is given by + +1 +1 +2 +2 +max +2 +1 +4 +16 +3 +lim +16 +36 +1 2 +n +n +n +P +n +n +n +P +A +P + ++ + + + + + + + + + + + + +→ +− + + + + + + + + + + + + + + +− += += + + + + + + +  + + + + ++ + + + + + + + + + + + + +. + (26) + + + +19 + +From eq. (24), as  →  , the areas of the triangles approach zero (as shown in Figure 17), as they +take the form of a straight line. Also, from eq. (24) (and as shown in Figure 16), triangles with a fixed P +will see their areas (as a function of γ) asymptotically increase as n → . Those areas are denoted +P +A +and are given by + +( +) +2 +2 +2 +4 +1 +1 +4 +2 +1 +P +A +P + + + + + +− += + + ++ + + + + +, + (27) + + + +Figure 16: Area (in dimensionless units) of Pythagorean triangles with unit perimeter. + + +Figure 17: Area (in dimensionless units) of Pythagorean triangles with unit perimeter. + + + + + +20 + +4.2 The Maximum Areas of Triangles for the n-Degree Pythagorean Theorem with Negative +Real Exponents + +For n +− + +, a plot of eq. (16) is shown in Figure 18. The area relationship for positive real exponents +also applies to the case of negative real exponents. However, for all +( ) +crit +n +n + + +, the vertex angles are +real and acute, and if a and γ are fixed, the maximum area triangle has the largest vertex angle. + + + +Figure 18: Triangle area (in dimensionless units) with unit side length (a = 1). The missing section at the top right +and on the right side of figure are due to n > ncrit(γ) for a given value of γ; the border of this section has the equation +ncrit(γ) as shown in Figure 9. + +To determine the triangle with the maximum area, consider eqs. (15) and (9). + + + +( +) + + +( +) +( +) +( +) +1 +2 +2 +2 +1 +2 +2 +2 +2 +2 +2 +max +1 +1 +1 +1 +1 +max sin +1 +min cos +1 +min +2 +2 +2 +2 +n +n +A +a +a +a + + + + + + + + + + + + + + + + ++ − ++ + + + + + + += += +− += +− + + + + + + + + + + + + + + + + + + + + +. + (28) + +In the special case of an isosceles triangle ( +) +1 + = +, +( +) +2 +2 +2 +1 +1 +min +1 2 +2 +n +n +n +n + + + +− + + + + + + + + ++ − ++ + + = − + + + + + + + for +1 +n  − +because +( +) +crit +1 +n + = + does not exist. + + + +Triangle Area +0.5 +2 +0.45 +3 +0.4 +Pythagorean Exponent +0.35 +4 +0.3 +4 +0.25 +0.2 +0.15 +8 +0.1 +-9 +0.05 +-10 +1.2 +1.4 +1.6 +1.8 +2 +Side Ratio21 + +The maximum triangle area is then + +1 +1 +1 +2 +1 +2 +2 +max +1 +sin +2 +1 4 +2 +n +n +n +n +A +a +a + + + +− +− + + + + + + + + += + + + + + + + += += +− + + + + + + + + + +. + (29) + +If the negative infinite degree case of eq. (28) is considered, the following results. + +( +) +2 +2 +1 +1 +lim +min +2 +2 +n +n +n + + + + +→− + + + + ++ − ++ + + + + = + + + + + + + + + + + + + for any side ratio that conforms to 1 +2 + + + +. For this negative +infinite degree case, the maximum triangle area is + +2 +2 +2 +max +1 +1 +sin +4 +2 +4 +A +a +a + + + + +− = += +− +. + (30) + +Equation (30) necessarily complies with the requirement that 1 +2 + + + +. Additionally, three interesting +results emerge about this infinite set of negative infinite degree triangles: +1. In this set, there are two values of γ that produce triangles with equal areas: +1 + = and +3 + = +, +and that maximum area is +1, +3 +2 +2 +max +1 +3 +sin +2 +4 +A +a +a + + + + +− += += += += +. +2. +2 + = + necessarily gives zero area because the triangle has collapsed into a straight line. +3. In this infinite set, the triangle with the largest area is the member for which +2 + = +, and its area +is +max +2 +2 +max +1 +1 +sin +2 +2 +A +a +a + + +− = += +. + + +4.2.1 +The Area of a Triangle with a Fixed Perimeter (n < 0) + +If, on the other hand, the perimeter is fixed, eq. (24) applies for 1 +2 + + + +, and the largest area +triangle will be isosceles, and it will have an area of +2 +3 +36 P . This area is expected because the conditions +that +1 + = and n → − result in equilateral triangle in which +3 +P +a += +, and therefore, +2 +2 +3 +3 +36 +4 +P +a += +. + +From eq. (24), if lim +P +n +A +→− + is taken, eq. (31) results. As seen in Figure 19 (a plot of eq. (31)), the area +will asymptotically increase as n → − and will asymptotically decrease toward zero as +2 + → +. Eq. +(31) determines the area for a fixed perimeter, arbitrary side ratio, and infinite degree triangle. + + +22 + +( +) +2 +2 +2 +1 +4 +4 +2 +P +A +P + + + +− + + += +− + + ++ + + + + +, + (31) +where +P +A− is the asymptotic area of the triangles with a fixed perimeter and for which n → − . + +Among the infinite set of triangles whose areas are given by eq. (31), the triangle with the largest area +is isosceles (as stated above), and its area is +2 +3 +36 P . + + +Figure 19: Area of Pythagorean triangles with unit perimeter. + + +Figure 20: Area of Pythagorean triangles with unit perimeter. + + +23 + +5. Summary + +The Pythagorean Theorem has been extended to positive and negative real exponents unfettered by +the physical requirement of dimension. The relationship between the ratio of the adjacent sides and the +vertex angle was determined for a given degree. It was found that for positive exponents, the stipulation +that +1 +  can be applied to all degrees, and no complex vertex angles arose. For 1 +2 +n + + +, an obtuse +triangle results, and if +2 +n  +, the triangle is acute. However, for negative exponents to produce real +vertex angles, the restriction 1 +2 + + + + is necessary but not sufficient. The additional requirement that if +1 +2 + + + +, then +( ) +crit +n +n + + + needs to be imposed. + +The areas of the associated triangles for positive and negative real exponents were explored. With +fixed a and γ values, the areas for +| +1 +n +n ++ + + are maximized when the triangle is right isosceles +requiring +1 + = and +2 +n = +. Additionally, triangle areas increase as n → . Alternatively, if the +perimeter of a triangle is kept constant, the triangle area approaches a maximum value with increasing n +and approaches 0 for decreasing γ. + +For the n +− + + case, as n → − , the triangle with a fixed perimeter and the maximum area has a +side ratio of +2 + = +. In contrast, if the degree is finite and +( ) +crit +n +n + + + (if +1 +  ), the maximum area +occurs when the vertex angle ( +) +,n +  + is a maximum. + + + + + + +24 + +Works Cited +1. Faltings, G., The proof of Fermat’s last theorem by R. Taylor and A. Wiles. Notices of the AMS, 1995. +42(7): p. 743-746. +2. Agarwal, R.P., Pythagorean theorem before and after Pythagoras. Adv. Stud. Contemp. Math, 2020. +30: p. 357-389. +3. Amir-Moéz, A.R., R.E. Byerly, and R.R. Byerly, Pythagorean theorem in unitary spaces. Publikacije +Elektrotehničkog fakulteta. Serija Matematika, 1996: p. 85-89. +4. Atzema, E.J., Beyond Monge's Theorem: A Generalization of the Pythagorean Theorem. Mathematics +Magazine, 2000. 73(4): p. 293-296. +5. Conant, D. and W. Beyer, Generalized pythagorean theorem. The American Mathematical Monthly, +1974. 81(3): p. 262-265. +6. Cook, W.J., An n-dimensional Pythagorean theorem. The College Mathematics Journal, 2013. 44(2): +p. 98-101. +7. Czyzewska, K., Generalization of the Pythagorean theorem. Demonstratio Mathematica, 1991. 24(1- +2): p. 305-310. +8. Donchian, P.S. and H. Coxeter, 1142. An n-dimensional extension of Pythagoras' Theorem. The +Mathematical Gazette, 1935. 19(234): p. 206-206. +9. Drucker, D., A comprehensive Pythagorean theorem for all dimensions. The American Mathematical +Monthly, 2015. 122(2): p. 164-168. +10. Eifler, L. and N.H. Rhee, The n-dimensional Pythagorean theorem via the divergence theorem. The +American Mathematical Monthly, 2008. 115(5): p. 456-457. +11. Kadison, R.V., The Pythagorean theorem: I. The finite case. Proceedings of the National Academy of +Sciences, 2002. 99(7): p. 4178-4184. +12. Lee, J.-R., The law of cosines in a tetrahedron. The Pure and Applied Mathematics, 1997. 4(1): p. 1-6. +13. Quadrat, J.-P., J.B. Lasserre, and J.-B. Hiriart-Urruty, Pythagoras' theorem for areas. The American +Mathematical Monthly, 2001. 108(6): p. 549-551. +14. Veljan, D., The 2500-year-old Pythagorean theorem. Mathematics Magazine, 2000. 73(4): p. 259-272. +15. Yeng, S., T. Lin, and Y.-F. Lin, The n-dimensional Pythagorean theorem. Linear and multilinear +algebra, 1990. 26(1-2): p. 9-13. + + diff --git a/0tE0T4oBgHgl3EQfuAGv/content/tmp_files/load_file.txt b/0tE0T4oBgHgl3EQfuAGv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9193b808c20381697b0bfdd6cc9a959ade55f6a --- /dev/null +++ b/0tE0T4oBgHgl3EQfuAGv/content/tmp_files/load_file.txt @@ -0,0 +1,697 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf,len=696 +page_content='1 The Real Number n-Degree Pythagorean Theorem Jeffrey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Lee1,2,4 Gerald B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Cleaver1,3 1Early Universe Cosmology and Strings Group 2Space Physics Numerical Modeling Group Center for Astrophysics, Space Physics, and Engineering Research 3Department of Physics 4Department of Geosciences Baylor University One Bear Place Waco, TX 76706 Jeff_Lee@Baylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='edu Gerald_Cleaver@Baylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='edu Key Words: ∞-degree Pythagorean Theorem, Law of Cosines, maximum triangle area, minimum triangle area Word Count: 4203 Abstract This paper extends the Pythagorean Theorem to positive and negative real exponents to take the form n n n a b c + = and makes use of the definition 1 b a \uf067 = \uf0b3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For the case of n + \uf0ce , 1 n \uf0b3 is necessary for the vertex angle to be real, and there are no restrictions on γ beyond its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, for n − \uf0ce , two significant restrictions that are necessary for n n n a b c + = to yield real vertex angles have been discovered by this work: 1 2 \uf067 \uf0a3 \uf03c , and n cannot exceed a critical value which is γ-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Additionally, the areas of the associated triangles have been determined as well as the conditions for those areas to be maxima or minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Introduction For centuries, the Pythagorean Theorem has been significant in the foundation of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Although the finally proven Fermat’s Last Theorem [1] definitively establishes the non-existence of Pythagorean Triples for n n n a b c + = with | 2 n n \uf0ce \uf03e , the Pythagorean Theorem can be extended to higher degrees which are not required to be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, only positive integers possess the physical representation of dimension for n n n a b c + = (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 n = defines a scalar sum of straight line segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 n = defines a scalar sum of areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3 n = defines a scalar sum of volumes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' and 4 n \uf0b3 defines a scalar sum of hypervolumes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 0 n = is mathematically meaningless because it results in 0 0 0 2 1 a b c + = → = ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 The extension of the Pythagorean Theorem to higher dimensions using a variety of methods has been extensively addressed in the literature [2-15], most frequently by considering a formulation such as 2 2 Total 1 n k k a a = =\uf0e5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Rather, the n-degree Pythagorean Theorem contains a relationship between the ratio of the adjacent side lengths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', γ) and the vertex angle (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' n\uf0ce indicates that the triangles to which n n n a b c + = is being applied are not necessarily right angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 1 \uf067 \uf0b3 is defined as a precondition with no loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If 0 1 \uf067 \uf03c \uf0a3 , the 1 \uf067 \uf0b3 triangle has merely been rotated within its plane, and 0 \uf067 \uf03c implies the geometrically uninteresting imposition upon the triangle of non-physical side lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The vertex angle θ is a function of γ and n and is therefore denoted ( ) ,n \uf071 \uf067 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For 0 1 n \uf03c \uf03c , the vertex angle is complex (not addressed in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For 1 2 n \uf03c \uf03c , the triangle is obtuse with ( ) 90 , 180 n \uf071 \uf067 \uf03c \uf03c , and for 2 n \uf03e , the triangle is acute with ( ) , 90 n \uf071 \uf067 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In each instance of | 1 n n + \uf0ce \uf0b3 , ( ) 0 , 180 n \uf071 \uf067 \uf0a3 \uf0a3 , and there are no restrictions on the ratio of the adjacent side lengths (other than 1 \uf067 \uf0b3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' When n − \uf0ce , the situation changes significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In order that ( ) ,n \uf071 \uf067 \uf0ce , the restriction 1 2 \uf067 \uf0a3 \uf03c must be imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, even with 1 2 \uf067 \uf0a3 \uf03c , there is one additional compulsory restriction for ( ) ,n \uf071 \uf067 \uf0ce : if 1 \uf067 \uf0b9 , the degree of the negative real exponent Pythagorean Theorem must not exceed a critical value which is dependent on γ ( ) ( ) ( ) crit crit i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 , and thus, 1 n n n n \uf067 \uf067 \uf0a3 \uf0b9 \uf0b3 \uf0b9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If ( ) crit 1 n n \uf067 \uf03e \uf0b9 , the vertex angle is complex (also not addressed in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For | 1 n n + \uf0ce \uf0b3 , the areas of the associated triangles, with fixed a and γ values, reach a maximum which occurs when the triangle is right isosceles ( ) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 and 90 \uf067 \uf071 = = ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' the areas increase as n → \uf0a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Conversely, if the perimeter of a triangle is fixed, the triangle area approaches a maximum value for increasing n and approaches 0 for decreasing γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For ( ) crit | if 1 n n n \uf067 \uf067 − \uf0ce \uf0a3 \uf0b9 , the areas of the associated triangles with a fixed perimeter are maximized for 2 \uf067 = and as n → −\uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For finite n, the maximum area occurs when ( ) ,n \uf071 \uf067 is a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The n-Degree Pythagorean Theorem with Positive Real Exponents For the general scalene triangle in Figure 1, the n-degree Pythagorean Theorem can be written as n n n a b c + = (1) which must also conform to the standard Law of Cosines, 2 2 2 2 cos c a b ab \uf071 = + − , (2) 3 with n + \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In this work, the extension of the Pythagorean exponents to numbers other than n + \uf0ce does not extend with it the Law of Cosines because unlike in [10, 12] because an n-dimensional simplex does not arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Here, the objective is the solution for the vertex angle θ such that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (1) conforms to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 1: A scalene triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Using 1 b a \uf067 = \uf0b3 , rewriting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (1) as ( ) 2 2 n n n c a b = + , (3) and equating eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (2) and (3) yields ( ) 2 2 1 1 1 cos 2 n n \uf067 \uf067 \uf071 \uf067 + − \uf0e9 \uf0f9 + − + \uf0ea \uf0fa = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb , (4) where \uf071 + denotes that the vertex angle arises from a version of the n-degree Pythagorean Theorem in which n + \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If 0 n = , as stated above, no triangle exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' there is no 0-degree (or 0-dimensional) Pythagorean Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If 0 1 n \uf03c \uf03c , \uf071 is a complex angle for all side ratios γ and is not applicable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, if 1 n \uf0b3 , \uf071 is always a real angle, and therefore, a corresponding real triangle does exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As expected, for the 1 n = case, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) becomes ( ) ( ) 1 1 cos 1 180o n \uf071 + − = = − = which is sensible because the triangle has collapsed into a straight line which is independent of the side ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For the 2 n = case, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) θb c a b θa θ 4 becomes ( ) ( ) 1 2 cos 0 90o n \uf071 + − = = = , and the traditional Pythagorean Theorem with a 90o vertex angle, also independent of the side ratio, is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' It is important to note that the vertex angle exists only for combinations of n and γ such that the argument of the inverse cosine function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) is not greater than 1 or less than -1, ( ) 2 2 1 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 2 n n \uf067 \uf067 \uf067 \uf0e6 \uf0f6 + − + \uf0e7 \uf0f7 \uf0a3 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, the stipulation that 1 \uf067 \uf0b3 ensures that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) will always result in real angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, there are no forbidden side ratios, and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) applies without restriction for 1 n \uf0b3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The 1 ≤ n ≤ 2 Case If 1 2 n \uf0a3 \uf0a3 , the range of vertex angles is between 90o and 180o (as shown above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Plots of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) are shown in Figure 2 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 2: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for 1 ≤ n ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Vertex Angle (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') 2 180 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 170 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 160 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 150 Pythagorean Exponent 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 140 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 130 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 120 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 100 06 2 3 4 5 6 8 6 10 Side Ratio5 Figure 3: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for 1 ≤ n ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For any given value of n, there is a corresponding value of γ that results in a maximum vertex angle which is seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' That vertex angle and the side ratio that gives rise to it are found by equating the first derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( )( ) 2 2 0 1 1 1 0 n n n n d d \uf071 \uf067 \uf067 \uf067 \uf067 + − = \uf0de − + − + = (5) The solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (5) for ( ) n \uf067 is not analytic, and the total number of complex solutions grows rapidly with increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, it is clear that for all values of n, 1 \uf067 = is a solution – it is the only positive real solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, the largest vertex angle occurs in an isosceles triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, by substituting 1 \uf067 = into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4), the maximum vertex angle can be found (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 1 max cos 1 2 n n \uf071 + − − \uf0e6 \uf0f6 = − \uf0e7 \uf0f7 \uf0e8 \uf0f8 , (6) where max \uf071 + denotes the maximum value of \uf071 + for any degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The second derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) is extremely unruly, and therefore, the confirmation that max \uf071 + is a maximum angle was performed numerically with Maple®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' A plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (6) is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 6 Figure 4: Maximum vertex angle versus Pythagorean exponent for 1 ≤ n ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 The n ≥ 2 Case For 2 n \uf0b3 , plots of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) are shown in Figure 5 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In this case, the vertex angles do not exceed 90o, and there is a minimum vertex angle for which eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' That vertex angle and the side length that gives rise to it are also found by equating the first derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) to zero (as was done above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Given that for all values of n, 1 \uf067 = is once again the only positive real solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (5), and by substituting 1 \uf067 = into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4), the minimum vertex angle can be found (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, as was the case in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 for the largest vertex angle for 1 2 n \uf0a3 \uf0a3 , the smallest vertex angle for 2 n \uf03e occurs when the triangle is isosceles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 1 min cos 1 2 n n \uf071 + − − \uf0e6 \uf0f6 = − \uf0e7 \uf0f7 \uf0e8 \uf0f8 , (7) where min \uf071 + denotes the minimum value of \uf071 + for any degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 180 170 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') 160 Maximum Vertex Angle 150 140 130 120 110 100 90 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 2 Pythagorean Exponent7 Figure 5: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 6: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The side ratio is extended to the excluded regime of γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The confirmation that min \uf071 + is a minimum angle was also performed numerically with Maple®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' A plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (7) is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Vertex Angle (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') 10 06 6 85 Pythagorean Exponent 80 6 75 70 4 65 3 2 09 1 2 3 4 5 7 8 6 10 Side Ratio8 Figure 7: Minimum vertex angle versus Pythagorean exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In the limit that the Pythagorean exponent becomes infinite, ( ) 2 2 1 1 1 lim lim cos 2 n n n n \uf067 \uf067 \uf071 \uf071 \uf067 − \uf0a5 →\uf0a5 →\uf0a5 \uf0ec \uf0fc \uf0e9 \uf0f9 + − + \uf0ef \uf0ef \uf0ea \uf0fa = = \uf0ed \uf0fd \uf0ea \uf0fa \uf0ef \uf0ef \uf0ea \uf0fa \uf0eb \uf0fb \uf0ee \uf0fe , (8) which must be evaluated as a piecewise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='Case 1 ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='γ < 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='0 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='lim cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf071 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='→\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='+ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf03c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='Case 2 ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1 1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='lim cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf071 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='→\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='+ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='Case 3 ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf03e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='lim cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf071 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='→\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='+ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf03e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='i This case is included for completeness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' even though it was specified that γ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 06 85 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') Minimum Vertex Angle 80 75 70 65 60 2 3 4 5 6 7 6 10 11 12 13 14 15 Pythagorean Exponent9 Case 2, illustrated graphically by 100 n = in Figure 6, indicates that the infinite degree Pythagorean Theorem generates an equilateral triangle, and expectedly, it is the only case that does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For case 3 with an infinite side ratio, ( ) lim lim 1 90o n \uf067 \uf071 \uf067 \uf0a5 →\uf0a5 →\uf0a5 \uf0e9 \uf0f9 \uf03e = \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, the infinite degree Pythagorean Theorem for a triangle with an infinite side ratio (a straight line) requires the same vertex angle as the standard ( ) 2 n = Pythagorean Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' A plot of the behavior of the three cases of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (8) is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 8: Plot of the infinite degree Pythagorean Theorem which was produced with n = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The n-Degree Pythagorean Theorem with Negative Real Exponents Section 2 can be adapted to examine real triangles that conform to the negative exponent n-degree Pythagorean Theorem, and a very different picture emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Once again, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (2) and (3) can be equated to form ( ) 2 2 1 1 1 cos 2 n n \uf067 \uf067 \uf071 \uf067 − − \uf0e9 \uf0f9 + − + \uf0ea \uf0fa = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb , (9) where \uf071 − denotes that the vertex angle arises from a version of the n-degree Pythagorean Theorem in which n − \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Unlike the positive exponents case, there is an infinite set of ( ) ,n \uf067 for which the always positive argument of the inverse cosine function exceeds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If 1 \uf067 = , then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) yields ( ) 2 2 2 1 1 1 1 1 1 1 cos cos 1 2 2 n n \uf071 − \uf0e6 \uf0f6 − \uf0e7 \uf0f7 − − \uf0e8 \uf0f8 \uf0e9 \uf0f9 \uf0e6 \uf0f6 + − + \uf0ea \uf0fa = = − \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0ea \uf0fa \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (10) 10 2 1 1 2 1 n \uf0e6 \uf0f6 − \uf0e7 \uf0f7 \uf0e8 \uf0f8 − \uf03c for all n, and isosceles triangles are permitted for all degrees of the negative real exponent Pythagorean Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, as γ increases, ( ) 2 2 1 1 2 n n \uf067 \uf067 \uf067 + − + (the argument of the inverse cosine function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9)) also increases and can exceed 1, and the vertex angle becomes complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, there exists a γ- dependent critical value of the Pythagorean degree, ( ) crit n \uf067 , which is the largest exponent for a given side ratio that will produce a real vertex angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' This occurs when ( ) crit crit 2 2 1 1 1 2 n n \uf067 \uf067 \uf067 + − + = , which can be simplified as ( ) crit crit 1 1 1 n n \uf067 \uf067 + = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (11) Equation (11) must be solved numerically, and its solution, ( ) crit n \uf067 , is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 9: The Pythagorean critical degree as a function of the side ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, if ( ) crit n n \uf067 \uf03e , then \uf071 is a complex angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If, on the other hand, ( ) crit n n \uf067 \uf0a3 , \uf071 will be real, and a corresponding real triangle exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The upper limit of γ is not immediately apparent, but consider the argument of the inverse cosine function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For \uf071 − to be a real angle, ( ) 2 2 1 1 1 2 n n \uf067 \uf067 \uf067 + − + \uf0a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As such, ( ) ( ) 2 2 1 1 n n \uf067 \uf067 − \uf0a3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, the term ( ) 2 1 1 n n \uf067 + \uf0b3 for all γ and all n including n → −\uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Side Ratio 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 2 0 1 2 Critical Degree 3 4 5 6 7 811 This requires that ( ) 2 min 1 1 \uf067 \uf0e9 \uf0f9 − \uf03c \uf0eb \uf0fb (for n \uf0b9 −\uf0a5), and therefore, 2 \uf067 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Consequently, the n-degree Pythagorean Theorem with negative real exponents cannot be applied to real triangles with side ratios greater than or equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As indicated by Figure 10 and Figure 11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2 \uf067 \uf03c \uf03c is required to produce a real vertex angle for ( ) crit n n \uf067 \uf0a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, by definition 1 \uf067 \uf0b3 , and therefore, 1 2 \uf067 \uf0a3 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Also different from the 1 n = and 2 n = cases is that the 1 n = − and 2 n = − cases produce non- constant but equal γ-dependent angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9): when 1 n = − or 2 n = − , ( ) 2 1 2 1 cos 2 2 1 \uf067 \uf067 \uf071 \uf067 \uf067 − − \uf0e9 \uf0f9 + = − \uf0ea \uf0fa + \uf0ea \uf0fa \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The n ≤ ncrit(γ) Case For ( ) crit n n \uf067 \uf0a3 , plots of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) are shown in Figure 10 and Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Additionally, a maximum value of the vertex angle exists for a given side ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1, that vertex angle and the side ratio that gives rise to it are found by equating the first derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) (instead of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4)) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 10: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for n ≤ ncrit(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The missing section at the top and on the right side of the figure are due to n > ncrit(γ) for a given value of γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' the border of this section has the equation ncrit(γ) as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The side ratio is extended to the excluded regime of γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Vertex Angle (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') 0 60 1 50 Pythagorean Exponent 40 30 20 8 10 9 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 2 Side Ratio12 Figure 11: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The gaps between the plots and the horizontal axis are due to n > ncrit(γ) at those values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The side ratio is extended to the excluded regime of γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As was the case with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (5), the solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (12) for ( ) n \uf067 is not analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, it is clear that for all values of n, 1 \uf067 = is once again a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' By substituting 1 \uf067 = into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9), the vertex angle can be found (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (13)) which expectedly has the same mathematical form as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( )( ) 2 2 0 1 1 1 0 n n n n d d \uf071 \uf067 \uf067 \uf067 \uf067 − − = \uf0de − + − + = (12) 2 1 max cos 1 2 n n \uf071 − − − \uf0e6 \uf0f6 = − \uf0e7 \uf0f7 \uf0e8 \uf0f8 , (13) where max \uf071 − denotes the maximum value of \uf071 − for a given degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The second derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) is the same as the second derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4), and therefore, the mathematical confirmation that max \uf071 − is a maximum angle was also performed numerically with Maple®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' A plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (13) is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 60 n=-1 n=-2 50 n=-3 n = -100 40 30 20 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 Side Ratio13 Figure 12: Maximum vertex angle versus Pythagorean exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In the limit that the Pythagorean exponent becomes infinitely negative, ( ) 2 2 1 1 1 lim lim cos 2 n n n n \uf067 \uf067 \uf071 \uf071 \uf067 − −\uf0a5 →−\uf0a5 →−\uf0a5 \uf0ec \uf0fc \uf0e9 \uf0f9 + − + \uf0ef \uf0ef \uf0ea \uf0fa = = \uf0ed \uf0fd \uf0ea \uf0fa \uf0ef \uf0ef \uf0ea \uf0fa \uf0eb \uf0fb \uf0ee \uf0fe , (14) which is like the case with positive exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Even with the imposed condition that 1 2 \uf067 \uf0a3 \uf03c , eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (14) must be evaluated as a piecewise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Case 1 ( ) = γ 1 ( ) 1 60o \uf071 \uf067 −\uf0a5 = = Case 2 ( ) 1 < γ < 2 ( ) 1 1 2 cos 2 \uf067 \uf071 \uf067 − −\uf0a5 \uf0e6 \uf0f6 \uf03c \uf03c = \uf0e7 \uf0f7 \uf0e8 \uf0f8 Case 3 ( ) − → γ 2 ( ) 2 0 \uf071 \uf067 − −\uf0a5 → = 60 50 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') Maximum Vertex Angle 40 30 20 10 0 15 12 6- 6 3 0 Pythagorean Exponent14 Case 1 produces the same result as did case 2 for positive exponents ( ) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 \uf067 = – a maximum vertex angle of 60o and thus, an equilateral triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Also, like the positive exponents case, the equilateral triangle is never actually realized because although γ can equal 1, n cannot be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As illustrated in Figure 13, the negative infinite degree Pythagorean Theorem can produce real triangles, and the requirement that 1 2 \uf067 \uf0a3 \uf03c is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Furthermore, a straight line will result when ( ) ( ) ( ) crit crit 2 2 1 1 1 lim lim cos 0 2 n n n n n n \uf067 \uf067 \uf067 \uf067 \uf071 \uf067 − → → \uf0ec \uf0fc \uf0e9 \uf0f9 + − + \uf0ef \uf0ef \uf0ea \uf0fa = = \uf0ed \uf0fd \uf0ea \uf0fa \uf0ef \uf0ef \uf0ea \uf0fa \uf0eb \uf0fb \uf0ee \uf0fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 13: Plot of the negative infinite degree Pythagorean Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The Areas of the Associated Triangles The area of the scalene triangle in Figure 1, with a real vertex angle, is 1 sin 2 A ab \uf071 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Its side lengths are a, b a \uf067 = , and from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (1), ( ) 1 1 n n c a \uf067 = + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, its area is 2 1 sin 2 A a \uf067 \uf071 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (15) Applying 2 2 cos 1 sin \uf071 \uf071 = − and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) or eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) depending on whether n + \uf0ce or n − \uf0ce , respectively, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (15) can be written in terms of n and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 15 ( ) 2 2 2 2 2 4 1 1 2 n n a A \uf067 \uf067 \uf067 \uf0e9 \uf0f9 \uf0e6 \uf0f6 = − + − + \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) The restrictions on γ remain: 1 \uf067 \uf0b3 for n + \uf0ce , and 1 2 \uf067 \uf0a3 \uf03c for n − \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, there are no restrictions on a (other than 0 a \uf03e ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The Maximum Areas of Triangles for the n-Degree Pythagorean Theorem with Positive Real Exponents A plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) is shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' It is clear as if a is constant and γ increases, the area of the associated triangle increases without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Also, if a is constant, increasing n increases the area, but this is a negligible effect compared to increasing γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, the effect that n has on the area increases as γ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 14: Triangle area (in dimensionless units) with unit side length (a = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In the left-side figure, the effect of n on the area is difficult to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, in the right-side figure, the effect of n on the area is clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Determining the conditions that maximize the area of a triangle can be done with an “angular” approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (15), the maximum area of a triangle with given values of a and γ clearly occurs when 90 \uf071 = (corresponding to 2 n = ) and is 2 1 2 A a \uf067 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In summary, If γ and n are fixed, increasing a will increase the triangle’s area without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If a and n are fixed, increasing γ will increase the triangle’s area without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, no absolute maximum area triangle exists because the triangle will experience infinite dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The only minimum area is the trivial solution ( ) 0 A = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Most significant from this analysis is that a right triangle, with fixed values of a and γ, has the maximum area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Triangle Area Triangle Area 10 5 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 8 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 Pythagorean Exponent 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 Pythag orean Exponent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 0 2 3 4 5 6 7 8 9 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 Side Ratio Side Ratio16 The above result can, of course, be determined from the dependence of \uf071 + on n and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4), ( ) 2 2 1 1 1 cos 90 2 n n \uf067 \uf067 \uf071 \uf067 + − \uf0e9 \uf0f9 + − + \uf0ea \uf0fa = = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb which gives ( ) 2 2 1 1 n n \uf067 \uf067 + = + (17) because ( ) 2 2 1 1 0 2 n n \uf067 \uf067 \uf067 + − + = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The only real solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (17) for n is 2 n = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' An alternative and more rigorous approach to finding the degree which gives rise to the maximum area case for a given value of γ is from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 0 A n \uf0b6 = \uf0b6 gives ( ) ( ) ( ) 2 2 2 1 1 1 1 ln ln 1 0 1 n n n n n n n n \uf067 \uf067 \uf067 \uf067 \uf067 \uf067 \uf067 \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e6 \uf0f6 + + − + − + = \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0ea \uf0fa + \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (18) But ( ) 2 1 0 n n \uf067 + \uf0b9 for any value of γ or n, and therefore, ( ) ( ) 2 2 1 1 1 ln ln 1 0 1 n n n n n n \uf067 \uf067 \uf067 \uf067 \uf067 \uf067 \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e6 \uf0f6 + − + − + = \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0ea \uf0fa + \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (19) ( ) 2 2 1 1 0 n n \uf067 \uf067 \uf0e9 \uf0f9 + − + = \uf0ea \uf0fa \uf0eb \uf0fb iff 2 n = (as shown with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( ) 1 ln ln 1 0 1 n n n n \uf067 \uf067 \uf067 \uf067 \uf0e9 \uf0f9 \uf0e6 \uf0f6 − + \uf0b9 \uf0ea \uf0fa \uf0e7 \uf0f7 + \uf0e8 \uf0f8 \uf0eb \uf0fb , regardless of the value of γ, although it asymptotically approaches zero for increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As expected, this somewhat more complex approach gives the same result as the “angular approach” used above – the maximum area triangle for positive real exponents occurs when 2 n = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, the n-degree Pythagorean Theorem that produces triangles with the largest area for a given a and γ is the standard Pythagorean Theorem, and the triangles it produces are right angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Even though the triangles that result from 1 2 n \uf0a3 \uf03c have vertex angles which are larger than 90o, they have smaller areas than 2 1 2 A a \uf067 = because sin 1 \uf071 \uf03c for obtuse \uf071 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 17 For a given a and a given n (not necessarily 2 n = ), the value of γ that produces the triangle with the maximum area would be calculated from the derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 0 A \uf067 \uf0e6 \uf0f6 \uf0b6 = \uf0e7 \uf0f7 \uf0b6 \uf0e8 \uf0f8 which yields ( ) ( ) 2 2 1 2 2 1 1 1 1 4 n n n n n \uf067 \uf067 \uf067 \uf067 \uf0e6 \uf0f6 − \uf0e7 \uf0f7 − \uf0e8 \uf0f8 \uf0e9 \uf0f9 \uf0e9 \uf0f9 + − + − + = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (20) However, equation (20) has no solution for 0 n \uf03e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' This is physically reasonable because increasing γ for a given value of a continuously dilates the triangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' therefore, there is no finite maximum area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The smallest of all of the areas of these continuously enlarging triangles (with a fixed side length a) occurs when 1 \uf067 = and is 3 2 1 2 2 min 2 1 4 n n n n A a − − \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e9 \uf0f9 = − \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (21) Equation (21) (a plot of which is shown in Figure 15) represents the areas of the members of an infinite set of isosceles triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Each of these triangles has the minimum possible area among all of the various triangles of the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The triangle in this infinite set with the largest area occurs from the infinite degree case, is denoted by max min A \uf0a5 , and is given by the limit of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' max 2 min min lim 6 n A A a \uf0a5 →\uf0a5 = = , (22) Figure 15: The minimum area (in dimensionless units) of an isosceles triangle with unit side length (a = 1) for the n-degree Pythagorean Theorem with positive real exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The red line indicates max min A \uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The Area of a Triangle with a Fixed Perimeter (n > 0) To further examine these ideas, it is sensible to consider the area of a triangle with a fixed perimeter P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Then, determine the values of γ and n that produce the maximum area for the n + \uf0ce case (and then in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2, for the n − \uf0ce case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The triangle perimeter is trivially ( ) 1 1 1 n n P a b c a \uf067 \uf067 \uf0e6 \uf0f6 = + + = + + + \uf0e7 \uf0f7 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (23) The area, in terms of the fixed perimeter, is denoted P A , and it results from combining eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( ) ( ) 2 2 2 2 2 1 4 1 1 2 1 1 P n n n n P A \uf067 \uf067 \uf067 \uf067 \uf067 \uf0ec \uf0fc \uf0ef \uf0ef \uf0e9 \uf0f9 \uf0ef \uf0ef = − + − + \uf0ed \uf0fd \uf0ea \uf0fa \uf0e9 \uf0f9 \uf0eb \uf0fb \uf0ef \uf0ef + + + \uf0ea \uf0fa \uf0ef \uf0ef \uf0eb \uf0fb \uf0ee \uf0fe (24) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24): for a fixed P, and as a function of n, the area of the triangle is a maximum when 1 \uf067 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' This area is denoted by max P A and is given by 1 1 2 max 2 1 4 16 16 1 2 n n n P n n P A + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 − \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 − = \uf0e7 \uf0f7 \uf0e7 \uf0e6 \uf0f6 \uf0f7 + \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 , (25) Equation (25) represents the areas of an infinite set of maximum area isosceles triangles with a fixed perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The triangle in that set with the largest area is the infinite degree member which is denoted by max P A \uf0a5 , and its area is given by 1 1 2 2 max 2 1 4 16 3 lim 16 36 1 2 n n n P n n n P A P \uf0a5 + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 →\uf0a5 − \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e6 \uf0f6 − = = \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0e6 \uf0f6 \uf0f7 \uf0e8 \uf0f8 + \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (26) 19 From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24), as \uf067 → \uf0a5 , the areas of the triangles approach zero (as shown in Figure 17), as they take the form of a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Also, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24) (and as shown in Figure 16), triangles with a fixed P will see their areas (as a function of γ) asymptotically increase as n → \uf0a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Those areas are denoted P A\uf0a5 and are given by ( ) 2 2 2 4 1 1 4 2 1 P A P \uf067 \uf067 \uf0a5 \uf0e9 \uf0f9 − = \uf0ea \uf0fa + \uf0ea \uf0fa \uf0eb \uf0fb , (27) Figure 16: Area (in dimensionless units) of Pythagorean triangles with unit perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 17: Area (in dimensionless units) of Pythagorean triangles with unit perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 The Maximum Areas of Triangles for the n-Degree Pythagorean Theorem with Negative Real Exponents For n − \uf0ce , a plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) is shown in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The area relationship for positive real exponents also applies to the case of negative real exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, for all ( ) crit n n \uf067 \uf0a3 , the vertex angles are real and acute, and if a and γ are fixed, the maximum area triangle has the largest vertex angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 18: Triangle area (in dimensionless units) with unit side length (a = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The missing section at the top right and on the right side of figure are due to n > ncrit(γ) for a given value of γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' the border of this section has the equation ncrit(γ) as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' To determine the triangle with the maximum area, consider eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (15) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' \uf05b \uf05d ( ) \uf05b \uf05d ( ) ( ) ( ) 1 2 2 2 1 2 2 2 2 2 2 max 1 1 1 1 1 max sin 1 min cos 1 min 2 2 2 2 n n A a a a \uf067 \uf067 \uf067 \uf071 \uf067 \uf071 \uf067 \uf067 \uf0ec \uf0fc \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0ef \uf0ef + − + \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0ef \uf0ef = = − = − \uf0ed \uf0fd \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0ef \uf0ef \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0ef \uf0ef \uf0ee \uf0fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (28) In the special case of an isosceles triangle ( ) 1 \uf067 = , ( ) 2 2 2 1 1 min 1 2 2 n n n n \uf067 \uf067 \uf067 − \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e6 \uf0f6 + − + \uf0e7 \uf0f7 = − \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 for 1 n \uf0a3 − because ( ) crit 1 n \uf067 = does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Triangle Area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='45 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 Pythagorean Exponent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='35 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='15 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='05 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 2 Side Ratio21 The maximum triangle area is then 1 1 1 2 1 2 2 max 1 sin 2 1 4 2 n n n n A a a \uf067 \uf067 \uf071 − − \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 = \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e6 \uf0f6\uf0e6 \uf0f6 = = − \uf0e7 \uf0f7\uf0e7 \uf0f7 \uf0e7 \uf0f7\uf0e7 \uf0f7 \uf0e8 \uf0f8\uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (29) If the negative infinite degree case of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (28) is considered, the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( ) 2 2 1 1 lim min 2 2 n n n \uf067 \uf067 \uf067 \uf067 →−\uf0a5 \uf0ec \uf0fc \uf0e6 \uf0f6 + − + \uf0ef \uf0ef \uf0e7 \uf0f7 = \uf0ed \uf0fd \uf0e7 \uf0f7 \uf0ef \uf0ef \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0ee \uf0fe for any side ratio that conforms to 1 2 \uf067 \uf0a3 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For this negative infinite degree case, the maximum triangle area is 2 2 2 max 1 1 sin 4 2 4 A a a \uf067 \uf071 \uf067 \uf067 −\uf0a5 = = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (30) Equation (30) necessarily complies with the requirement that 1 2 \uf067 \uf0a3 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Additionally, three interesting results emerge about this infinite set of negative infinite degree triangles: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In this set, there are two values of γ that produce triangles with equal areas: 1 \uf067 = and 3 \uf067 = , and that maximum area is 1, 3 2 2 max 1 3 sin 2 4 A a a \uf067 \uf067 \uf067 \uf071 −\uf0a5 = = = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 \uf067 = necessarily gives zero area because the triangle has collapsed into a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In this infinite set, the triangle with the largest area is the member for which 2 \uf067 = , and its area is max 2 2 max 1 1 sin 2 2 A a a \uf067 \uf071 −\uf0a5 = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The Area of a Triangle with a Fixed Perimeter (n < 0) If, on the other hand, the perimeter is fixed, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24) applies for 1 2 \uf067 \uf0a3 \uf03c , and the largest area triangle will be isosceles, and it will have an area of 2 3 36 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' This area is expected because the conditions that 1 \uf067 = and n → −\uf0a5 result in equilateral triangle in which 3 P a = , and therefore, 2 2 3 3 36 4 P a = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24), if lim P n A →−\uf0a5 is taken, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (31) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As seen in Figure 19 (a plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (31)), the area will asymptotically increase as n → −\uf0a5 and will asymptotically decrease toward zero as 2 \uf067 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (31) determines the area for a fixed perimeter, arbitrary side ratio, and infinite degree triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 22 ( ) 2 2 2 1 4 4 2 P A P \uf067 \uf067 \uf067 −\uf0a5 \uf0e9 \uf0f9 = − \uf0ea \uf0fa + \uf0ea \uf0fa \uf0eb \uf0fb , (31) where P A−\uf0a5 is the asymptotic area of the triangles with a fixed perimeter and for which n → −\uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Among the infinite set of triangles whose areas are given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (31), the triangle with the largest area is isosceles (as stated above), and its area is 2 3 36 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 19: Area of Pythagorean triangles with unit perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 20: Area of Pythagorean triangles with unit perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Summary The Pythagorean Theorem has been extended to positive and negative real exponents unfettered by the physical requirement of dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The relationship between the ratio of the adjacent sides and the vertex angle was determined for a given degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' It was found that for positive exponents, the stipulation that 1 \uf067 \uf0b3 can be applied to all degrees, and no complex vertex angles arose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For 1 2 n \uf03c \uf03c , an obtuse triangle results, and if 2 n \uf03e , the triangle is acute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, for negative exponents to produce real vertex angles, the restriction 1 2 \uf067 \uf0a3 \uf03c is necessary but not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The additional requirement that if 1 2 \uf067 \uf03c \uf03c , then ( ) crit n n \uf067 \uf0a3 needs to be imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The areas of the associated triangles for positive and negative real exponents were explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' With fixed a and γ values, the areas for | 1 n n + \uf0ce \uf03e are maximized when the triangle is right isosceles requiring 1 \uf067 = and 2 n = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Additionally, triangle areas increase as n → \uf0a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Alternatively, if the perimeter of a triangle is kept constant, the triangle area approaches a maximum value with increasing n and approaches 0 for decreasing γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For the n − \uf0ce case, as n → −\uf0a5 , the triangle with a fixed perimeter and the maximum area has a side ratio of 2 \uf067 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In contrast, if the degree is finite and ( ) crit n n \uf067 \uf0a3 (if 1 \uf067 \uf0b9 ), the maximum area occurs when the vertex angle ( ) ,n \uf071 \uf067 is a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 24 Works Cited 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Faltings, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', The proof of Fermat’s last theorem by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Taylor and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Wiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Notices of the AMS, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 42(7): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 743-746.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 259-272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Yeng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Lin, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Lin, The n-dimensional Pythagorean theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Linear and multilinear algebra, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 26(1-2): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 9-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} diff --git a/19FST4oBgHgl3EQfXDjl/vector_store/index.faiss b/19FST4oBgHgl3EQfXDjl/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4ed5accc5bd4be1a3f9c5f20cff2177aaa5fba1f --- /dev/null +++ b/19FST4oBgHgl3EQfXDjl/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Email: morlokan@uni-muenster.de, Tel. +49-251-83-39069 +Aleksandra Stojic, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: +a.stojic@uni-muenster.de; +Isabelle Dittmar, Hochschule Emden/Leer, Constantiaplatz 4, 26723 Emden, Germany, Email: +isabelle.dittmar@hs-emden-leer.de +Harald Hiesinger, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: +hiesinger@uni-muenster.de +Manuel Ahmedi, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: +ahmedi@uni-muenster.de +Martin Sohn, Hochschule Emden/Leer, Constantiaplatz 4, 26723 Emden, Germany, Email: +martin.sohn@hs-emden-leer.de +Iris Weber, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany. Email: +sonderm@uni-muenster.de +Joern Helbert, Institute for Planetary Research, DLR, Rutherfordstrasse 2, 12489 Berlin, Germany, Email: +joern.helbert@dlr.de +© 2016 This manuscript version is made available under the CC-BY-NC-ND 4.0 + + + + +2 + +ABSTRACT +This study is part of an effort to build a mid-infrared database (7-14µm) of spectra for MERTIS (Mercury +Radiometer and Thermal Infrared Spectrometer), an instrument onboard of the ESA/JAXA BepiColombo +space probe to be launched to Mercury in 2017. +Mercury was exposed to abundant impacts throughout its history. This study of terrestrial impactites can +provide estimates of the effects of shock metamorphism on the mid-infrared spectral properties of +planetary materials. +In this study, we focus on the Nördlinger Ries crater in Southern Germany, a well preserved and easily +accessible impact crater with abundant suevite impactites. Suevite and melt glass bulk samples from +Otting and Aumühle, as well as red suevite from Polsingen were characterized and their reflectance +spectra in mid-infrared range obtained. In addition, in-situ mid-infrared spectra were made from glasses +and matrix areas in thin sections. The results show similar, but distinguishable spectra for both bulk +suevite and melt glass samples, as well as in-situ measurements. +Impact melt glass from Aumühle and Otting have spectra dominated by a Reststrahlen band at 9.3-9.6 +µm. Bulk melt rock from Polsingen and bulk suevite and fine-grained matrix have their strongest band +between 9.4 to 9.6 µm. There are also features between 8.5 and 9 µm, and 12.5 - 12.8 µm associated +with crystalline phases. There is evidence of weathering products in the fine-grained matrix, such as +smectites. Mercury endured many impacts with impactors of all sizes over its history. So spectral +characteristics observed for impactites formed only in a single impact like in the Ries impact event can be +expected to be very common on planetary bodies exposed to many more impacts in their past. We +conclude that in mid-infrared remote sensing data the surface of Mercury can be expected to be +dominated by features of amorphous materials. + + + + +3 + +Keywords: Spectroscopy, Impact processes, Infrared observations, Instrumentation + + + + + + + + + + + + + + + + + + + + +4 + +1. Introduction +The aim of this study is to provide mid-infrared reflectance spectra with special emphasis on the region +from 7µm – 14µm for a range of impactites for the application in planetary remote sensing. We generate +these spectra for a database for the ESA/JAXA BepiColombo mission to Mercury (Benkhoff et al., 2010). +Onboard is a mid-infrared spectrometer (MERTIS-Mercury Radiometer and Thermal Infrared +Spectrometer). This unique device allows mapping spectral features in the 7-14 µm range, with a spatial +resolution of ~500 m (Helbert et al., 2009; Benkhoff et al., 2010; Hiesinger et al., 2010). +Infrared spectroscopy provides a means to characterize the mineralogy of rocks via remote +sensing, in contrast to Gamma-Ray, Neutron and X-ray spectrometers, which determine elemental +compositions (Pieters and Englert, 1993). Thus, IR spectroscopy provides a central analytical tool to +determine the mineralogy of remote planetary surfaces. In order to correctly interpret the remote +sensing data, laboratory spectra of natural, synthetic rocks and minerals have to be collected to compare +them to the spectra that will be obtained from MERTIS, once BepiColombo enters the Hermean orbit +(Maturilli et al., 2008; Hiesinger et al., 2010). Comparable earlier instruments were the Thermal Emission +Spectrometer (TES) on the Mars Global Surveyor (Christensen et al., 2005) and the Thermal Emission +Imaging System (THEMIS) on the Mars Odyssey orbiter (Christensen et al., 2004). The lunar surface was +mapped in the mid-infrared with the DIVINER Lunar Radiometer Instrument on the Lunar +Reconnaissance Orbiter (Paige et al., 2010). The OSIRIS-REx Thermal Emission Spectrometer (OTES) will +map asteroid Bennu (1999 RQ36) (Hamilton and Christensen, 2014), and the Thermal Infrared Imager +(TIR) onboard Hayabusa 2 will map asteroid 1999JU3 (Okada et al., 2015). +The Ries crater in southern Germany (Fig.1) provides well preserved layers of impactites and is +one of the best studied impact sites in the world (von Engelhardt, 1995). The 14.6 Myr old crater with a +diameter of 24 kilometers (Buchner et al., 2010) offers the whole range of impact-associated rocks and + + + +5 + +minerals (von Engelhardt, 1990). Large impact events play a key role in surface modification processes in +effect on most terrestrial planets and their moons (Hörz and Cintala, 1997). Thus, the investigation of +terrestrial impact processed rocks and understanding how these processes affect the spectral properties +of the resulting impact generated rocks and melt glass is important for the interpretation of infrared data +from the surfaces of other planetary bodies. Given the characteristics of surface regolith, we need +spectral data from different grain size fractions. This is necessary because grain size variations affect the +corresponding spectra immensely by reducing the spectral contrast of features and creating additional +features at small grain sizes (e.g., Salisbury and Eastes, 1985; Salisbury and Wald, 1992; Mustard and +Hayes, 1997; Ruff and Christensen, 2002). +Reflectance and emission studies about spectral features in the mid-infrared in minerals +undergoing high shock metamorphism including formation of melt glass were made on experimentally +shocked samples, e.g., on anorthosite, pyroxenite, basalt, and feldspar (Johnson et al. , 2002, 2003, 2007, +2012; and Jaret et al. ,2015). In these studies, especially feldspar-rich material showed loss and +degradation of features, as well as band shifts with increasing structural disorder resulting from +increasing shock pressure. Moroz et al. (2010) analyzed impact glasses from laser pulse experiments with +Martian soil analogue JSC Mars-1. Similar, Basilevsky et al. (2000) and Morris et al. (2000) spectrally +studied melt glass also made from Martian soil analogs. Byrnes et al. (2007), and Lee et al.,(2010) +measured synthetic quartzofeldspathic glasses and Dufresne et al. (2009), Minitti et al. (2002), Minitti +and Hamilton (2010) measured synthetic glass with basaltic to intermediate composition. In general, the +resulting mid-infrared reflectance or emission spectra display a dominant feature in the 9.2-10.5 µm +range. Pollack et al. (1973), Crisp et al., 1990; Nash and Salisbury (1991) and Wyatt et al. (2001), studied +obsidian or basalt. Further infrared reflectance and emission spectra of natural impact glass and tektites +were made, e.g., by Thomson and Schultz (2002), Gucsik et al. (2004), Faulques et al. (2005) and Palomba +et al. (2006). Wright et al. (2011), Basavaiah and Chavan (2013), and Jaret et al. (2013) investigated + + + +6 + +shocked bulk material from the Lonar Crater, India, showing mainly a simple spectrum with a dominating +feature in the ~9.4 - 10.2 µm range. +Complementary infrared transmission and absorbance spectra of related materials were also +made, e.g, of experimentally shocked feldspar materials (Stöffler and Hornemann, 1972; Ostertag, 1983) +and of CM2 chondrite Murchison (Morlok et al., 2010). Glasses with feldspathic composition produced in +static high pressure experiments were measured by, e.g., Iiishi et al. (1971), Velde et al. (1987), and +Williams and Jeanloz (1988, 1989). In addition, transmission infrared analyses of natural impact glass and +tektites for studying water contents were made by Beran and Koeberl (1997), for identification purposes +(Fröhlich et al., 2013) or for astrophysical studies of circumstellar dust (Morlok et al., 2014). King et al. +(2004) provided an overview of silicate glass analysis with various mid-infrared techniques. +Furthermore, there are also numerous studies how impact shock affects the spectral properties +in the visible and near-infrared, such as Johnson and Hörz (2003), Adams et al. (1979), and Bruckenthal +and Pieters (1984) for experimentally shocked feldspars or enstatite. Moroz et al. (2009) studied +synthetic glasses with Martian soil composition as impact melt analogs, and Cannon and Mustard (2015) +identified glass-rich impactites on Mars. Bell et al. (1976) and Stockstill-Cahill et al. (2014) analyzed lunar +glass analogues, while Keppler (1992) studied synthetic silicate glasses with albite and diopside +composition. Schulz and Mustard (2004) studied terrestrial impact melt rocks. +Shock metamorphism begins with fracturing and brecciation of the rocks at lower pressures; +above 2 GPa the mineral phases start to change. Shatter cones and conical fracturing patterns form. +Between 8 and 25 GPa, planar deformation features appear along with microscopic changes in the +crystal structure due to impact shock in, e.g., quartz and feldspar. At pressures >25 GPa, shock +metamorphism continues with the solid-state transformation of minerals into diaplectic glasses, which +are amorphous materials generated at pressures up to 40 GPa. At pressures over 35 GPa, partial melting +of phases begins and at over 60 GPa rocks completely melt, followed by vaporization at pressures above +100 GPa (e.g., Stöffler, 1966, 1971, 1984; Chao, 1967; von Engelhardt and Stöffler, 1968; Stöffler and + + + +7 + +Langenhorst, 1994; French, 1998). Also, high pressure mineral polymorphs form; stishovite and coesite +form from quartz at pressures from 12-15 GPa and over 30 GPa, respectively. Carbon is transformed into +diamond at 13 GPa (Stöffler and Langenhorst, 1994; French, 1998). +In the impactite suevite, shock metamorphism can be divided into stages 0 – IV. Stage 0 ranges +from 0-10 GPa, stage I (10-35 GPa) and II (35-60 GPa) cover changes in crystal structure and partial +melting; stage IV pressures over 60 GPa result in melt glasses (Stöffler, 1971 and 1984; French, 1998; +Stöffler and Grieve, 2007; Stöffler et al., 2013). Suevite consists of materials of all five shock stages +(Stöffler et al., 2013). Thus the infrared spectra obtained from the suevites can be expected to contain a +mixture of spectral features depending on the level of shock metamorphism the material underwent. +In this study, we focus on impactites from the Ries crater, since the crater site allows easy access +to a range of naturally shocked rocks and minerals (von Engelhardt, 1990, 1995). Prior to the impact +event, the Ries area was covered with sediments mainly consisting of limestones. The underlying +crystalline basement is dominated by granites and gneisses, with significant amounts of amphibolite (von +Engelhardt and Graup, 1984; von Engelhardt, 1997). The average chemical composition of very +homogeneous impact melts (shock stage IV; Stöffler et al, 2013) is similar to the composition of the +modelled crystalline basement rock clasts based on the fallout suevite, identifying these rocks as the +main source of the homogenized melt (Staehle, 1972; von Engelhardt et al., 1984 , 1995, 1997; +Vennemann et al., 2001). +We focus on the suevite, the top layer of ejecta in the Ries that was deposited on all other +impact ejecta, Bunte Breccia (e.g. Abadian, 1972; Hörz et al., 1983), megablocks (e.g. Sturm et al., 2015) +and polymict crystalline breccia (e.g. von Engelhardt, 1997). The most voluminous ejecta is the Bunte +Breccia, consisting mainly of sedimentary material. Megablocks are rocks in the size range from 25 m to +kilometers, consisting of sedimentary (limestone) and crystalline materials. Both Bunte Breccia and + + + +8 + +megablocks show low degrees of shock metamorphism. Polymict crystalline breccias are mixtures of +basement rocks with shock stages up to stage II. Suevite is the only ejecta layer to contain material in all +stages of shock metamorphism (e.g., von Engelhardt, 1969, 1990, 1997). Moldavites, a type of tektites +formed in the Ries impact are an additional type of ejecta that was deposited 350 km east of the impact +site (e.g., von Engelhardt, 1987). +The mostly granitic and felsic petrology of the crystalline basement of the Ries, which controlled +the composition of the suevite (von Engelhardt et al., 1990, 1997), is quite different compared to that of +Mercury and other planetary surfaces in the Solar System. Also, granite in general is very rare outside +Earth (Bonin, 2012). Based on MESSENGER data, the best terrestrial analogues for the surface of Mercury +are basalts or ultramafic komatiites (Nittler et al., 2011; Stockstill-Cahill et al., 2012; Charlier et al., 2013; +Maturilli et al., 2014), but natural shocked forms are rare on Earth (e.g. Wright et al., 2011). However, a +spectral study of naturally shocked granitic material is of interest for our remote sensing purposes +because it gives insight into the spectral characteristics of impactites and their components, while +naturally shocked basalts are rare (Wright et al., 2011). Furthermore, the easy access to the Ries site +gives access to larger, representative amounts of sample material. Although rare, granitic materials have +been recognized on other planetary bodies. Granitic material has been found as fragments and clasts in +lunar samples (e.g., Warren et al., 1983; Jolliff et al., 1999; Shervais and McGee, 1999; Seddio et al., +2015). Granitoid or felsic materials may also occur on the mostly basaltic surface of Mars (e.g., +Christensen et al., 2005; Bandfield, 2006, Ehlmann and Edwards, 2014; Sautter et al., 2014, 2015). There +are also indications for felsic, granitoid material on Venus (Müller et al., 2008; Gilmore, 2015). +Suevites are divided into crater or fallback suevite, occurring inside the inner ring of the Ries, and +outer suevite, encompassing suevite found outside of the inner part of the crater (Pohl et al., 1977, +Stöffler, 1977, 2013). Normal suevite in this study is outer suevite that originates from the Otting and +Aumühle quarries (Fig.1, 2a,b). It mainly consists of three components: a porous matrix of fine-grained + + + +9 + +rock, melt glass, and crystalline basement rocks displaying all stages of shock metamorphism ( Stöffler, +1966; von Engelhardt and Graup, 1984; Stöffler et al., 2013). Red suevite (Fig. 2c) from the Polsingen +quarry is a rare variation of the outer suevite. Its red color is a characteristic owing to high hematite +content (Stöffler et al., 2013). In the red suevite, the actual groundmass consists of impact melt, in which +fragments are embedded (Reimold et al., 2010). Also, chemical differences between normal suevite and +red suevite (Na2O, K2O) were reported by Reimold et al. (2013) and Stöffler (2013). In addition to the +bulk materials, we are also interested in the components of the suevite. In particular, we studied the +melt glasses (Fig.2a, b), i.e., completely shock-melted and quenched rock material. Furthermore, we +looked at the fine-grained matrix itself, which was affected by weathering processes (von Engelhardt, +1995; Stöffler et al., 2013). + +2. Samples and Techniques +2.1 Sample Selection +We selected three bulk suevite samples from three different Ries localities of the suevite layer +(von Engelhardt and Stöffler, 1974; von Engelhardt et al., 1995, 1997; Bayerisches Geologisches +Landesamt, 2004): normal crater suevite from the Otting (samples Otting Bulk 1-3) and Aumühle +(Aumühle Bulk4, Bulk 18, Bulk 18 Matrix) quarries, and red suevite (samples Polsingen 2-4) from a small +outcrop in Polsingen (outer suevite) (von Engelhardt, 1997; Reimold et al., 2010). In addition, we used +material from three separated impact melt glasses: from a large, 30 cm sized glass ‘bomb’ (Otting +Glasbombe) and a separated glass ‘Flädle’ from Otting (Otting Glas 1), and another glass separated from +the Aumühle suevite (Aumühle 13). +In addition, for quantitative chemical and additional IR-microscopy analyses, polished thin +sections of Aumühle, Otting and Polsingen were produced from the bulk sample rocks. The vitreous + + + +10 + +state of the glass was confirmed using polarized light microscopy by the total extinction of the glass +under crossed polarizers (Fig.2a-c). However, the two Otting samples show brownish halos or ‘Schlieren’, +signs of incipient devitrification by the formation of small crystallites of pyroxenes, feldspar, oxides and +possibly incipient signs of alteration phases like clays (see discussion)(von Engelhardt et al., 1995). + +2.2 Sample Preparation +We used aliquots of larger amounts of material to avoid bias due to larger +fragments/components thus ensuring a homogeneous and representative sample. The original sample +masses were greater than 100 grams for each sample. Bulk samples were first powdered in steel and +agate mortars. Subsequently, the powders were cleaned in acetone and dry sieved into four size +fractions: 0-25 µm, 25-63 µm, 63-125 µm and 125-250 µm. This was done using an automatic Retsch Tap +Sieve; each size fraction was dry sieved for at least one hour. In order to remove clinging fines, the larger +two fractions were cleaned with acetone. +For additional in situ measurements involving optical microscopy, micro-FTIR, and Scanning +Electron Microscopy (SEM) measurements, we used thin sections of representative blocks of the +samples. The thin sections were polished to ~30 µm using standard procedures for petrological thin +sections, ensuring a high specular reflectance from a flat surface from which especially microscopic IR +investigations benefit greatly. + +2.3 Optical Microscopy +Overview images of the samples in normal light and between crossed polarizers were obtained with the +KEYENCE Digital Microscope VHX-500F. Light microscopy allows rapid assessment of the general + + + +11 + +homogeneity, as well as first mineral identification in the samples. Images were made using a +magnification of 10. + +2.4 Infrared Analyses +2.4.1 Diffuse Reflectance Powder Analyses +To ensure diffuse reflectance, the size fractions were gently placed in aluminum sample cups (1 +cm diameter), and the surface flattened with a spatula following a similar procedure described by +Mustard and Hayes (1997). For mid-infrared analyses from 2-20 µm, we used a Bruker Vertex 70 infrared +system with a MCT detector at the Infrared and Raman for Interplanetary Spectroscopy (IRIS) laboratory +at the Institut für Planetologie in Münster. To avoid sample surface disturbance due to pore collapse +during evacuation, we avoided analyses under near vacuum and measured under normal atmosphere. As +a consequence, water and atmosphere related features slightly affect the spectral range of interest near +7 µm. To ensure a high signal-to-noise ratio, we accumulated 512 scans for each size fraction. For +background calibration a diffuse gold standard was applied. For the MERTIS database, we obtained +analyses in a variable geometry stage (Bruker A513) in order to emulate various observational +geometries of the orbiter. The data presented here were obtained at 30° incidence (i) and 30° +emergence angle (e). +The spectra of this study are intended to be compared with remote sensing data in the thermal +infrared. Emission and reflectance spectra can be compared using Kirchhoff’s law: ε = 1 – R +(R=Reflectance, ε = Emission) (Nicodemus, 1965). This relation works very well for the comparison of +directional emissivity and directional hemispherical reflectance (Hapke, 1993, Salisbury et al., 1994). + + + +12 + +However, in order to directly relate directional emissivity with reflectance by using Kirchhoff’s law, the +reflected light in all directions has to be collected (Thomson and Salisbury, 1993). A bi-directional, +variable mirror set-up was used for this study without a hemisphere integrating all radiation. This has to +be kept in mind when comparing the results in a quantitative manner with emission data (Salisbury et al., +1991; Christensen et al., 2000). In the case of the spectra obtained with the micro-FTIR in specular +reflectance mode, the diffusely scattered part of the light is very small, since highly polished glass was +analyzed. Although specular reflectance is not entirely relatable to the directed emission, as required by +Kirchhoff’s law, the differences will be mainly in spectral contrast, otherwise (e.g. band shape and +positions) the data will be comparable (Ramsey and Fink, 1999; Byrnes et al, 2007; Lee et al., 2010). +Although the spectral range of interest for the database is from 7-14 µm, we measured our +powder from 7-20 µm (Fig. 4a-c), since features of interest can appear at longer wavelengths. The signal +of the detector used becomes weak at wavelengths above 18 µm, resulting in a low signal to noise ratio. +Above 19 µm, the spectral features are mainly random noise. The analyses were conducted under +ambient pressure, which possibly affected the water bands at ~3 and ~6 µm. We therefore only present +representative features from 2-7 µm for each sample in Fig.4d. The impactites analyzed in this study are +often mixtures of various mineral phases. So in order to identify specific mineral bands in our laboratory +spectra, we used spectra from the Arizona State University Thermal Emission laboratory (Christensen et +al., 2000) and the Johns Hopkins ASTER laboratory (Baldridge et al. 2009). + +2.4.2 In-situ FTIR Microspectroscopy +For in situ analyses, we used a Bruker Hyperion 2000 IR microscope attached to the external port +of a Bruker Vertex 70v at the Hochschule Emden/Leer. Here a 1000×1000 µm2 sized aperture was +applied to obtain analyses of interesting features with in situ reflectance spectroscopy on polished thin + + + +13 + +sections. For each spectrum, 128 scans were added. A gold mirror was used for background calibration. +The analyses were made in the range from 2-15 µm. Since all features below 7 µm are very weak, we +present results in the range of interest, 7-14 µm (Fig.5). +The small shift between powder and microscope analyses observed (see 3. Results) is probably +due to the inherent differences among the samples (polished thin sections compared to powders) and +the different optical set-ups of the techniques. Differences in the positions between the CF in powdered +and solid samples were also observed by Cooper et al. (2002). + +2.5 SEM EDX Analyses +In order to document the suevites and their components, we used a JEOL 6610-LV Scanning +electron microscope equipped with a silicon drift Oxford EDX (Energy Dispersive X-Ray Spectroscopy) +system to obtain micrographs of particular areas of interest and perform quantitative chemical analyses. +Each chemical analysis was quantified with an ASTIMEX™ standard set for major elements prior to the +measurement. Beam current stability was controlled and measured before each analysis using a Faraday +cup. The calibration was confirmed by re-analyzing the standards after the calibration procedure. For +analyses of the chemical composition of melt glass and fine-grained matrix areas, we analyzed areas of +100 x 100 µm2 using 90 seconds integration times. The rather small area was chosen to maintain +comparability between amorphous parts and fine-grained matrix areas. Otherwise it would be difficult to +obtain representative measurements of larger areas due to abundant cracks, gaps, holes, and veins, +which are often re-filled by secondary alteration phases. A broad beam and shorter integration times are +helpful to measure volatile elements correctly. + + + + +14 + +3. Results +3.1 Optical Images +Optical images (Fig2a-c) give an overview of the thin sections for the three investigated samples. +The Otting (Fig.2a) and Aumühle (Fig.2b) samples in the transmitted light show the components of the +normal suevite, the darker melt glasses, embedded in a brighter matrix, which consists of fine fragments +of glass, rock and also secondary alteration products. Under crossed polarizers, the glasses are +recognizable by their nearly black appearance due to complete extinction, while the fine mixture of the +matrix appears brighter. The Polsingen sample (Fig.2c), being a coherent melt rock, consists entirely of a +red groundmass with abundant larger and smaller fragments in transmitted light. Brownish rims around +minerals and holes indicate significant weathering. In the image under crossed polarizers, the +groundmass appears dark, a few crystalline fragments are visible by their brighter appearance. + +3.2. SEM/EDX Analyses +Melt glass and fine-grained matrix area measurements in the suevites are shown in Table 1. Due +to porosity and volatile contents (water) the total analysis of matrix material in weight (wt)% is lower +than 100 wt%. The melt glasses also show lower totals, which is due to crystallized or oxidized inclusions +as well as inevitable cracks and veins. +For a better comparison with earlier data, we plot characteristic oxide ratios for MgO, FeO, CaO, +Na2O, K2O of our SEM/EDX results with those from the literature (Fig. 3) (Stöffler et al., 2013). Results for +melt glass in Aumühle and Otting are very similar to each other and to earlier studies of glass from outer +suevite. The results for the fine-grained matrix analyses cluster in an area overlapping with the +composition of outer suevite (Fig. 3) (Stöffler et al., 2013). The analyses plotting outside the area are + + + +15 + +explained by the lack of larger rock fragments within the analyzed areas. This resulted in a higher +content of fine grained secondary phases typical for the matrix, such as clay minerals, which is also +indicated by elevated Al2O3 contents (Tab. 1) (Stöffler et al., 2013). Otherwise, the compositions of the +fine-grained matrices can be explained as a mixture of the endmember glass and various basement rock +fragments. Melt rock from the red suevite in Polsingen plots slightly below the studies for bulk red +suevite (Stöffler et al., 2013). This is probably due to the high degree of weathering of the samples from +the only accessible outcrops, resulting in increased alkali (Na2O, K2O, CaO) concentrations (Reimold et al., +2013; Stöffler, 2013). + +3.3 Mid-Infrared Analyses of Powders +The powder spectra of the bulk normal suevite from Otting and Aumühle (Fig.4a, Tab. 2a) show +very similar band shapes and intensities for all samples, reflecting the high chemical homogeneity +already indicated by SEM/EDX. The Christiansen Features (CF) of the bulk Otting and Aumühle bulk +suevites are between 7.4 and 8.1 µm, respectively, for all size fractions within one sample group. In the +finest fraction from 0-25 µm, the CF is usually at longer wavelengths, from 7.9-8.1 µm indicating a +mineralogical heterogeneity among the finer fraction. Clays are extremely fine-grained and likely remain +in the smallest grain size fraction, which could account for the slight shift in the CF (see discussion). The +Transparency features (TF) appear at 10.4 - 11.7 µm usually in the smallest size fractions of the bulk +suevites. The strongest feature in this region is between 11.6 and 11.7µm. The strong Reststrahlen band +(RB) at 9.4 µm again indicates a highly amorphous and homogeneous internal structure of the minerals +comprising the sample. Minor bands or shoulders are found on the slope of the RB at 8.5-8.6 µm and +8.8-8.9 µm, indicating crystalline species (see Discussion). Further RB or potential TF are observed in the +bulk suevites at 10.4-10.5 µm and 11.1-11.2 µm (overlapping with the transparency feature in the + + + +16 + +smallest size fractions) and 12.5-12.9 µm. Broad features in the 18-19 µm region are also typical. All bulk +suevite spectra show very strong water features at 2.8 µm and 6.1 µm (Fig. 4d). These volatile bands +probably result from clay minerals, but analysis under atmospheric conditions could also have affected +the spectra. A spectrum of separated fine-grained matrix from Aumühle 18 (Fig. 4a, Tab. 2a) is very +similar in band shape and peak positions to the bulk spectrum Aumühle 18, indicating a high abundance +of melt and crystalline clasts in the fine-grained matrix. +Melt glasses from Aumühle and Otting (Fig. 4b; Tab. 2b) are also very homogeneous. The +Christiansen Features (CF) are at 7.6-7.9 µm, the Transparency Features (TF) at 11.7-11.8 µm in the finest +size fraction. The strongest Reststrahlen band is at 9.3-9.4 µm, with a weak shoulder in the finest size +fraction at 8.6 µm. This all confirms the amorphous nature of the material. Only weak bands are found in +the 18-19 µm region. Water features occur at 6.1 µm and 2.8-2.9 µm (Fig. 4d). +In the red suevite (Polsingen) (Fig. 4c; Tab. 2c), the CF occurs between 7.6 and 7.9 µm. A strong +TF band is located at 11.8-12.0 µm, and the strongest Reststrahlen band is at 9.4-9.5 µm. Further, weak +Reststrahlen bands or shoulders are between 8.2 and 8.8 µm in red suevite samples. Also, the slopes +between 10 and 12 µm in Polsingen 3 and 4 are less steep than in the normal suevite. Broad bands at +~17 µm, and between 18-19 µm are also typical. All red suevites show strong water features at 6.1 and +2.7-3.0 µm, which are probably caused by weathering phases (Fig.4d). + +3.4 In Situ Analyses +Fine-grained matrix analyses for Aumühle (Fig.5, Table 3) show CFs between 7.6 and 8.3 µm, those for +Otting are located at 7.6-8 µm. The only powder analysis of separated matrix from Aumühle 18 overlaps +with these ranges (7.4-7.9µm). The strongest Reststrahlen band is at 9.5-9.6 µm in the Aumühle matrix, +at slightly longer wavelengths than in the powdered sample (9.4 µm). In the fine-grained Otting matrix, + + + +17 + +the Reststrahlen band is very similar (between 9.4 and 9.5 µm) for both in situ and powder +measurements. There are several additional Reststrahlen bands for fine grained matrix in the in situ +analyses: at 8.5-8.6 µm, 8.9-9.0 µm, and from 12.5 to 12.9 µm. The intensities vary from shoulders to +clear bands. Here the band positions are very similar to the powdered matrix from Aumühle 18. +The in-situ analyses of glasses in the polished sections of Aumühle and Otting (Fig.5, Tab.3) have +Christiansen Features between 7.7 and 7.9 µm, identical to the powder measurements. The dominating +Reststrahlen bands are at 9.4 to 9.6 µm, thus slightly shifted to longer wavelengths compared to the +powder data. The Polsingen samples where the ‘matrix’ consists of melt glass, show a high homogeneity: +the CF is at 7.9-8 µm, the Reststrahlen band at 9.6 µm. Characteristic is a shoulder/feature at 8.8 µm. +Again, a slight shift for the position of the Reststrahlen band was observed compared to the powder +analyses, whereas the other features occur at similar wavelength positions. + +4. Discussion +The investigated impact melt glasses in Aumühle and Otting do not have significant bands of +crystalline features, and are dominated by the Reststrahlen band at 9.3-9.4 µm in the powdered samples +and 9.4-9.6 µm in the micro-FTIR analyses. This range is similar to quartzofeldspathic glasses or glass +with granitic/rhyolitic composition (9.1-9.8 µm) with a mafic component (9.4-10.5 µm) (Wyatt et al., +2001; Byrnes et al, 2007; Johnson et al., 2007; DuFresne et al., 2009; Lee et al., 2010; Minitti and +Hamilton, 2010; Wright et al., 2011). This mixture reflects the starting composition of the Ries impact +site, which was dominated by gneiss, granite and about 13% mafic rock (amphibolite; von Engelhardt, +1997). The CF of the melt glasses (7.6-7.9 µm) also falls into the region for acidic and intermediate rocks, +while mafic minerals and rocks tend to have the CF greater than 8 µm (Pieters and Englert, 1993; +Salisbury and Walter, 1989; Cooper et al., 2002). + + + +18 + +Bulk melt rock, i.e. the red suevite from Polsingen, shows the main RB band slightly shifted by about 0.1 +µm to longer wavelengths. The similarity of the various samples even on millimeter scale in in-situ +analyses confirms that the red suevite is a homogeneous melt rock. Compared to the melt glasses in +normal suevite, the spectra from Polsingen show weak, but clear RB or shoulders of crystalline materials. +These indicate higher contents of crystalline components from granite and gneiss fragments (e.g., Hecker +et al., 2012). +Bulk suevites also form a homogeneous group, only the relative intensity of the mostly weak +crystalline bands hints at varying contents of partially shocked fragments. Separated matrix is spectrally +similar to the bulk material, indicating a high amount of fine-grained rock fragments in the matrix. This is +confirmed by the in-situ analyses of the fine-grained matrix, which have the same/similar band positions, +but varying intensities of the Restrahlen bands. The Reststrahlen bands in bulk powder suevite and fine- +grained matrix are a mixture of features characteristic for glasses and crystalline components. +The dominating band in the bulk samples between 9.4 µm (powders) and 9.4 to 9.6 µm (micro- +FTIR) is similar to the melt glasses (Byrnes et al, 2007; Lee et al., 2010; Wright et al., 2011). However, +crystalline features between 8.5-8.6 µm, 8.8-9.0 µm, and from 12.5 to 12.9 µm in the bulk and fine- +grained matrices are identical to that of quartz (e.g., Wenrich and Christensen, 1996; Christensen et al., +2000; Michalski et al., 2003; Baldridge et al., 2009), a major component of biotite-rich granite and gneiss +as found in the basement of the Ries Crater (von Engelhardt, 1997; Hecker et al., 2012). +The ranges for the position of the Christiansen Features in glass and bulk suevite overlap, but +some of the bulk samples have their CF at significantly shorter wavelengths compared to the glass. This +probably reflects the increased content of a crystalline quartz component. The lowest CF are between +7.4 and 7.5 µm, close to that of quartz with 7.35-7.4 µm (Tab. 2a)(e.g., Christensen et al., 2000; Michalski +et al., 2003; Baldridge et al., 2009; Tappert et al., 2013). This in turn would point to a significant + + + +19 + +crystalline component at least in some of the bulk suevites. A comparison with CF positions of terrestrial +rocks (Salisbury and Walter, 1989; Cooper et al., 2002) also indicates that the bulk powder suevites (7.4- +8.1µm) cover the range for acidic rocks (7.6-7.8 µm) and intermediate rocks (7.8-8.2 µm). +The variations in the positions of the CF are probably due to slight compositional variations as a +result of sieving. A higher content of quartz drives the CF to shorter wavelengths (e.g., Christensen et al., +2000; Michalski et al., 2003; Baldridge et al., 2009; Tappert et al., 2013), while an increase of clay +minerals or a mafic component (amphibolite) in the finest fraction during sieving could shift the position +of the CF towards longer wavelengths. Hornblende, a major phase of amphibolite, which comprises the +mafic component in the suevite (von Engelhardt, 1997), has a high CF from 8.2-8.5 µm (Salisbury, 1992; +1993; Baldridge et al., 2009). The CF of alteration phase montmorillionite falls between 7.9 and 8.2 µm +and would be difficult to identify directly in a mixture with rocks of intermediate composition +(Christiansen et al., 2000; Cooper et al., 2002; Koeppen et al., 2005; Michalski et al., 2005; Baldridge, +2009).The CF shift is also pronounced in the micro-FTIR spectra of the Aumühle samples, where the +intensity of the crystalline quartz features also varied in the analyzed areas (Fig.5). +In summary, in the mid-infrared range, suevites can be identified by their highly amorphous +band shape and few crystalline features. This could help to identify impactites like suevite in remote +sensing data, but also provides information about the shocked basement rock. However, glassy, +amorphous spectral features can also be expected from volcanic glasses like obsidian, so additional +geologic information about an observed area is necessary to distinguish the sources of glassy material +(Hamilton et al., 2001; Moroz et al., 2009; Wright et al., 2011). +While the starting composition of Mercury was probably not like the granitic basement of the +impact site in the Ries, i.e., in terms of mineralogy, it is still possible to draw conclusions for future +observations of the Mercurian surface. Mercury is a planet that underwent massive impact cratering in + + + +20 + +its early history (e.g., Hiesinger et al. 2010; Strom et al., 2011; Fassett et al., 2012). As a consequence, the +resulting future mid-infrared observations that will be made of Mercury by MERTIS on BepiColombo +might show only a few strong crystalline features. While the Ries crater represents only the effects of +one impact, surface material on Mercury underwent many impacts resulting in regolith gardening +(Domingue et al., 2014). The low degree of crystallinity of all involved rocks in suevite after only one +impact event indicates that the degree of amorphization can be expected to be much higher on the +surface of Mercury. + It has to be taken into account that the suevite is not the only type of ejecta observed in the +Ries. By volume, it is only a relatively small component – the Bunte Breccia has an estimated volume of +95 km3, the megablocks about 47 km3, and up to 22 km3 of suevite (Stöffler et al., 2013; Sturm et al, +2015). However, the suevite is the uppermost ejecta layer (e.g., von Engelhardt, 1990), and so is the +material most likely to be observed in remote sensing. Also, in a continuing impact gardening of a +planetary surface, the less shocked rocks will also increasingly experience higher degrees of shock +metamorphism. This might be even more enhanced by higher impact velocities on Mercury compared to +Earth, which result in the production of larger amounts of impact melt (e.g., Fassett et al., 2012). +In addition, the effects of space weathering will damage the crystalline structure of the remaining +material even more (Domingue et al., 2014). Observations in the ultraviolet, visible and near-infrared +range by MESSENGER do not show much variation over the surface, which could also point towards a +complete amorphization of the surface minerals (Izenberg et al., 2014). However, constant reprocessing +and gardening of the surface may also allow formation of new crystallites, resulting in a mixture of +crystallized and amorphous material. In addition volcanic activity (Thomas et al., 2014) will produce +crystalline phases. Consequently, ground based spectroscopic observations of larger areas on Mercury +(e.g., Sprague et al., 2000) show features of crystalline species such as pyroxene and feldspar. + + + +21 + +Furthermore, for our studied samples, alteration of pristine impact material by weathering has +to be taken into account. The fine-grained parts of the matrix contain clay minerals, mainly +montmorillonite, but also minor illite and halloysite (Stöffler et al, 2013). A strong band of +montmorillonite is found at 9.4-9.5 µm, overlapping with the position for the strong feature of +amorphous silicates. The strong feature of illite and halloysite is at slightly longer wavelengths, 9.4-9.7 +µm (Christensen et al., 2000; Koeppen et al., 2005; Michalski et al., 2006; Baldridge et al., 2009). A +further feature that could indicate montmorillonite is a weak band at about 8.8 µm (Christensen et al., +2000; Michalski et al., 2006; Baldridge et al., 2009), occurring in most bulk suevites and the Polsingen +sample (Tab.2). However, this band is also characteristic for quartz (e.g., Christensen et al., 2000; +Michalski et al., 2003; Tappert et al., 2013). +Another way to identify clay minerals are spectral features at longer wavelengths, in the 17-19 +µm regions. Clay minerals have features in the 18-19 µm region, montmorillionite from 18.7-19 µm, illite +and halloysite from 18-19 µm (Christensen et al., 2000; Baldridge et al., 2006; Michalski et al., 2006). +Here the Aumühle and Otting glasses show only a few, weak features. The suevite bulk samples show +more features in the 18.0-19 µm region, especially from 18.4-19 µm. This points towards a clay +component, but quartz can also have a feature at 18.3 µm (Baldridge et al., 2009). In contrast to the +Aumühle and Otting glasses, the Polsingen melt rocks show clear bands in the 18-19 µm region, +indicating a stronger degree of weathering compared to the glasses. However, this part is at the limit of +the spectral range of the spectrometer used, so exact band positions are difficult to obtain. +The strongest transparency feature of the bulk suevites (11.6-11.7µm) and red suevite (12.0µm) +is similar to that of the Aumühle and Otting glasses (at between 11.7 – 11.8 µm), and is also typical for +acidic and intermediate rocks (Salisbury and Walter, 1989; Cooper et al., 2002). A potential TF of +montmorillionite and illite at 12.1-12.7 µm is not visible in the smallest size fractions, except possibly the +Red Suevite from Polsingen. Illite has another TF at 11.5-11.7, which would overlap with the glass TF + + + +22 + +(Baldridge et al., 2006). A potential TF in bulk suevite at 11.1-11.2 µm is similar to one of the +montmorillonite TF found at 11.2-11.5 µm (Baldridge et al., 2006). However, quartz has its TF also in this +region (10.9-11.1µm) (Salisbury 1992; 1993; Baldridge et al., 2009). +Water bands could also potentially allow identifying secondary phases, but analyses were +conducted under ambient air pressure. The water bands for the samples in this study are all very similar +to each other, which indicate that adsorbed water could have influenced the spectra. This would render +the features difficult to use for comparison. +Further hints for the occurrence of smectites are elevated aluminum contents in the matrices and the +Polsingen samples (Tab.1). Certainly, aqueous alteration due to volatiles appears implausible on Mercury +at large scales. But application of the data from this study for remote sensing observations of bodies +where impactites were affected by alteration, like Earth or Mars, is still valid, as (minor to moderate) +alteration features do not overprint the spectral signature of the impactites. + +5. Conclusion +Despite the general similarity of their bulk spectra, bulk suevite, red suevite melt rock, and impact melts +are distinguishable by their mid-infrared features, although they are all dominated by amorphous +materials. Suevite glass from Aumühle and Otting shows simple spectra, dominated by an amorphous +feature. Suevites have clear bands of crystalline materials, while the Polsingen impact melt falls between +the two groups. +Secondary phases like clay minerals have features overlapping with other components (lithic rock clasts, +amorphous material) and are difficult to identify in the laboratory bulk spectra. This shows that low to +moderate amounts of alteration may not significantly affect the study of impactites on remote bodies. + + + +23 + +The samples of Polsingen impact melt also show signs of weathering, in contrast to the melt glasses from +Aumühle and Otting. +On the basis of our observations, we conclude that in mid-infrared remote sensing data, the surface +layer of Mercury will be dominated by features of amorphous materials. Because of the high degree of +amorphization that occurs after only one impact, as inferred from the Ries event, and following impact +gardening, even remaining crystalline materials will undergo high degrees of shock metamorphism. +However, the suevite represents only the uppermost impact layer in the Ries, while the underlying, more +voluminous ejecta show higher degrees of crystallinity. + +Acknowledgements +Many thanks to Prof. Alexander Deutsch (Münster) and Gisela Pösges (Rieskrater Museum, Nördlingen) +for precious help with the samples. 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(2011) Laboratory thermal emission spectroscopy of shocked +basalt from Lonar Crater, India, and implications for Mars orbital and sample data. Journal of +Geophysical Research 116. +Wyatt M.B., Christensen P.R., Taylor L.A. (2001) Analysis of terrestrial and Martian volcanic compositions +using thermal emission spectroscopy: 1. Determination of mineralogy, chemistry, and classification +strategies. Journal of Geophysical Research, Volume 106, Issue E7, p. 14711-14732 + + + + + + + + + + + + + + + + + + + + +35 + +Figure Captions +Figure 1: Map of the Ries area (adapted from Stöffler, 2013). Samples for this study were taken in the +Otting, Aumühle and Polsingen locations. +Figure 2a-c: Micrographs of representative areas of locations sampled for this study under normal +transmitted light (top) and crossed polarizers (bottom). The Otting (a) and Aumühle (b) samples show +the melt glasses embedded in the fine grained matrix. Spots (1 mm2 each) analyzed in situ with a FTIR- +microscope are marked with black boxes. Red suevite from Polsingen (c) does not clearly show the +distinction between amorphous material and matrix, it is probably a coherent melt rock. +Figure 3: Comparison of SEM/EDAX data for melt glass and matrix. Data from this study: squares = +Aumühle, circles = Otting, triangles = Polsingen. Empty symbols = glass; filled symbols = fine-grained +matrix. Literature data (Stöffler et al., 2013) are marked with encircled areas (see legend). Chemical +data for impact melt glass in suevite from our study fall into an area for earlier analyses. Analyses of fine- +grained matrix overlap with bulk–suevite data, but also fall outside the range, due to higher contents of +secondary phases. +Figure 4a-d. Mid-infrared reflectance spectra of samples in the size fractions 0-25µm (blue), 25-63µm +(pink), 63-125µm (red) and 125-250µm (brown) of (a) Suevite bulk and matrix analyses from Aumühle +and Otting, (b) Spectra of separated melt glasses from Aumühle and Otting, (c) Samples of red suevite +melt glass from Polsingen. Vertical lines mark characteristic Christiansen, Reststrahlen and Transparency +features. (d) Water features in the range from 2-7 µm for selected samples. Spectra are off-set for +clarity. Blue: melt glasses, Red: red suevite from Polsingen, Purple: suevite bulk samples. +Figure 5: Infrared and Raman for Interplanetary Spectroscopy (IRIS) laboratory In situ mid-infrared +spectra obtained from polished sections of suevite using a FTIR microscope. Each area was 1 mm2 in size; +the locations are shown in Figure2 a-c. + +Figure 1 + +Drilling +& Outcrop +Structural +IN +Aumuhle +Inner ring +Polsingen +Centeri +i +Potting +4 +- +- +Enkingen +Itzing 0 +@Altenburg +Mauren +Seelbronn +Bollstadt +5 km +Amerdingen3 mm +Glass2 +Glass3 +Glass1 +Matrix1 +Matrix2 +Matrix4 +Matrix3 +Matrix5 +Otting +Glass2 +Glass3 +Glass1 +Matrix1 +Matrix2 +Matrix4 +Matrix3 +Matrix5 +Transmitted Light +Cross Polarized +Figure 2a + +工 +1.00 mm2 mm3 mm +Matrix4 +Matrix3 +Matrix1 +Matrix2 +Glass1 +Glass2 +Glass3 +Aumühle +Matrix4 +Matrix3 +Matrix1 +Matrix2 +Glass1 +Glass2 +Glass3 +Transmitted Light +Cross Polarized +Figure 2b + +2 mm +Polsingen1 +Polsingen2 +Polsingen3 +Polsingen +Polsingen1 +Polsingen2 +Polsingen3 +Transmitted Light +Cross Polarized +Figure 2c + +TI +1.00mm2mmMatrix +Glass +Impact Melt Rock +Basement Rocks +Melt/Suevite Crater +Melt Outer Suevite +Outer Suevite +Polsingen (Melt rock) +Aumühle +Otting +FeO+MgO +CaO+Na2O +K2O +Figure 3 + +7 +8 +9 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Corresponding Author: Andreas Morlok, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Email: morlokan@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='de, Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' +49-251-83-39069 Aleksandra Stojic, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Email: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='stojic@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Isabelle Dittmar, Hochschule Emden/Leer, Constantiaplatz 4, 26723 Emden, Germany, Email: isabelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='dittmar@hs-emden-leer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='de Harald Hiesinger, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Email: hiesinger@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='de Manuel Ahmedi, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Email: ahmedi@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='de Martin Sohn, Hochschule Emden/Leer, Constantiaplatz 4, 26723 Emden, Germany, Email: martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='sohn@hs-emden-leer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='de Iris Weber, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Email: sonderm@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='de Joern Helbert, Institute for Planetary Research, DLR, Rutherfordstrasse 2, 12489 Berlin, Germany, Email: joern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='helbert@dlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='de © 2016 This manuscript version is made available under the CC-BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='0 2 ABSTRACT This study is part of an effort to build a mid-infrared database (7-14µm) of spectra for MERTIS (Mercury Radiometer and Thermal Infrared Spectrometer), an instrument onboard of the ESA/JAXA BepiColombo space probe to be launched to Mercury in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Mercury was exposed to abundant impacts throughout its history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This study of terrestrial impactites can provide estimates of the effects of shock metamorphism on the mid-infrared spectral properties of planetary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In this study, we focus on the Nördlinger Ries crater in Southern Germany, a well preserved and easily accessible impact crater with abundant suevite impactites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Suevite and melt glass bulk samples from Otting and Aumühle, as well as red suevite from Polsingen were characterized and their reflectance spectra in mid-infrared range obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In addition, in-situ mid-infrared spectra were made from glasses and matrix areas in thin sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The results show similar, but distinguishable spectra for both bulk suevite and melt glass samples, as well as in-situ measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Impact melt glass from Aumühle and Otting have spectra dominated by a Reststrahlen band at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='3-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Bulk melt rock from Polsingen and bulk suevite and fine-grained matrix have their strongest band between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' There are also features between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 and 9 µm, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm associated with crystalline phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' There is evidence of weathering products in the fine-grained matrix, such as smectites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Mercury endured many impacts with impactors of all sizes over its history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' So spectral characteristics observed for impactites formed only in a single impact like in the Ries impact event can be expected to be very common on planetary bodies exposed to many more impacts in their past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' We conclude that in mid-infrared remote sensing data the surface of Mercury can be expected to be dominated by features of amorphous materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 3 Keywords: Spectroscopy, Impact processes, Infrared observations, Instrumentation 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Introduction The aim of this study is to provide mid-infrared reflectance spectra with special emphasis on the region from 7µm – 14µm for a range of impactites for the application in planetary remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' We generate these spectra for a database for the ESA/JAXA BepiColombo mission to Mercury (Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Onboard is a mid-infrared spectrometer (MERTIS-Mercury Radiometer and Thermal Infrared Spectrometer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This unique device allows mapping spectral features in the 7-14 µm range, with a spatial resolution of ~500 m (Helbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Hiesinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Infrared spectroscopy provides a means to characterize the mineralogy of rocks via remote sensing, in contrast to Gamma-Ray, Neutron and X-ray spectrometers, which determine elemental compositions (Pieters and Englert, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Thus, IR spectroscopy provides a central analytical tool to determine the mineralogy of remote planetary surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In order to correctly interpret the remote sensing data, laboratory spectra of natural, synthetic rocks and minerals have to be collected to compare them to the spectra that will be obtained from MERTIS, once BepiColombo enters the Hermean orbit (Maturilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Hiesinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Comparable earlier instruments were the Thermal Emission Spectrometer (TES) on the Mars Global Surveyor (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2005) and the Thermal Emission Imaging System (THEMIS) on the Mars Odyssey orbiter (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The lunar surface was mapped in the mid-infrared with the DIVINER Lunar Radiometer Instrument on the Lunar Reconnaissance Orbiter (Paige et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The OSIRIS-REx Thermal Emission Spectrometer (OTES) will map asteroid Bennu (1999 RQ36) (Hamilton and Christensen, 2014), and the Thermal Infrared Imager (TIR) onboard Hayabusa 2 will map asteroid 1999JU3 (Okada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Ries crater in southern Germany (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1) provides well preserved layers of impactites and is one of the best studied impact sites in the world (von Engelhardt, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 Myr old crater with a diameter of 24 kilometers (Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010) offers the whole range of impact-associated rocks and 5 minerals (von Engelhardt, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Large impact events play a key role in surface modification processes in effect on most terrestrial planets and their moons (Hörz and Cintala, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Thus, the investigation of terrestrial impact processed rocks and understanding how these processes affect the spectral properties of the resulting impact generated rocks and melt glass is important for the interpretation of infrared data from the surfaces of other planetary bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Given the characteristics of surface regolith, we need spectral data from different grain size fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This is necessary because grain size variations affect the corresponding spectra immensely by reducing the spectral contrast of features and creating additional features at small grain sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Salisbury and Eastes, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Salisbury and Wald, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Mustard and Hayes, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Ruff and Christensen, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Reflectance and emission studies about spectral features in the mid-infrared in minerals undergoing high shock metamorphism including formation of melt glass were made on experimentally shocked samples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', on anorthosite, pyroxenite, basalt, and feldspar (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' , 2002, 2003, 2007, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' and Jaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' ,2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In these studies, especially feldspar-rich material showed loss and degradation of features, as well as band shifts with increasing structural disorder resulting from increasing shock pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Moroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2010) analyzed impact glasses from laser pulse experiments with Martian soil analogue JSC Mars-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Similar, Basilevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2000) and Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2000) spectrally studied melt glass also made from Martian soil analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Byrnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2007), and Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=',(2010) measured synthetic quartzofeldspathic glasses and Dufresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2009), Minitti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2002), Minitti and Hamilton (2010) measured synthetic glass with basaltic to intermediate composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In general, the resulting mid-infrared reflectance or emission spectra display a dominant feature in the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Pollack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (1973), Crisp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Nash and Salisbury (1991) and Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2001), studied obsidian or basalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Further infrared reflectance and emission spectra of natural impact glass and tektites were made, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', by Thomson and Schultz (2002), Gucsik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2004), Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2005) and Palomba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2011), Basavaiah and Chavan (2013), and Jaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2013) investigated 6 shocked bulk material from the Lonar Crater, India, showing mainly a simple spectrum with a dominating feature in the ~9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Complementary infrared transmission and absorbance spectra of related materials were also made, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g, of experimentally shocked feldspar materials (Stöffler and Hornemann, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Ostertag, 1983) and of CM2 chondrite Murchison (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Glasses with feldspathic composition produced in static high pressure experiments were measured by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Iiishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (1971), Velde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (1987), and Williams and Jeanloz (1988, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In addition, transmission infrared analyses of natural impact glass and tektites for studying water contents were made by Beran and Koeberl (1997), for identification purposes (Fröhlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013) or for astrophysical studies of circumstellar dust (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2004) provided an overview of silicate glass analysis with various mid-infrared techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Furthermore, there are also numerous studies how impact shock affects the spectral properties in the visible and near-infrared, such as Johnson and Hörz (2003), Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (1979), and Bruckenthal and Pieters (1984) for experimentally shocked feldspars or enstatite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Moroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2009) studied synthetic glasses with Martian soil composition as impact melt analogs, and Cannon and Mustard (2015) identified glass-rich impactites on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (1976) and Stockstill-Cahill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2014) analyzed lunar glass analogues, while Keppler (1992) studied synthetic silicate glasses with albite and diopside composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Schulz and Mustard (2004) studied terrestrial impact melt rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Shock metamorphism begins with fracturing and brecciation of the rocks at lower pressures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' above 2 GPa the mineral phases start to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Shatter cones and conical fracturing patterns form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Between 8 and 25 GPa, planar deformation features appear along with microscopic changes in the crystal structure due to impact shock in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', quartz and feldspar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' At pressures >25 GPa, shock metamorphism continues with the solid-state transformation of minerals into diaplectic glasses, which are amorphous materials generated at pressures up to 40 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' At pressures over 35 GPa, partial melting of phases begins and at over 60 GPa rocks completely melt, followed by vaporization at pressures above 100 GPa (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Stöffler, 1966, 1971, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Chao, 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' von Engelhardt and Stöffler, 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stöffler and 7 Langenhorst, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' French, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Also, high pressure mineral polymorphs form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' stishovite and coesite form from quartz at pressures from 12-15 GPa and over 30 GPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Carbon is transformed into diamond at 13 GPa (Stöffler and Langenhorst, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' French, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In the impactite suevite, shock metamorphism can be divided into stages 0 – IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stage 0 ranges from 0-10 GPa, stage I (10-35 GPa) and II (35-60 GPa) cover changes in crystal structure and partial melting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' stage IV pressures over 60 GPa result in melt glasses (Stöffler, 1971 and 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' French, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stöffler and Grieve, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Suevite consists of materials of all five shock stages (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Thus the infrared spectra obtained from the suevites can be expected to contain a mixture of spectral features depending on the level of shock metamorphism the material underwent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In this study, we focus on impactites from the Ries crater, since the crater site allows easy access to a range of naturally shocked rocks and minerals (von Engelhardt, 1990, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Prior to the impact event, the Ries area was covered with sediments mainly consisting of limestones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The underlying crystalline basement is dominated by granites and gneisses, with significant amounts of amphibolite (von Engelhardt and Graup, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' von Engelhardt, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The average chemical composition of very homogeneous impact melts (shock stage IV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stöffler et al, 2013) is similar to the composition of the modelled crystalline basement rock clasts based on the fallout suevite, identifying these rocks as the main source of the homogenized melt (Staehle, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' von Engelhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1984 , 1995, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Vennemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' We focus on the suevite, the top layer of ejecta in the Ries that was deposited on all other impact ejecta, Bunte Breccia (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Abadian, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Hörz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1983), megablocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2015) and polymict crystalline breccia (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' von Engelhardt, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The most voluminous ejecta is the Bunte Breccia, consisting mainly of sedimentary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Megablocks are rocks in the size range from 25 m to kilometers, consisting of sedimentary (limestone) and crystalline materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Both Bunte Breccia and 8 megablocks show low degrees of shock metamorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Polymict crystalline breccias are mixtures of basement rocks with shock stages up to stage II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Suevite is the only ejecta layer to contain material in all stages of shock metamorphism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', von Engelhardt, 1969, 1990, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Moldavites, a type of tektites formed in the Ries impact are an additional type of ejecta that was deposited 350 km east of the impact site (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', von Engelhardt, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The mostly granitic and felsic petrology of the crystalline basement of the Ries, which controlled the composition of the suevite (von Engelhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1990, 1997), is quite different compared to that of Mercury and other planetary surfaces in the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Also, granite in general is very rare outside Earth (Bonin, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Based on MESSENGER data, the best terrestrial analogues for the surface of Mercury are basalts or ultramafic komatiites (Nittler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stockstill-Cahill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Charlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Maturilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2014), but natural shocked forms are rare on Earth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, a spectral study of naturally shocked granitic material is of interest for our remote sensing purposes because it gives insight into the spectral characteristics of impactites and their components, while naturally shocked basalts are rare (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Furthermore, the easy access to the Ries site gives access to larger, representative amounts of sample material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Although rare, granitic materials have been recognized on other planetary bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Granitic material has been found as fragments and clasts in lunar samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Warren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Jolliff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Shervais and McGee, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Seddio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Granitoid or felsic materials may also occur on the mostly basaltic surface of Mars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Bandfield, 2006, Ehlmann and Edwards, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Sautter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2014, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' There are also indications for felsic, granitoid material on Venus (Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Gilmore, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Suevites are divided into crater or fallback suevite, occurring inside the inner ring of the Ries, and outer suevite, encompassing suevite found outside of the inner part of the crater (Pohl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1977, Stöffler, 1977, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Normal suevite in this study is outer suevite that originates from the Otting and Aumühle quarries (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1, 2a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' It mainly consists of three components: a porous matrix of fine-grained 9 rock, melt glass, and crystalline basement rocks displaying all stages of shock metamorphism ( Stöffler, 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' von Engelhardt and Graup, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Red suevite (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2c) from the Polsingen quarry is a rare variation of the outer suevite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Its red color is a characteristic owing to high hematite content (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In the red suevite, the actual groundmass consists of impact melt, in which fragments are embedded (Reimold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Also, chemical differences between normal suevite and red suevite (Na2O, K2O) were reported by Reimold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2013) and Stöffler (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In addition to the bulk materials, we are also interested in the components of the suevite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In particular, we studied the melt glasses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2a, b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', completely shock-melted and quenched rock material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Furthermore, we looked at the fine-grained matrix itself, which was affected by weathering processes (von Engelhardt, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Samples and Techniques 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 Sample Selection We selected three bulk suevite samples from three different Ries localities of the suevite layer (von Engelhardt and Stöffler, 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' von Engelhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1995, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Bayerisches Geologisches Landesamt, 2004): normal crater suevite from the Otting (samples Otting Bulk 1-3) and Aumühle (Aumühle Bulk4, Bulk 18, Bulk 18 Matrix) quarries, and red suevite (samples Polsingen 2-4) from a small outcrop in Polsingen (outer suevite) (von Engelhardt, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Reimold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In addition, we used material from three separated impact melt glasses: from a large, 30 cm sized glass ‘bomb’ (Otting Glasbombe) and a separated glass ‘Flädle’ from Otting (Otting Glas 1), and another glass separated from the Aumühle suevite (Aumühle 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In addition, for quantitative chemical and additional IR-microscopy analyses, polished thin sections of Aumühle, Otting and Polsingen were produced from the bulk sample rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The vitreous 10 state of the glass was confirmed using polarized light microscopy by the total extinction of the glass under crossed polarizers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, the two Otting samples show brownish halos or ‘Schlieren’, signs of incipient devitrification by the formation of small crystallites of pyroxenes, feldspar, oxides and possibly incipient signs of alteration phases like clays (see discussion)(von Engelhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2 Sample Preparation We used aliquots of larger amounts of material to avoid bias due to larger fragments/components thus ensuring a homogeneous and representative sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The original sample masses were greater than 100 grams for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Bulk samples were first powdered in steel and agate mortars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Subsequently, the powders were cleaned in acetone and dry sieved into four size fractions: 0-25 µm, 25-63 µm, 63-125 µm and 125-250 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This was done using an automatic Retsch Tap Sieve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' each size fraction was dry sieved for at least one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In order to remove clinging fines, the larger two fractions were cleaned with acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' For additional in situ measurements involving optical microscopy, micro-FTIR, and Scanning Electron Microscopy (SEM) measurements, we used thin sections of representative blocks of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The thin sections were polished to ~30 µm using standard procedures for petrological thin sections, ensuring a high specular reflectance from a flat surface from which especially microscopic IR investigations benefit greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='3 Optical Microscopy Overview images of the samples in normal light and between crossed polarizers were obtained with the KEYENCE Digital Microscope VHX-500F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Light microscopy allows rapid assessment of the general 11 homogeneity, as well as first mineral identification in the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Images were made using a magnification of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 Infrared Analyses 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 Diffuse Reflectance Powder Analyses To ensure diffuse reflectance, the size fractions were gently placed in aluminum sample cups (1 cm diameter), and the surface flattened with a spatula following a similar procedure described by Mustard and Hayes (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' For mid-infrared analyses from 2-20 µm, we used a Bruker Vertex 70 infrared system with a MCT detector at the Infrared and Raman for Interplanetary Spectroscopy (IRIS) laboratory at the Institut für Planetologie in Münster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' To avoid sample surface disturbance due to pore collapse during evacuation, we avoided analyses under near vacuum and measured under normal atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' As a consequence, water and atmosphere related features slightly affect the spectral range of interest near 7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' To ensure a high signal-to-noise ratio, we accumulated 512 scans for each size fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' For background calibration a diffuse gold standard was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' For the MERTIS database, we obtained analyses in a variable geometry stage (Bruker A513) in order to emulate various observational geometries of the orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The data presented here were obtained at 30° incidence (i) and 30° emergence angle (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The spectra of this study are intended to be compared with remote sensing data in the thermal infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Emission and reflectance spectra can be compared using Kirchhoff’s law: ε = 1 – R (R=Reflectance, ε = Emission) (Nicodemus, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This relation works very well for the comparison of directional emissivity and directional hemispherical reflectance (Hapke, 1993, Salisbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 12 However, in order to directly relate directional emissivity with reflectance by using Kirchhoff’s law, the reflected light in all directions has to be collected (Thomson and Salisbury, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A bi-directional, variable mirror set-up was used for this study without a hemisphere integrating all radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This has to be kept in mind when comparing the results in a quantitative manner with emission data (Salisbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In the case of the spectra obtained with the micro-FTIR in specular reflectance mode, the diffusely scattered part of the light is very small, since highly polished glass was analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Although specular reflectance is not entirely relatable to the directed emission, as required by Kirchhoff’s law, the differences will be mainly in spectral contrast, otherwise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' band shape and positions) the data will be comparable (Ramsey and Fink, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Byrnes et al, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Although the spectral range of interest for the database is from 7-14 µm, we measured our powder from 7-20 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 4a-c), since features of interest can appear at longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The signal of the detector used becomes weak at wavelengths above 18 µm, resulting in a low signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Above 19 µm, the spectral features are mainly random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The analyses were conducted under ambient pressure, which possibly affected the water bands at ~3 and ~6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' We therefore only present representative features from 2-7 µm for each sample in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The impactites analyzed in this study are often mixtures of various mineral phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' So in order to identify specific mineral bands in our laboratory spectra, we used spectra from the Arizona State University Thermal Emission laboratory (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000) and the Johns Hopkins ASTER laboratory (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2 In-situ FTIR Microspectroscopy For in situ analyses, we used a Bruker Hyperion 2000 IR microscope attached to the external port of a Bruker Vertex 70v at the Hochschule Emden/Leer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Here a 1000×1000 µm2 sized aperture was applied to obtain analyses of interesting features with in situ reflectance spectroscopy on polished thin 13 sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' For each spectrum, 128 scans were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A gold mirror was used for background calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The analyses were made in the range from 2-15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Since all features below 7 µm are very weak, we present results in the range of interest, 7-14 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The small shift between powder and microscope analyses observed (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Results) is probably due to the inherent differences among the samples (polished thin sections compared to powders) and the different optical set-ups of the techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Differences in the positions between the CF in powdered and solid samples were also observed by Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 SEM EDX Analyses In order to document the suevites and their components, we used a JEOL 6610-LV Scanning electron microscope equipped with a silicon drift Oxford EDX (Energy Dispersive X-Ray Spectroscopy) system to obtain micrographs of particular areas of interest and perform quantitative chemical analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Each chemical analysis was quantified with an ASTIMEX™ standard set for major elements prior to the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Beam current stability was controlled and measured before each analysis using a Faraday cup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The calibration was confirmed by re-analyzing the standards after the calibration procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' For analyses of the chemical composition of melt glass and fine-grained matrix areas, we analyzed areas of 100 x 100 µm2 using 90 seconds integration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The rather small area was chosen to maintain comparability between amorphous parts and fine-grained matrix areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Otherwise it would be difficult to obtain representative measurements of larger areas due to abundant cracks, gaps, holes, and veins, which are often re-filled by secondary alteration phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A broad beam and shorter integration times are helpful to measure volatile elements correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 Optical Images Optical images (Fig2a-c) give an overview of the thin sections for the three investigated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Otting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2a) and Aumühle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2b) samples in the transmitted light show the components of the normal suevite, the darker melt glasses, embedded in a brighter matrix, which consists of fine fragments of glass, rock and also secondary alteration products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Under crossed polarizers, the glasses are recognizable by their nearly black appearance due to complete extinction, while the fine mixture of the matrix appears brighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Polsingen sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2c), being a coherent melt rock, consists entirely of a red groundmass with abundant larger and smaller fragments in transmitted light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Brownish rims around minerals and holes indicate significant weathering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In the image under crossed polarizers, the groundmass appears dark, a few crystalline fragments are visible by their brighter appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' SEM/EDX Analyses Melt glass and fine-grained matrix area measurements in the suevites are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Due to porosity and volatile contents (water) the total analysis of matrix material in weight (wt)% is lower than 100 wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The melt glasses also show lower totals, which is due to crystallized or oxidized inclusions as well as inevitable cracks and veins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' For a better comparison with earlier data, we plot characteristic oxide ratios for MgO, FeO, CaO, Na2O, K2O of our SEM/EDX results with those from the literature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 3) (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Results for melt glass in Aumühle and Otting are very similar to each other and to earlier studies of glass from outer suevite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The results for the fine-grained matrix analyses cluster in an area overlapping with the composition of outer suevite (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 3) (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The analyses plotting outside the area are 15 explained by the lack of larger rock fragments within the analyzed areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This resulted in a higher content of fine grained secondary phases typical for the matrix, such as clay minerals, which is also indicated by elevated Al2O3 contents (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 1) (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Otherwise, the compositions of the fine-grained matrices can be explained as a mixture of the endmember glass and various basement rock fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Melt rock from the red suevite in Polsingen plots slightly below the studies for bulk red suevite (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This is probably due to the high degree of weathering of the samples from the only accessible outcrops, resulting in increased alkali (Na2O, K2O, CaO) concentrations (Reimold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Stöffler, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='3 Mid-Infrared Analyses of Powders The powder spectra of the bulk normal suevite from Otting and Aumühle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4a, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2a) show very similar band shapes and intensities for all samples, reflecting the high chemical homogeneity already indicated by SEM/EDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Christiansen Features (CF) of the bulk Otting and Aumühle bulk suevites are between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 µm, respectively, for all size fractions within one sample group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In the finest fraction from 0-25 µm, the CF is usually at longer wavelengths, from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 µm indicating a mineralogical heterogeneity among the finer fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Clays are extremely fine-grained and likely remain in the smallest grain size fraction, which could account for the slight shift in the CF (see discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Transparency features (TF) appear at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7 µm usually in the smallest size fractions of the bulk suevites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The strongest feature in this region is between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The strong Reststrahlen band (RB) at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 µm again indicates a highly amorphous and homogeneous internal structure of the minerals comprising the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Minor bands or shoulders are found on the slope of the RB at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm, indicating crystalline species (see Discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Further RB or potential TF are observed in the bulk suevites at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2 µm (overlapping with the transparency feature in the 16 smallest size fractions) and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Broad features in the 18-19 µm region are also typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' All bulk suevite spectra show very strong water features at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' These volatile bands probably result from clay minerals, but analysis under atmospheric conditions could also have affected the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A spectrum of separated fine-grained matrix from Aumühle 18 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 4a, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2a) is very similar in band shape and peak positions to the bulk spectrum Aumühle 18, indicating a high abundance of melt and crystalline clasts in the fine-grained matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Melt glasses from Aumühle and Otting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 4b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2b) are also very homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Christiansen Features (CF) are at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm, the Transparency Features (TF) at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm in the finest size fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The strongest Reststrahlen band is at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='3-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 µm, with a weak shoulder in the finest size fraction at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This all confirms the amorphous nature of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Only weak bands are found in the 18-19 µm region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Water features occur at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 µm and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In the red suevite (Polsingen) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 4c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2c), the CF occurs between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A strong TF band is located at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='0 µm, and the strongest Reststrahlen band is at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Further, weak Reststrahlen bands or shoulders are between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm in red suevite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Also, the slopes between 10 and 12 µm in Polsingen 3 and 4 are less steep than in the normal suevite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Broad bands at ~17 µm, and between 18-19 µm are also typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' All red suevites show strong water features at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='0 µm, which are probably caused by weathering phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 In Situ Analyses Fine-grained matrix analyses for Aumühle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5, Table 3) show CFs between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='3 µm, those for Otting are located at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6-8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The only powder analysis of separated matrix from Aumühle 18 overlaps with these ranges (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The strongest Reststrahlen band is at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm in the Aumühle matrix, at slightly longer wavelengths than in the powdered sample (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In the fine-grained Otting matrix, 17 the Reststrahlen band is very similar (between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm) for both in situ and powder measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' There are several additional Reststrahlen bands for fine grained matrix in the in situ analyses: at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='0 µm, and from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The intensities vary from shoulders to clear bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Here the band positions are very similar to the powdered matrix from Aumühle 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The in-situ analyses of glasses in the polished sections of Aumühle and Otting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='3) have Christiansen Features between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm, identical to the powder measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The dominating Reststrahlen bands are at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm, thus slightly shifted to longer wavelengths compared to the powder data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Polsingen samples where the ‘matrix’ consists of melt glass, show a high homogeneity: the CF is at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9-8 µm, the Reststrahlen band at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Characteristic is a shoulder/feature at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Again, a slight shift for the position of the Reststrahlen band was observed compared to the powder analyses, whereas the other features occur at similar wavelength positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Discussion The investigated impact melt glasses in Aumühle and Otting do not have significant bands of crystalline features, and are dominated by the Reststrahlen band at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='3-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 µm in the powdered samples and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm in the micro-FTIR analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This range is similar to quartzofeldspathic glasses or glass with granitic/rhyolitic composition (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm) with a mafic component (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm) (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Byrnes et al, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' DuFresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Minitti and Hamilton, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This mixture reflects the starting composition of the Ries impact site, which was dominated by gneiss, granite and about 13% mafic rock (amphibolite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' von Engelhardt, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The CF of the melt glasses (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm) also falls into the region for acidic and intermediate rocks, while mafic minerals and rocks tend to have the CF greater than 8 µm (Pieters and Englert, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Salisbury and Walter, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 18 Bulk melt rock, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' the red suevite from Polsingen, shows the main RB band slightly shifted by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1 µm to longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The similarity of the various samples even on millimeter scale in in-situ analyses confirms that the red suevite is a homogeneous melt rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Compared to the melt glasses in normal suevite, the spectra from Polsingen show weak, but clear RB or shoulders of crystalline materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' These indicate higher contents of crystalline components from granite and gneiss fragments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Hecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Bulk suevites also form a homogeneous group, only the relative intensity of the mostly weak crystalline bands hints at varying contents of partially shocked fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Separated matrix is spectrally similar to the bulk material, indicating a high amount of fine-grained rock fragments in the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This is confirmed by the in-situ analyses of the fine-grained matrix, which have the same/similar band positions, but varying intensities of the Restrahlen bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Reststrahlen bands in bulk powder suevite and fine- grained matrix are a mixture of features characteristic for glasses and crystalline components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The dominating band in the bulk samples between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 µm (powders) and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm (micro- FTIR) is similar to the melt glasses (Byrnes et al, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, crystalline features between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6 µm, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='0 µm, and from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 µm in the bulk and fine- grained matrices are identical to that of quartz (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Wenrich and Christensen, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Michalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009), a major component of biotite-rich granite and gneiss as found in the basement of the Ries Crater (von Engelhardt, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Hecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The ranges for the position of the Christiansen Features in glass and bulk suevite overlap, but some of the bulk samples have their CF at significantly shorter wavelengths compared to the glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This probably reflects the increased content of a crystalline quartz component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The lowest CF are between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm, close to that of quartz with 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='35-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4 µm (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2a)(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Michalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Tappert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This in turn would point to a significant 19 crystalline component at least in some of the bulk suevites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A comparison with CF positions of terrestrial rocks (Salisbury and Walter, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2002) also indicates that the bulk powder suevites (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1µm) cover the range for acidic rocks (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm) and intermediate rocks (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The variations in the positions of the CF are probably due to slight compositional variations as a result of sieving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A higher content of quartz drives the CF to shorter wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Michalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Tappert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013), while an increase of clay minerals or a mafic component (amphibolite) in the finest fraction during sieving could shift the position of the CF towards longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Hornblende, a major phase of amphibolite, which comprises the mafic component in the suevite (von Engelhardt, 1997), has a high CF from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm (Salisbury, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The CF of alteration phase montmorillionite falls between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2 µm and would be difficult to identify directly in a mixture with rocks of intermediate composition (Christiansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Koeppen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Michalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='The CF shift is also pronounced in the micro-FTIR spectra of the Aumühle samples, where the intensity of the crystalline quartz features also varied in the analyzed areas (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In summary, in the mid-infrared range, suevites can be identified by their highly amorphous band shape and few crystalline features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This could help to identify impactites like suevite in remote sensing data, but also provides information about the shocked basement rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, glassy, amorphous spectral features can also be expected from volcanic glasses like obsidian, so additional geologic information about an observed area is necessary to distinguish the sources of glassy material (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Moroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' While the starting composition of Mercury was probably not like the granitic basement of the impact site in the Ries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', in terms of mineralogy, it is still possible to draw conclusions for future observations of the Mercurian surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Mercury is a planet that underwent massive impact cratering in 20 its early history (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Hiesinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Strom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Fassett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' As a consequence, the resulting future mid-infrared observations that will be made of Mercury by MERTIS on BepiColombo might show only a few strong crystalline features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' While the Ries crater represents only the effects of one impact, surface material on Mercury underwent many impacts resulting in regolith gardening (Domingue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The low degree of crystallinity of all involved rocks in suevite after only one impact event indicates that the degree of amorphization can be expected to be much higher on the surface of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' It has to be taken into account that the suevite is not the only type of ejecta observed in the Ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' By volume, it is only a relatively small component – the Bunte Breccia has an estimated volume of 95 km3, the megablocks about 47 km3, and up to 22 km3 of suevite (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Sturm et al, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, the suevite is the uppermost ejecta layer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', von Engelhardt, 1990), and so is the material most likely to be observed in remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Also, in a continuing impact gardening of a planetary surface, the less shocked rocks will also increasingly experience higher degrees of shock metamorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This might be even more enhanced by higher impact velocities on Mercury compared to Earth, which result in the production of larger amounts of impact melt (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Fassett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In addition, the effects of space weathering will damage the crystalline structure of the remaining material even more (Domingue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Observations in the ultraviolet, visible and near-infrared range by MESSENGER do not show much variation over the surface, which could also point towards a complete amorphization of the surface minerals (Izenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, constant reprocessing and gardening of the surface may also allow formation of new crystallites, resulting in a mixture of crystallized and amorphous material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In addition volcanic activity (Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2014) will produce crystalline phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Consequently, ground based spectroscopic observations of larger areas on Mercury (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000) show features of crystalline species such as pyroxene and feldspar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 21 Furthermore, for our studied samples, alteration of pristine impact material by weathering has to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The fine-grained parts of the matrix contain clay minerals, mainly montmorillonite, but also minor illite and halloysite (Stöffler et al, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A strong band of montmorillonite is found at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm, overlapping with the position for the strong feature of amorphous silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The strong feature of illite and halloysite is at slightly longer wavelengths, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7 µm (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Koeppen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Michalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A further feature that could indicate montmorillonite is a weak band at about 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Michalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009), occurring in most bulk suevites and the Polsingen sample (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, this band is also characteristic for quartz (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Michalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Tappert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Another way to identify clay minerals are spectral features at longer wavelengths, in the 17-19 µm regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Clay minerals have features in the 18-19 µm region, montmorillionite from 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7-19 µm, illite and halloysite from 18-19 µm (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Michalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Here the Aumühle and Otting glasses show only a few, weak features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The suevite bulk samples show more features in the 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='0-19 µm region, especially from 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='4-19 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This points towards a clay component, but quartz can also have a feature at 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='3 µm (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' In contrast to the Aumühle and Otting glasses, the Polsingen melt rocks show clear bands in the 18-19 µm region, indicating a stronger degree of weathering compared to the glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, this part is at the limit of the spectral range of the spectrometer used, so exact band positions are difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The strongest transparency feature of the bulk suevites (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='6-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7µm) and red suevite (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='0µm) is similar to that of the Aumühle and Otting glasses (at between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='8 µm), and is also typical for acidic and intermediate rocks (Salisbury and Walter, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A potential TF of montmorillionite and illite at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7 µm is not visible in the smallest size fractions, except possibly the Red Suevite from Polsingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Illite has another TF at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='7, which would overlap with the glass TF 22 (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' A potential TF in bulk suevite at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2 µm is similar to one of the montmorillonite TF found at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='2-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='5 µm (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, quartz has its TF also in this region (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='9-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1µm) (Salisbury 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Water bands could also potentially allow identifying secondary phases, but analyses were conducted under ambient air pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The water bands for the samples in this study are all very similar to each other, which indicate that adsorbed water could have influenced the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This would render the features difficult to use for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Further hints for the occurrence of smectites are elevated aluminum contents in the matrices and the Polsingen samples (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Certainly, aqueous alteration due to volatiles appears implausible on Mercury at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' But application of the data from this study for remote sensing observations of bodies where impactites were affected by alteration, like Earth or Mars, is still valid, as (minor to moderate) alteration features do not overprint the spectral signature of the impactites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Conclusion Despite the general similarity of their bulk spectra, bulk suevite, red suevite melt rock, and impact melts are distinguishable by their mid-infrared features, although they are all dominated by amorphous materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Suevite glass from Aumühle and Otting shows simple spectra, dominated by an amorphous feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Suevites have clear bands of crystalline materials, while the Polsingen impact melt falls between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Secondary phases like clay minerals have features overlapping with other components (lithic rock clasts, amorphous material) and are difficult to identify in the laboratory bulk spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This shows that low to moderate amounts of alteration may not significantly affect the study of impactites on remote bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 23 The samples of Polsingen impact melt also show signs of weathering, in contrast to the melt glasses from Aumühle and Otting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' On the basis of our observations, we conclude that in mid-infrared remote sensing data, the surface layer of Mercury will be dominated by features of amorphous materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Because of the high degree of amorphization that occurs after only one impact, as inferred from the Ries event, and following impact gardening, even remaining crystalline materials will undergo high degrees of shock metamorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' However, the suevite represents only the uppermost impact layer in the Ries, while the underlying, more voluminous ejecta show higher degrees of crystallinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Acknowledgements Many thanks to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Alexander Deutsch (Münster) and Gisela Pösges (Rieskrater Museum, Nördlingen) for precious help with the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' We also thank the editor, Will M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Grundy, Steve Ruff and another anonymous reviewer for helping to improve the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' This work is supported by the DLR funding 50 QW 1302 in the framework of the BepiColombo mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 24 5.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', Taylor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (2001) Analysis of terrestrial and Martian volcanic compositions using thermal emission spectroscopy: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Determination of mineralogy, chemistry, and classification strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Journal of Geophysical Research, Volume 106, Issue E7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' 14711-14732 35 Figure Captions Figure 1: Map of the Ries area (adapted from Stöffler, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Samples for this study were taken in the Otting, Aumühle and Polsingen locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Figure 2a-c: Micrographs of representative areas of locations sampled for this study under normal transmitted light (top) and crossed polarizers (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' The Otting (a) and Aumühle (b) samples show the melt glasses embedded in the fine grained matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Spots (1 mm2 each) analyzed in situ with a FTIR- microscope are marked with black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Red suevite from Polsingen (c) does not clearly show the distinction between amorphous material and matrix, it is probably a coherent melt rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Figure 3: Comparison of SEM/EDAX data for melt glass and matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Data from this study: squares = Aumühle, circles = Otting, triangles = Polsingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Empty symbols = glass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' filled symbols = fine-grained matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Literature data (Stöffler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=', 2013) are marked with encircled areas (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Chemical data for impact melt glass in suevite from our study fall into an area for earlier analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Analyses of fine- grained matrix overlap with bulk–suevite data, but also fall outside the range, due to higher contents of secondary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Figure 4a-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Mid-infrared reflectance spectra of samples in the size fractions 0-25µm (blue), 25-63µm (pink), 63-125µm (red) and 125-250µm (brown) of (a) Suevite bulk and matrix analyses from Aumühle and Otting, (b) Spectra of separated melt glasses from Aumühle and Otting, (c) Samples of red suevite melt glass from Polsingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Vertical lines mark characteristic Christiansen, Reststrahlen and Transparency features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' (d) Water features in the range from 2-7 µm for selected samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Spectra are off-set for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Blue: melt glasses, Red: red suevite from Polsingen, Purple: suevite bulk samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Figure 5: Infrared and Raman for Interplanetary Spectroscopy (IRIS) laboratory In situ mid-infrared spectra obtained from polished sections of suevite using a FTIR microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Each area was 1 mm2 in size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' the locations are shown in Figure2 a-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content=' Figure 1 Drilling & Outcrop Structural IN Aumuhle Inner ring Polsingen Centeri i Potting 4 Enkingen Itzing 0 @Altenburg Mauren Seelbronn Bollstadt 5 km Amerdingen3 mm Glass2 Glass3 Glass1 Matrix1 Matrix2 Matrix4 Matrix3 Matrix5 Otting Glass2 Glass3 Glass1 Matrix1 Matrix2 Matrix4 Matrix3 Matrix5 Transmitted Light Cross Polarized Figure 2a 工 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 mm2 mm3 mm Matrix4 Matrix3 Matrix1 Matrix2 Glass1 Glass2 Glass3 Aumühle Matrix4 Matrix3 Matrix1 Matrix2 Glass1 Glass2 Glass3 Transmitted Light Cross Polarized Figure 2b 2 mm Polsingen1 Polsingen2 Polsingen3 Polsingen Polsingen1 Polsingen2 Polsingen3 Transmitted Light Cross Polarized Figure 2c TI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00mm2mmMatrix Glass Impact Melt Rock Basement Rocks Melt/Suevite Crater Melt Outer Suevite Outer Suevite Polsingen (Melt rock) Aumühle Otting FeO+MgO CaO+Na2O K2O Figure 3 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='10 Reflectance Micron (µm) Aumühle 18 Matrix 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='10 Reflectance Micron (µm) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='10 Reflectance Micron (µm) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='10 Reflectance Micron (µm) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='10 Reflectance Micron (µm) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='10 Reflectance Micron (µm) Otting Bulk 1 Aumühle 4 Bulk Aumühle 18 Bulk Otting Bulk 2 Otting Bulk 3 Christiansen Feature Transparency Feature Reststrahlen Bands 125 250 µm 25 63 µm 0 25 µm 63 125 µm Figure 4a 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='05 0.' metadata={'source': 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+page_content='05 Reflectance Micron (µm) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQft_01/content/2301.01683v1.pdf'} +page_content='05 Reflectance Micron (µm) Polsingen Bulk 3 Polsingen Bulk 4 Polsingen Bulk 2 Figure 4c 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b/1NFRT4oBgHgl3EQfljc6/content/tmp_files/2301.13599v1.pdf.txt @@ -0,0 +1,551 @@ +An AMM minimizing user-level extractable +value and loss-versus-rebalancing +Conor McMenamin1 and Vanesa Daza1,2 +1 Department of Information and Communication Technologies, Universitat Pompeu +Fabra, Barcelona, Spain +2 CYBERCAT - Center for Cybersecurity Research of Catalonia +Abstract. We present V0LVER, an AMM protocol which solves an in- +centivization trilemma between users, passive liquidity providers, and +block producers. V0LVER enables users and passive liquidity providers +to interact without paying MEV or incurring uncontrolled loss-versus- +rebalancing to the block producer. V0LVER is an AMM protocol built +on an encrypted transaction mempool, where transactions are decrypted +after being allocated liquidity by the AMM. V0LVER ensures this liq- +uidity, given some external market price, is provided at that price in ex- +pectancy. This is done by providing just enough loss-versus-rebalancing +profits to the block producer, incentivizing the block producer to move +the pool price to the external market price. With this, users transact +in expectancy at the external market price in exchange for a fee, with +AMMs providing liquidity in expectancy at the external market price. +Under block producer and liquidity provider competition, all of the fees +in V0LVER approach zero. Without block producer arbitrage, V0LVER +guarantees fall back to those of an AMM capable of processing encrypted +transactions, free from loss-versus-rebalancing or user-level MEV. +Keywords: Extractable Value · Decentralized Exchange · Incentives · Blockchain +1 +Introduction +In this paper we introduce V0LVER 3, an AMM which provides arbitrarily high +protection against user-level MEV and LVR. V0LVER is the first AMM to align +the incentives of the three, typically competing, entities in AMMs; the user, the +pool, and the block producer. This is done by ensuring that at all times, a block +producer is incentivized to move the pool to the price maximizing LVR. When +the block producer chooses a price, the block producer is forced to assert this is +correct, a technique introduced in [9]. Unfortunately, the protocol in [9] gives the +This Technical Report is part of a project that has received funding from the +European Union’s Horizon 2020 research and innovation programme under +grant agreement number 814284 +3 near-0 Extractable Value and Loss-Versus-Rebalancing ⇝ V0LVER +arXiv:2301.13599v1 [cs.GT] 31 Jan 2023 + +2 +McMenamin and Daza +block producer total power to extract value from users, due to order information +being revealed to the block producer before it is allocated a trading price in the +blockchain. To address this, V0LVER is built on encrypted mempools. +Modern cryptographic tools allow us to encrypt the mempool using zero- +knowledge based collateralized commit-reveal protocols [7,2,8,13], delay encryp- +tion [4,5] and/or threshold encryption [1]. If a block producer adding an order to +an AMM is forced to replicate the payoff of the AMM, we demonstrate the block +producer maximizes her own utility by showing liquidity centred around the ex- +ternal market price.4 Providing users with an AMM where the expected trade +price is the external market price, excluding fees, is a significant advancement +and the main contribution of this paper. Although batching orders against AMM +liquidity has been proposed as a defense against LVR [12], naively batching or- +ders against an AMM still allows a block producer to extract LVR by censoring +user orders. In V0LVER, block producers are effectively forced to immediately +repay LVR, while always being incentivized to include order commitments in the +blockchain, and eventually allocate AMM liquidity to these orders. +2 +Related Work +As the phenomenon of LVR has only recently been identified, there are only +two academic papers on the subject of LVR protection [6,9] to the best of our +knowledge, with no work protecting against both LVR and user-level MEV. +In [6], the AMM must receive the price of a swap from a trusted oracle before +users can interact with the pool. Such sub-block time price data requires central- +ized sources which are prone to manipulation, or require the active participation +of AMM representatives, a contradiction of the passive nature of AMMs and +their liquidity providers. We see this as an unsatisfactory dependency for DeFi +protocols. +Our work is based on some of the techniques of the Diamond protocol as +introduced in [9]. The Diamond protocol requires block producers to effectively +attest to the final price of the block given the orders that are to be proposed to +the AMM within the block. This technique requires the block producer to know +exactly what orders are going to be added to the blockchain. This unfortunately +gives the block producer total freedom to extract value from users submitting +orders to the AMM. With V0LVER, we address this issue while keeping the LVR +protection guarantees of Diamond. +Encrypting the transaction mempool using threshold encryption controlled +by a committee has been proposed in [1] and applied in [11]. In [11], a DEX +involving an AMM and based on frequent batch auctions [3] is proposed. This +DEX does not provide LVR resistance, and incentivizes transaction censorship +when a large LVR opportunity arises on the DEX. This is protected against in +V0LVER. +4 This holds true in many CFMMs, including the famous Uniswap V2 protocol. + +V0LVER +3 +3 +Preliminaries +This section introduces the key terminology and definitions needed to understand +LVR, and the proceeding analysis. In this work we are concerned with a single +swap between token x and token y. We use x and y subscripts when referring +to quantities of the respective tokens. The external market price of a swap is +denoted by ϵ, with the price of a swap quoted as the quantity of token x per +token y. +3.1 +Constant Function Market Makers +A CFMM is characterized by reserves (Rx, Ry) ∈ R2 ++ which describes the total +amount of each token in the pool. The price of the pool is given by pool price +function P : R2 ++ → R taking as input pool reserves (Rx, Ry). P() has the +following properties: +(a) P() is everywhere differentiable, with ∂P +∂Rx +> 0, +∂P +∂Ry +< 0. +(b) +lim +Rx→0 P = 0, +lim +Rx→∞ P = ∞, +lim +Ry→0 P = ∞, +lim +Ry→∞ P = 0. +(c) If P(Rx, Ry) = p, then P(Rx + cp, Ry + c) = p, ∀c > 0. +(1) +For a CFMM, the feasible set of reserves C is described by: +C = {(Rx, Ry) ∈ R2 ++ : f(Rx, Ry) = k} +(2) +where f : R2 ++ → R is the pool invariant and k ∈ R is a constant. The pool is +defined by a smart contract which allows any player to move the pool reserves +from the current reserves (Rx,0, Ry,0) ∈ C to any other reserves (Rx,1, Ry,1) ∈ C +if and only if the player provides the difference (Rx,1 − Rx,0, Ry,1 − Ry,0). To +reason about V0LVER, we assume the existence of an external market price +between x and y, which we define as follows, based on the definition of [3]. +Whenever an arbitrageur interacts with an AMM pool, say at time t with +reserves (Rx,t, Ry,t), we assume as in [10] that the arbitrageur always moves the +pool reserves to a point which maximizes arbitrageur profits, exploiting the dif- +ference between P(Rx,t, Ry,t) and the external market price at time t, denoted ϵt. +In this paper, we consider only the subset of CFMMs in which, given the LVR ex- +tracted in block Bt+1 corresponds to reserves (Rx,t+1, Ry,t+1), P(Rx,t+1, Ry,t+1) += ϵt+1. This holds for Uniswap V2 pools, among others. +3.2 +LVR-resistant AMM +We provide here an overview of the most important features of Diamond [9], an +AMM protocol which is proved to provide arbitrarily high LVR protection under +competition to capture LVR among block producers. In V0LVER, we adapt these +features for use on an encrypted transaction mempool. + +4 +McMenamin and Daza +A Diamond pool Φ is described by reserves (Rx, Ry), a pricing function P(), a +pool invariant function f(), an LVR-rebate parameter β ∈ (0, 1), and conversion +frequency T ∈ N. The authors also define a corresponding CFMM pool of Φ, +denoted CFMM(Φ). CFMM(Φ) is the CFMM pool with reserves (Rx, Ry) whose +feasible set is described by pool invariant function f() and pool constant k = +f(Rx, Ry). Conversely, Φ is the corresponding V0LVER pool of CFMM(Φ). The +authors note that CFMM(Φ) changes every time the Φ pool reserves change. The +protocol progresses in blocks, with one reserve update possible per block. +For an arbitrageur wishing to move the price of CFMM(Φ) to p from starting +reserves (Rx,0, Ry,0), let this require ∆y > 0 tokens to be added to CFMM(Φ), +and ∆x > 0 tokens to be removed from CFMM(Φ). The same price in Φ is +achieved by the following process: +1. Adding (1 − β)∆y tokens to Φ and removing (1 − β)∆x tokens. +2. Removing δx > 0 tokens such that: +P(Rx,0 − (1 − β)∆x − δx, Ry,0 + (1 − β)∆y) = p. +(3) +These δx tokens are added to the vault of Φ. +Vault tokens are periodically re-entered into Φ through what is effectively an +auction process, where the tokens being re-added are in a ratio which approxi- +mates the external market price at the time. The main result of [9] is the proving +that given a block producer interacts with Φ when the LVR parameter is β, and +there is an LVR opportunity of LV R in CFMM(Φ), the maximum LVR in Φ is +(1 − β)LV R. This results is stated formally therein as follows: +Theorem 1. For a CFMM pool CFMM(Φ) with LVR of L > 0, the LVR of Φ, +the corresponding pool in Diamond, has expectancy of at most (1 − β)L. +In this paper we use the same base functionality of Diamond to restrict the +LVR of block producers. Given a block producer wants to move the price of +CFMM(Φ) to some price p to extract maximal LVR LV R, the maximal LVR +in Φ of (1 − β)LV R is also achieved by moving the price to p. An important +point to note about applying LVR rebates as done in [9], is that directly after +tokens are placed in the vault, the pool constant drops. This must be considered +when calculating the profitability of an arbitrageur extracting LVR from a Dia- +mond pool. We do this when analyzing the profitability of V0LVER in Section +5. Importantly, tokens are eventually re-added to the pool, and over time the +expected value of the pool constant is increasing, as demonstrated in [9]. +4 +Our Protocol +We now outline the model in which we construct V0LVER, followed by a detailed +description of V0LVER. + +V0LVER +5 +4.1 +Model +In this paper we consider a blockchain in which all transactions are attempting +to interact with a single V0LVER pool between tokens x and y. +1. A transaction submitted by a player for addition to the blockchain while +observing blockchain height H, is finalized in a block of height at most +H + T, for some known T > 0. +2. The token swap has an external market price ϵ, which follows a Martingale +process. +3. There exists a population of arbitrageurs able to frictionlessly trade at exter- +nal market prices, who continuously monitor and interact with the blockchain. +4. Encrypted orders are equally likely to buy or sell tokens at ϵ, distributed +symmetrically around ϵ. +4.2 +Protocol Framework +This section outlines the terminology and functionalities used in V0LVER. It is +intended as a reference point to understand the core V0LVER protocol. Specifi- +cally, we describe the possible transactions in V0LVER, the possible states that +V0LVER orders/order commitments can be in, and the possible actions of block +producers. +As in the protocol of Section 3.2, a V0LVER pool Φ with reserves (Rx, Ry) +is defined with respect to a CFMM pool, denoted CFMM(Φ), with reserves +(Rx, Ry), a pricing function P(), under the restrictions of Section 3.1, and a +pool invariant function f(). +Orders in V0LVER are intended to interact with the AMM pool with some +delay due to commit-reveal. Therefore, we need to introduce the concept of +allocated funds to be used when orders eventually get revealed. To do this, we +define an allocated pool Φa. For a player committing to an order of size either +sizex or sizey known to be of maximum size maxx or maxy, the allocation pool +consists of (∆x, ∆y) tokens such that: +f(Rx, Ry) = f(Rx + maxx, Ry − ∆y) = f(Rx − ∆x, Ry + maxy). +(4) +Moreover, if f(Rx, Ry) = f(Rx+sizex, Ry−δy), meaning the CFMM pool would +sell δy tokens in return for sizex, the allocation pool also sells δy for sizex. +There are three types of transaction in our protocol. To define these trans- +actions, we need an LVR rebate function β : [0, 1, ..., Z, Z +1] → [0, 1]. It suffices +to consider β() as a strictly decreasing function with β(z) = 0 ∀z ≥ Z. +1. Order. These are straightforward buy or sell orders indicating a limit price, +size and direction to be traded. WLOG, we assume all orders in our system +are executable. +2. Order commitment transaction (OCT). These are encrypted orders +known to be collateralized by either maxx or maxy tokens. The exact size, +direction, price, and sender of an OCT sent from player Pi is hidden from all + +6 +McMenamin and Daza +other players. This is possible using anonymous ZK proofs of collateral such +as used in [8,13,7]), which can be implemented on a blockchain in conjunction +with a user-lead commit-reveal protocol, delay encryption scheme [4,5] or +threshold encryption scheme [1,11]. +3. Update transaction. These transactions are executed in a block before +any OCT is allowed to interact with the protocol pool. Let the current block +height be H. Update transactions take as input an allocation block height +Ha ≤ H, some number of transactions to allocate Ta ∈ Z≥0, and pool price +p. Given an allocation block height of H′ +a in the previous update transaction, +valid update transactions require: +(a) Ha > H′ +a. +(b) The number of OCTs in blocks [H′ +a + 1, ..., Ha] equals Ta. +Given inputs (Ha, Ta, p), the block producer moves the price of the pool to +p. The producer receives (1−β()) of the implied change in reserves from this +price move, as is done in [9]. +The producer then deposits (Taβ(H − Ha)maxx, Taβ(H − Ha) maxx +p +) to an +allocation pool denoted ΦHa, with (Ta(1 − β(H − Ha))maxx, Ta(1 − β(H − +Ha)) maxx +p +) being added to ΦHa from the AMM reserves. +In other words, if an allocation pool requires up to (Tamaxx, Ta maxx +p +) tokens +to trade with orders corresponding to the Ta allocated orders, the block producer +is forced to provide β(H − Ha) of the tokens in the pool, with starting bid and +offer prices equal to the pool price set by the block producer. This is used to +incentivize the block producer to always choose a pool price equal to the external +market price. +Every block, a block producer has four possible actions to perform on OCTs +and their orders. Orders in our system are batch-settled with other orders allo- +cated at the same time, and the liquidity in the respective allocation pool. +1. Insert OCTs to the blockchain. +2. Allocate inserted OCTs. For a block producer adding a block at height H +to allocate any number (including 0) inserted OCTs with inserted height of +at most Hi, the block producer must: +(a) Allocate all unallocated OCTs with inserted height less than or equal to +Hi. Let there be x such inserted OCTs to allocate. +(b) Submit an update transaction with inputs (Ha = Hi, Ta = x, p), for +some p > 0. +3. Reveal allocated order. When a decrypted order corresponding to an OCT +at height Ha is finalized on the blockchain within T blocks after the corre- +sponding OCT is allocated, it is considered revealed. For allocation pool Φa +Ha +with reserves (Rx,Ha, Ry,Ha), the initial clearing price for orders allocated at +height Ha is equal to Rx,Ha +Ry,Ha . Upon order reveal, the clearing price for orders +is updated. This clearing price maximizes trade volume, as is done in [8]. +4. Execute revealed orders. T blocks after OCTs are allocated, any correspond- +ing revealed orders are executed at the last-updated clearing price for orders + +V0LVER +7 +allocated at the same time. The final tokens in the allocation pool are re- +distributed proportionally to the allocating block producer and V0LVER +reserves. +4.3 +Protocol Outline +Our protocol can be considered as two sub-protocols, an update protocol and an +execution protocol. The update protocol is analogous to the protocol in Section +3.2, while the execution protocol, and its combination with the update protocol +are novel and specific to this paper. Every round a block-producer can submit +an update transaction. There are two scenarios for an update transaction with +inputs (Ha, Ta, p) and block height of the previous update transaction H′ +a. Either +Ta = 0 or Ta > 0. If Ta = 0. the update transaction is equivalent to an arbi- +trageur operation on a Diamond pool with LVR-rebate parameter of β(Ha−H′ +a) +(see Section 3.2). +If Ta > 0, the arbitrageur must also deposit (Ta(1−β(H −Ha))maxx, Ta(1− +β(H−Ha)) maxx +p +) to the Ha allocation pool ΦHa, with (Taβ(H−Ha)maxx, Taβ(H− +Ha) maxx +p +) being added to ΦHa from the AMM reserves. +After an allocation pool is created for allocated OCTs {oct1, ..., octn}, the +orders corresponding to {oct1, ..., octn} can be revealed for up to T blocks. This +is sufficient time for any user whose OCT is contained in that set to reveal the +order corresponding to the OCT. To enforce revelation, tokens corresponding to +unrevealed orders are burned. After all orders have been revealed, or T blocks +have passed, any block producer can execute revealed orders against the alloca- +tion pool at a clearing price which maximizes volume traded. Specifically, given +an array of orders ordered by price, a basic smart-contract can verify that a +proposed clearing price maximizes volume traded, as is done in [8]. +The final tokens in the allocation pool are redistributed to the allocating +block producer and V0LVER reserves. Adding these tokens directly to the pool +(and not the vault as in the protocol from Section 3.2) allows the pool to update +its price to reflect the information of the revealed orders. +5 +Protocol Properties +The goal of this section is to show that the expected execution price of any user +order is the external market price when the order is allocated, excluding at most +impact and fees. Firstly, note that an update transaction prior to allocation +moves the pool reserves of a V0LVER pool identically to an LVR arbitrage +transaction in Section 3.2. If Ta = 0, from [9] we know the block producer moves +the pool price to the max LVR price which is the external market price, and the +result follows trivially. +Now instead, assume Ta > 0. Let the reserves of a V0LVER pool Φ before +the update transaction be (Rx,0, Ry,0). Given an external market price of ϵ, from +Section 3.1 we know the max LVR occurs by moving the pool reserves to some +(Rx,m, Ry,m) with Rx,m +Ry,m = ϵ. WLOG, let Rx,0 +Ry,0 < Rx,m +Ry,m . Let the block producer + +8 +McMenamin and Daza +move the pool price to p corresponding to reserves in the corresponding CFMM +pool of (Rx,p, Ry,p). For a non-zero β(), this means the tokens in Φ not in the +vault (as per the protocol in Section 3.2) are (R′ +x,p, R′ +y,p) = (bRx,p, bRy,p) for +some b < 1. This is because some tokens in Φ are removed from the pool and +placed in the vault, while maintaining +R′ +x,p +R′y,p = p. +There are three payoffs of interest here. For these, recall that by definition of +the external market price, the expected imbalance of an encrypted order in our +system is 0 at the external market price. +1. Payoff of block producer vs. AMM pool: (1−β())(Rx,0−Rx,p+(Ry,0− +Ry,p)ϵ). +2. Payoff of block producer vs. users: Against a block producer’s own or- +ders, the block producer has 0 expectancy. Against other player orders, the +block producer strictly maximizes her own expectancy when (Rx,p, Ry,p) = +(Rx,m, Ry,m). Otherwise the block producer is offering below ϵ against ex- +pected buyers, or bidding above ϵ to expected sellers. +3. Payoff of users vs. AMM pool: Consider a set of allocated orders ex- +ecuted against the allocation pool, corresponding to the pool receiving δx +and paying δy tokens. By definition of the allocation pool, this (δx, δy) is +the same token vector that would be applied to the CFMM pool with re- +serves (bRx,p, bRy,p) if those orders were batch executed directly against the +CFMM. Let these new reserves be (bRx,1, bRy,1). Thus the profit of these +orders is b(1 − β())(Rx,p − Rx,1 + (Ry,p − Ry,1)ϵ). +Optimal strategy for block producer Let the block producer account for +α ∈ [0, 1] of the orders executed against the allocation pool. The maximum +payoff of the block producer against the AMM pool is the maximum of the sum +of arbitrage profits and profits of block producer orders executed against the +pool. Thus, the functions to be maximized is: +(1−β())(Rx,0−Rx,p+(Ry,0−Ry,p)ϵ)+α +� +b(1−β())(Rx,p−Rx,1+(Ry,p−Ry,1)ϵ) +� +. +(5) +This is equal to +(1−αb)(1−β()) +� +Rx,0−Rx,p+(Ry,0−Ry,p)ϵ +� ++αb(1−β()) +� +Rx,0−Rx,1+(Rx,0−Ry,1)ϵ +� +. +(6) +We know the second term is maximized for (Rx,1, Ry,1) = (Rx,m, Ry,m), as +this corresponds to LVR. Similarly, the first term is maximized for (Rx,p, Ry,p) = +(Rx,m, Ry,m). +Now consider the payoff for the block producer against user orders (Payoff 2.). +We have already argued that this is maximized with (Rx,p, Ry,p) = (Rx,m, Ry,m). +As such, moving the pool price to ϵ is a dominant strategy for the block producer. +Given this, we can see that the expected execution price for a client is ϵ +excluding impact and fees, with impact decreasing in expectancy in the number +of orders allocated. The payoff for the AMM against the block producer via the + +V0LVER +9 +update transaction is (1 − β())LV R, while the payoff against other orders is at +least 0. +5.1 +Minimal LVR +In the previous section, it is demonstrated that user-level MEV is prevented in +V0LVER, with users trading at the external market price in expectancy, exclud- +ing fees. However, we have thus far only proved that LVR in a V0LVER pool is +(1 − β()) of the corresponding CFMM pool. As in [9], under sufficient competi- +tion among block producers the Nash Equilibrium for the LVR rebate function is +β(0). This is under the assumption that block producers take all non-negligible +extractable value opportunities, which we consider LVR to be. +However, in reality frictionless arbitrage against the external market price +in blockchain-based protocols is likely not possible, and so LVR extraction has +some cost. As such, the expected value for β() may be less than β(0). Deploying +V0LVER, and analyzing β() across different token pairs, and under varying costs +for block producers makes for interesting future work. +6 +Discussion +If a V0LVER pool allows an OCT to be allocated with β() = 0, V0LVER effec- +tively reverts to the corresponding CFMM pool, with MEV-proof batch settle- +ment for all simultaneously allocated OCTs, albeit without LVR protection for +the pool. To see this, note that as β() = 0, the block producer can fully extract +any existing LVR opportunity, without requiring a deposit to the allocation pool. +As such, the expected price of the allocation pool is the external market price, +with orders executed directly against the V0LVER reserves at the external mar- +ket price, excluding fees and impact. Importantly, there is never any way for the +block producer to extract any value from allocated orders. This is because the +settlement price for an OCT is effectively set when it allocated, before any price +or directional information is revealed. +Allocation of tokens to the allocation pool has an opportunity cost for both +the V0LVER pool and the block producer. Given the informational superiority of +the block producer, allocating tokens from the pool requires the upfront payment +of a fee to the pool. Doing this anonymously is important to avoid MEV-leakage +to the block producer. One possibility is providing an on-chain verifiable proof +of membership to set of players who have bought pool credits, where a valid +proof releases tokens to cover specific fees, as in [13,8]. Another possibility is +providing a proof to the block-producer that the user has funds to pay the fee, +with the block-producer paying the fee on behalf of the user. A final option +based on threshold encryption [11] is creating a state directly after allocation +before any more allocations are possible, in which allocated funds are either used +or de-allocated. All of these proposals have merits and limitations, but further +analysis of these are beyond the scope of this work. + +10 +McMenamin and Daza +7 +Conclusion +V0LVER is an AMM based on an encrypted transaction mempool in which LVR +and MEV are protected against. V0LVER aligns the incentives of users, passive +liquidity providers and block producers. This is done by ensuring the optimal +block producer strategy under competition among block producers simultane- +ously minimizes LVR against passive liquidity providers and MEV against users. +Interestingly, the exact strategy equilibria of V0LVER depend on factors be- +yond instantaneous token maximization for block producers. This is due to risks +associated with liquidity provision and arbitrage costs. On one hand, allocating +OCTs after setting the pool price to the external market price, and providing +some liquidity to OCTs around this price should be positive expectancy for block +producers. Similarly, increasing the number of OCTs should also reduce the vari- +ance of block producer payoffs. On the other hand, there are caveats in which all +OCTs are informed and uni-directional. Analyzing these trade-offs for various +risk profiles and trading scenarios makes for further interesting future work. +References +1. 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Tornado Cash: https://tornadocash.eth.link/, accessed: 31/01/2023 + diff --git a/1NFRT4oBgHgl3EQfljc6/content/tmp_files/load_file.txt b/1NFRT4oBgHgl3EQfljc6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..83451e135ec40a8c946fc4b8cf482eb89fba29f6 --- /dev/null +++ b/1NFRT4oBgHgl3EQfljc6/content/tmp_files/load_file.txt @@ -0,0 +1,381 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf,len=380 +page_content='An AMM minimizing user-level extractable value and loss-versus-rebalancing Conor McMenamin1 and Vanesa Daza1,2 1 Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain 2 CYBERCAT - Center for Cybersecurity Research of Catalonia Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' We present V0LVER, an AMM protocol which solves an in- centivization trilemma between users, passive liquidity providers, and block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' V0LVER enables users and passive liquidity providers to interact without paying MEV or incurring uncontrolled loss-versus- rebalancing to the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' V0LVER is an AMM protocol built on an encrypted transaction mempool, where transactions are decrypted after being allocated liquidity by the AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' V0LVER ensures this liq- uidity, given some external market price, is provided at that price in ex- pectancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is done by providing just enough loss-versus-rebalancing profits to the block producer, incentivizing the block producer to move the pool price to the external market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' With this, users transact in expectancy at the external market price in exchange for a fee, with AMMs providing liquidity in expectancy at the external market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Under block producer and liquidity provider competition, all of the fees in V0LVER approach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Without block producer arbitrage, V0LVER guarantees fall back to those of an AMM capable of processing encrypted transactions, free from loss-versus-rebalancing or user-level MEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Keywords: Extractable Value · Decentralized Exchange · Incentives · Blockchain 1 Introduction In this paper we introduce V0LVER 3, an AMM which provides arbitrarily high protection against user-level MEV and LVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' V0LVER is the first AMM to align the incentives of the three, typically competing, entities in AMMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' the user, the pool, and the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is done by ensuring that at all times, a block producer is incentivized to move the pool to the price maximizing LVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' When the block producer chooses a price, the block producer is forced to assert this is correct, a technique introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Unfortunately, the protocol in [9] gives the This Technical Report is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 814284 3 near-0 Extractable Value and Loss-Versus-Rebalancing ⇝ V0LVER arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='13599v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='GT] 31 Jan 2023 2 McMenamin and Daza block producer total power to extract value from users, due to order information being revealed to the block producer before it is allocated a trading price in the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' To address this, V0LVER is built on encrypted mempools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Modern cryptographic tools allow us to encrypt the mempool using zero- knowledge based collateralized commit-reveal protocols [7,2,8,13], delay encryp- tion [4,5] and/or threshold encryption [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' If a block producer adding an order to an AMM is forced to replicate the payoff of the AMM, we demonstrate the block producer maximizes her own utility by showing liquidity centred around the ex- ternal market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='4 Providing users with an AMM where the expected trade price is the external market price, excluding fees, is a significant advancement and the main contribution of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Although batching orders against AMM liquidity has been proposed as a defense against LVR [12], naively batching or- ders against an AMM still allows a block producer to extract LVR by censoring user orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' In V0LVER, block producers are effectively forced to immediately repay LVR, while always being incentivized to include order commitments in the blockchain, and eventually allocate AMM liquidity to these orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 2 Related Work As the phenomenon of LVR has only recently been identified, there are only two academic papers on the subject of LVR protection [6,9] to the best of our knowledge, with no work protecting against both LVR and user-level MEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' In [6], the AMM must receive the price of a swap from a trusted oracle before users can interact with the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Such sub-block time price data requires central- ized sources which are prone to manipulation, or require the active participation of AMM representatives, a contradiction of the passive nature of AMMs and their liquidity providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' We see this as an unsatisfactory dependency for DeFi protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Our work is based on some of the techniques of the Diamond protocol as introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The Diamond protocol requires block producers to effectively attest to the final price of the block given the orders that are to be proposed to the AMM within the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This technique requires the block producer to know exactly what orders are going to be added to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This unfortunately gives the block producer total freedom to extract value from users submitting orders to the AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' With V0LVER, we address this issue while keeping the LVR protection guarantees of Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Encrypting the transaction mempool using threshold encryption controlled by a committee has been proposed in [1] and applied in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' In [11], a DEX involving an AMM and based on frequent batch auctions [3] is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This DEX does not provide LVR resistance, and incentivizes transaction censorship when a large LVR opportunity arises on the DEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is protected against in V0LVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 4 This holds true in many CFMMs, including the famous Uniswap V2 protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' V0LVER 3 3 Preliminaries This section introduces the key terminology and definitions needed to understand LVR, and the proceeding analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' In this work we are concerned with a single swap between token x and token y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' We use x and y subscripts when referring to quantities of the respective tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The external market price of a swap is denoted by ϵ, with the price of a swap quoted as the quantity of token x per token y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='1 Constant Function Market Makers A CFMM is characterized by reserves (Rx, Ry) ∈ R2 + which describes the total amount of each token in the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The price of the pool is given by pool price function P : R2 + → R taking as input pool reserves (Rx, Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' P() has the following properties: (a) P() is everywhere differentiable, with ∂P ∂Rx > 0, ∂P ∂Ry < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (b) lim Rx→0 P = 0, lim Rx→∞ P = ∞, lim Ry→0 P = ∞, lim Ry→∞ P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (c) If P(Rx, Ry) = p, then P(Rx + cp, Ry + c) = p, ∀c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (1) For a CFMM, the feasible set of reserves C is described by: C = {(Rx, Ry) ∈ R2 + : f(Rx, Ry) = k} (2) where f : R2 + → R is the pool invariant and k ∈ R is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The pool is defined by a smart contract which allows any player to move the pool reserves from the current reserves (Rx,0, Ry,0) ∈ C to any other reserves (Rx,1, Ry,1) ∈ C if and only if the player provides the difference (Rx,1 − Rx,0, Ry,1 − Ry,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' To reason about V0LVER, we assume the existence of an external market price between x and y, which we define as follows, based on the definition of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Whenever an arbitrageur interacts with an AMM pool, say at time t with reserves (Rx,t, Ry,t), we assume as in [10] that the arbitrageur always moves the pool reserves to a point which maximizes arbitrageur profits, exploiting the dif- ference between P(Rx,t, Ry,t) and the external market price at time t, denoted ϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' In this paper, we consider only the subset of CFMMs in which, given the LVR ex- tracted in block Bt+1 corresponds to reserves (Rx,t+1, Ry,t+1), P(Rx,t+1, Ry,t+1) = ϵt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This holds for Uniswap V2 pools, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2 LVR-resistant AMM We provide here an overview of the most important features of Diamond [9], an AMM protocol which is proved to provide arbitrarily high LVR protection under competition to capture LVR among block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' In V0LVER, we adapt these features for use on an encrypted transaction mempool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 4 McMenamin and Daza A Diamond pool Φ is described by reserves (Rx, Ry), a pricing function P(), a pool invariant function f(), an LVR-rebate parameter β ∈ (0, 1), and conversion frequency T ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The authors also define a corresponding CFMM pool of Φ, denoted CFMM(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' CFMM(Φ) is the CFMM pool with reserves (Rx, Ry) whose feasible set is described by pool invariant function f() and pool constant k = f(Rx, Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Conversely, Φ is the corresponding V0LVER pool of CFMM(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The authors note that CFMM(Φ) changes every time the Φ pool reserves change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The protocol progresses in blocks, with one reserve update possible per block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' For an arbitrageur wishing to move the price of CFMM(Φ) to p from starting reserves (Rx,0, Ry,0), let this require ∆y > 0 tokens to be added to CFMM(Φ), and ∆x > 0 tokens to be removed from CFMM(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The same price in Φ is achieved by the following process: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Adding (1 − β)∆y tokens to Φ and removing (1 − β)∆x tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Removing δx > 0 tokens such that: P(Rx,0 − (1 − β)∆x − δx, Ry,0 + (1 − β)∆y) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (3) These δx tokens are added to the vault of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Vault tokens are periodically re-entered into Φ through what is effectively an auction process, where the tokens being re-added are in a ratio which approxi- mates the external market price at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The main result of [9] is the proving that given a block producer interacts with Φ when the LVR parameter is β, and there is an LVR opportunity of LV R in CFMM(Φ), the maximum LVR in Φ is (1 − β)LV R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This results is stated formally therein as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' For a CFMM pool CFMM(Φ) with LVR of L > 0, the LVR of Φ, the corresponding pool in Diamond, has expectancy of at most (1 − β)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' In this paper we use the same base functionality of Diamond to restrict the LVR of block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Given a block producer wants to move the price of CFMM(Φ) to some price p to extract maximal LVR LV R, the maximal LVR in Φ of (1 − β)LV R is also achieved by moving the price to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' An important point to note about applying LVR rebates as done in [9], is that directly after tokens are placed in the vault, the pool constant drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This must be considered when calculating the profitability of an arbitrageur extracting LVR from a Dia- mond pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' We do this when analyzing the profitability of V0LVER in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Importantly, tokens are eventually re-added to the pool, and over time the expected value of the pool constant is increasing, as demonstrated in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 4 Our Protocol We now outline the model in which we construct V0LVER, followed by a detailed description of V0LVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' V0LVER 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='1 Model In this paper we consider a blockchain in which all transactions are attempting to interact with a single V0LVER pool between tokens x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' A transaction submitted by a player for addition to the blockchain while observing blockchain height H, is finalized in a block of height at most H + T, for some known T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The token swap has an external market price ϵ, which follows a Martingale process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' There exists a population of arbitrageurs able to frictionlessly trade at exter- nal market prices, who continuously monitor and interact with the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Encrypted orders are equally likely to buy or sell tokens at ϵ, distributed symmetrically around ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2 Protocol Framework This section outlines the terminology and functionalities used in V0LVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' It is intended as a reference point to understand the core V0LVER protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Specifi- cally, we describe the possible transactions in V0LVER, the possible states that V0LVER orders/order commitments can be in, and the possible actions of block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' As in the protocol of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2, a V0LVER pool Φ with reserves (Rx, Ry) is defined with respect to a CFMM pool, denoted CFMM(Φ), with reserves (Rx, Ry), a pricing function P(), under the restrictions of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='1, and a pool invariant function f().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Orders in V0LVER are intended to interact with the AMM pool with some delay due to commit-reveal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Therefore, we need to introduce the concept of allocated funds to be used when orders eventually get revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' To do this, we define an allocated pool Φa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' For a player committing to an order of size either sizex or sizey known to be of maximum size maxx or maxy, the allocation pool consists of (∆x, ∆y) tokens such that: f(Rx, Ry) = f(Rx + maxx, Ry − ∆y) = f(Rx − ∆x, Ry + maxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (4) Moreover, if f(Rx, Ry) = f(Rx+sizex, Ry−δy), meaning the CFMM pool would sell δy tokens in return for sizex, the allocation pool also sells δy for sizex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' There are three types of transaction in our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' To define these trans- actions, we need an LVR rebate function β : [0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=', Z, Z +1] → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' It suffices to consider β() as a strictly decreasing function with β(z) = 0 ∀z ≥ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' These are straightforward buy or sell orders indicating a limit price, size and direction to be traded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' WLOG, we assume all orders in our system are executable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Order commitment transaction (OCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' These are encrypted orders known to be collateralized by either maxx or maxy tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The exact size, direction, price, and sender of an OCT sent from player Pi is hidden from all 6 McMenamin and Daza other players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is possible using anonymous ZK proofs of collateral such as used in [8,13,7]), which can be implemented on a blockchain in conjunction with a user-lead commit-reveal protocol, delay encryption scheme [4,5] or threshold encryption scheme [1,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Update transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' These transactions are executed in a block before any OCT is allowed to interact with the protocol pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Let the current block height be H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Update transactions take as input an allocation block height Ha ≤ H, some number of transactions to allocate Ta ∈ Z≥0, and pool price p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Given an allocation block height of H′ a in the previous update transaction, valid update transactions require: (a) Ha > H′ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (b) The number of OCTs in blocks [H′ a + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=', Ha] equals Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Given inputs (Ha, Ta, p), the block producer moves the price of the pool to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The producer receives (1−β()) of the implied change in reserves from this price move, as is done in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The producer then deposits (Taβ(H − Ha)maxx, Taβ(H − Ha) maxx p ) to an allocation pool denoted ΦHa, with (Ta(1 − β(H − Ha))maxx, Ta(1 − β(H − Ha)) maxx p ) being added to ΦHa from the AMM reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' In other words, if an allocation pool requires up to (Tamaxx, Ta maxx p ) tokens to trade with orders corresponding to the Ta allocated orders, the block producer is forced to provide β(H − Ha) of the tokens in the pool, with starting bid and offer prices equal to the pool price set by the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is used to incentivize the block producer to always choose a pool price equal to the external market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Every block, a block producer has four possible actions to perform on OCTs and their orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Orders in our system are batch-settled with other orders allo- cated at the same time, and the liquidity in the respective allocation pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Insert OCTs to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Allocate inserted OCTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' For a block producer adding a block at height H to allocate any number (including 0) inserted OCTs with inserted height of at most Hi, the block producer must: (a) Allocate all unallocated OCTs with inserted height less than or equal to Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Let there be x such inserted OCTs to allocate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (b) Submit an update transaction with inputs (Ha = Hi, Ta = x, p), for some p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Reveal allocated order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' When a decrypted order corresponding to an OCT at height Ha is finalized on the blockchain within T blocks after the corre- sponding OCT is allocated, it is considered revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' For allocation pool Φa Ha with reserves (Rx,Ha, Ry,Ha), the initial clearing price for orders allocated at height Ha is equal to Rx,Ha Ry,Ha .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Upon order reveal, the clearing price for orders is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This clearing price maximizes trade volume, as is done in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Execute revealed orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' T blocks after OCTs are allocated, any correspond- ing revealed orders are executed at the last-updated clearing price for orders V0LVER 7 allocated at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The final tokens in the allocation pool are re- distributed proportionally to the allocating block producer and V0LVER reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='3 Protocol Outline Our protocol can be considered as two sub-protocols, an update protocol and an execution protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The update protocol is analogous to the protocol in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2, while the execution protocol, and its combination with the update protocol are novel and specific to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Every round a block-producer can submit an update transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' There are two scenarios for an update transaction with inputs (Ha, Ta, p) and block height of the previous update transaction H′ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Either Ta = 0 or Ta > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' If Ta = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' the update transaction is equivalent to an arbi- trageur operation on a Diamond pool with LVR-rebate parameter of β(Ha−H′ a) (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' If Ta > 0, the arbitrageur must also deposit (Ta(1−β(H −Ha))maxx, Ta(1− β(H−Ha)) maxx p ) to the Ha allocation pool ΦHa, with (Taβ(H−Ha)maxx, Taβ(H− Ha) maxx p ) being added to ΦHa from the AMM reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' After an allocation pool is created for allocated OCTs {oct1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=', octn}, the orders corresponding to {oct1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=', octn} can be revealed for up to T blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is sufficient time for any user whose OCT is contained in that set to reveal the order corresponding to the OCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' To enforce revelation, tokens corresponding to unrevealed orders are burned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' After all orders have been revealed, or T blocks have passed, any block producer can execute revealed orders against the alloca- tion pool at a clearing price which maximizes volume traded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Specifically, given an array of orders ordered by price, a basic smart-contract can verify that a proposed clearing price maximizes volume traded, as is done in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The final tokens in the allocation pool are redistributed to the allocating block producer and V0LVER reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Adding these tokens directly to the pool (and not the vault as in the protocol from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2) allows the pool to update its price to reflect the information of the revealed orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 5 Protocol Properties The goal of this section is to show that the expected execution price of any user order is the external market price when the order is allocated, excluding at most impact and fees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Firstly, note that an update transaction prior to allocation moves the pool reserves of a V0LVER pool identically to an LVR arbitrage transaction in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' If Ta = 0, from [9] we know the block producer moves the pool price to the max LVR price which is the external market price, and the result follows trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Now instead, assume Ta > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Let the reserves of a V0LVER pool Φ before the update transaction be (Rx,0, Ry,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Given an external market price of ϵ, from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='1 we know the max LVR occurs by moving the pool reserves to some (Rx,m, Ry,m) with Rx,m Ry,m = ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' WLOG, let Rx,0 Ry,0 < Rx,m Ry,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Let the block producer 8 McMenamin and Daza move the pool price to p corresponding to reserves in the corresponding CFMM pool of (Rx,p, Ry,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' For a non-zero β(), this means the tokens in Φ not in the vault (as per the protocol in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2) are (R′ x,p, R′ y,p) = (bRx,p, bRy,p) for some b < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is because some tokens in Φ are removed from the pool and placed in the vault, while maintaining R′ x,p R′y,p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' There are three payoffs of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' For these, recall that by definition of the external market price, the expected imbalance of an encrypted order in our system is 0 at the external market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Payoff of block producer vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' AMM pool: (1−β())(Rx,0−Rx,p+(Ry,0− Ry,p)ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Payoff of block producer vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' users: Against a block producer’s own or- ders, the block producer has 0 expectancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Against other player orders, the block producer strictly maximizes her own expectancy when (Rx,p, Ry,p) = (Rx,m, Ry,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Otherwise the block producer is offering below ϵ against ex- pected buyers, or bidding above ϵ to expected sellers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Payoff of users vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' AMM pool: Consider a set of allocated orders ex- ecuted against the allocation pool, corresponding to the pool receiving δx and paying δy tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' By definition of the allocation pool, this (δx, δy) is the same token vector that would be applied to the CFMM pool with re- serves (bRx,p, bRy,p) if those orders were batch executed directly against the CFMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Let these new reserves be (bRx,1, bRy,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Thus the profit of these orders is b(1 − β())(Rx,p − Rx,1 + (Ry,p − Ry,1)ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Optimal strategy for block producer Let the block producer account for α ∈ [0, 1] of the orders executed against the allocation pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The maximum payoff of the block producer against the AMM pool is the maximum of the sum of arbitrage profits and profits of block producer orders executed against the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Thus, the functions to be maximized is: (1−β())(Rx,0−Rx,p+(Ry,0−Ry,p)ϵ)+α � b(1−β())(Rx,p−Rx,1+(Ry,p−Ry,1)ϵ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (5) This is equal to (1−αb)(1−β()) � Rx,0−Rx,p+(Ry,0−Ry,p)ϵ � +αb(1−β()) � Rx,0−Rx,1+(Rx,0−Ry,1)ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' (6) We know the second term is maximized for (Rx,1, Ry,1) = (Rx,m, Ry,m), as this corresponds to LVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Similarly, the first term is maximized for (Rx,p, Ry,p) = (Rx,m, Ry,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Now consider the payoff for the block producer against user orders (Payoff 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' We have already argued that this is maximized with (Rx,p, Ry,p) = (Rx,m, Ry,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' As such, moving the pool price to ϵ is a dominant strategy for the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Given this, we can see that the expected execution price for a client is ϵ excluding impact and fees, with impact decreasing in expectancy in the number of orders allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' The payoff for the AMM against the block producer via the V0LVER 9 update transaction is (1 − β())LV R, while the payoff against other orders is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='1 Minimal LVR In the previous section, it is demonstrated that user-level MEV is prevented in V0LVER, with users trading at the external market price in expectancy, exclud- ing fees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' However, we have thus far only proved that LVR in a V0LVER pool is (1 − β()) of the corresponding CFMM pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' As in [9], under sufficient competi- tion among block producers the Nash Equilibrium for the LVR rebate function is β(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is under the assumption that block producers take all non-negligible extractable value opportunities, which we consider LVR to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' However, in reality frictionless arbitrage against the external market price in blockchain-based protocols is likely not possible, and so LVR extraction has some cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' As such, the expected value for β() may be less than β(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Deploying V0LVER, and analyzing β() across different token pairs, and under varying costs for block producers makes for interesting future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 6 Discussion If a V0LVER pool allows an OCT to be allocated with β() = 0, V0LVER effec- tively reverts to the corresponding CFMM pool, with MEV-proof batch settle- ment for all simultaneously allocated OCTs, albeit without LVR protection for the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' To see this, note that as β() = 0, the block producer can fully extract any existing LVR opportunity, without requiring a deposit to the allocation pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' As such, the expected price of the allocation pool is the external market price, with orders executed directly against the V0LVER reserves at the external mar- ket price, excluding fees and impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Importantly, there is never any way for the block producer to extract any value from allocated orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is because the settlement price for an OCT is effectively set when it allocated, before any price or directional information is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Allocation of tokens to the allocation pool has an opportunity cost for both the V0LVER pool and the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Given the informational superiority of the block producer, allocating tokens from the pool requires the upfront payment of a fee to the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Doing this anonymously is important to avoid MEV-leakage to the block producer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' One possibility is providing an on-chain verifiable proof of membership to set of players who have bought pool credits, where a valid proof releases tokens to cover specific fees, as in [13,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Another possibility is providing a proof to the block-producer that the user has funds to pay the fee, with the block-producer paying the fee on behalf of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' A final option based on threshold encryption [11] is creating a state directly after allocation before any more allocations are possible, in which allocated funds are either used or de-allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' All of these proposals have merits and limitations, but further analysis of these are beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' 10 McMenamin and Daza 7 Conclusion V0LVER is an AMM based on an encrypted transaction mempool in which LVR and MEV are protected against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' V0LVER aligns the incentives of users, passive liquidity providers and block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is done by ensuring the optimal block producer strategy under competition among block producers simultane- ously minimizes LVR against passive liquidity providers and MEV against users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Interestingly, the exact strategy equilibria of V0LVER depend on factors be- yond instantaneous token maximization for block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' This is due to risks associated with liquidity provision and arbitrage costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' On one hand, allocating OCTs after setting the pool price to the external market price, and providing some liquidity to OCTs around this price should be positive expectancy for block producers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Similarly, increasing the number of OCTs should also reduce the vari- ance of block producer payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' On the other hand, there are caveats in which all OCTs are informed and uni-directional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Analyzing these trade-offs for various risk profiles and trading scenarios makes for further interesting future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Asayag, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=', Cohen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=', Grayevsky, I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='04929 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='04929, ac- cessed: 26/01/2023 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content=' Tornado Cash: https://tornadocash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='eth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} +page_content='link/, accessed: 31/01/2023' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFRT4oBgHgl3EQfljc6/content/2301.13599v1.pdf'} diff --git a/5tE3T4oBgHgl3EQfpQqt/vector_store/index.pkl b/5tE3T4oBgHgl3EQfpQqt/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..01e7536fd2a55f3904a401af1ee91bf5a8407b0f --- /dev/null +++ b/5tE3T4oBgHgl3EQfpQqt/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:171f3d4c46b3ad5a922bf7ace975522894f96ecf64b7f62c0ce6791f5c8969d2 +size 172469 diff --git a/69FKT4oBgHgl3EQfUC2Z/content/tmp_files/2301.11782v1.pdf.txt b/69FKT4oBgHgl3EQfUC2Z/content/tmp_files/2301.11782v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..950b10dfce609ee36afe8dac3cd86b74df252c9a --- /dev/null +++ b/69FKT4oBgHgl3EQfUC2Z/content/tmp_files/2301.11782v1.pdf.txt @@ -0,0 +1,787 @@ +arXiv:2301.11782v1 [math.NT] 27 Jan 2023 +A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS +ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT +Abstract. Given a sequence of frequencies {λn}n≥1, a corresponding generalized Dirichlet +series is of the form f(s) = � +n≥1 ane−λns. We are interested in multiplicatively generated +systems, where each number eλn arises as a finite product of some given numbers {qn}n≥1, +1 < qn → ∞, referred to as Beurling primes. In the classical case, where λn = log n, Bohr’s +theorem holds: if f converges somewhere and has an analytic extension which is bounded in a +half-plane {Re s > θ}, then it actually converges uniformly in every half-plane {Re s > θ +ε}, +ε > 0. We prove, under very mild conditions, that given a sequence of Beurling primes, a small +perturbation yields another sequence of primes such that the corresponding Beurling integers +satisfy Bohr’s condition, and therefore the theorem. Applying our result in conjunction with +work of Diamond–Montgomery–Vorhauer and Zhang, we find a system of Beurling primes for +which both Bohr’s theorem and the Riemann hypothesis are valid. We discuss the connections +between our work and Diophantine approximation with Beurling integers, as well as with a +conjecture of Helson concerning outer functions in Hardy spaces of generalized Dirichlet series. +1. Introduction +For an increasing sequence of positive frequencies λ = {λn}n≥1, and a generalized Dirichlet +series +f(s) = +� +n≥1 +ane−λns, +the abscissas σc, σu, and σa of point-wise, uniform, and absolute convergence are defined as in +the classical theory of Dirichlet series [10]. In this article we wish to find sets of frequencies such +that the analogue of a theorem of Bohr [4] holds: if σc(f) < ∞ and f has a bounded analytic +extension to a half-plane {Re s > θ}, then σu(f) ≤ θ. The problem of finding frequencies for +which the abscissas of bounded and uniform convergence always coincide, which originated with +Bohr and Landau [17], has recently been revisited [2, 19] with the context of Hardy spaces of +Dirichlet series in mind. +Indeed, Bohr’s theorem is essentially a necessity for a satisfactory +Hardy space theory, see [18, Ch. 6]. +An important class of frequencies were introduced by Beurling [3]. Given an arbitrary in- +creasing sequence q = {qn}n≥1, 1 < qn → ∞, such that {log qn}n≥1 is linearly independent over +Q, we will denote by Nq = {νn}n≥1 the set of numbers that can be written (uniquely) as finite +products with factors from q, ordered in an increasing manner. The numbers qn are known +as Beurling primes, and the numbers νn are Beurling integers. The corresponding generalized +Dirichlet series are of the form +f(s) = +� +n≥1 +anν−s +n . +There are a number of criteria to guarantee the validity of Bohr’s theorem for frequencies +{λn}n≥1. Bohr’s original condition asks for the existence of c1, c2 > 0 such that +(1) +λn+1 − λn ≥ c1e−c2λn+1, +n ∈ N. +Landau relaxed the condition somewhat: for every δ > 0 there should be a c > 0 such that +(2) +λn+1 − λn ≥ ce−eδλn+1 , +n ∈ N. +1 + +2 +ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT +Landau’s condition was recently relaxed further by Bayart [2]: for every δ > 0 there should be +a C > 0 such that +(3) +log +�λm + λn +λm − λn +� ++ (m − n) ≤ Ceδλn, +m > n ∈ N. +All of these conditions are usually very difficult to check for any given Beurling system, since +they involve the distances between the corresponding Beurling integers. Furthermore, while it +is easy to construct Beurling integers for which conditions (1)-(3) fail, it appears to be rather +difficult to construct Beurling systems which do satisfy the Bohr condition. This is especially +true if one wants to retain properties of the ordinary integers, such as the asympotic behaviour +of the counting function Nq(x) = � +νn≤x 1, see for example [9]. +One motivation for considering Beurling integers is to investigate the properties of the q-zeta +function +ζq(s) = +� +n≥1 +ν−s +n += +� +n≥1 +1 +1 − q−s +n +, +and their interplay with the counting functions +Nq(x) = +� +νn≤x +1, +πq(x) = +� +qn≤x +1. +As an example, Beurling [3] himself showed that the condition +(4) +Nq(x) = ax + O( +x +(log x)γ ), +for some γ > 3 +2, +implies the analogue of the prime number theorem, +(5) +πq(x) := +� +qn≤x +1 ∼ +x +log x. +We refer to [15] for an overview of further developments. +In Section 2 we begin with a preparatory result which is interesting in its own right. It states +that starting with the classical set of primes numbers we can add almost any finite sequence of +Beurling primes while retaining the validity of Bohr’s theorem. +Theorem 1.1. Let {pn}n≥1 be the sequence of ordinary prime numbers and let N ≥ 1. Then +Bohr’s condition (1) holds for the Beurling integers generated by the primes +q = {pn}n≥1 +� +{qj}N +j=1, +for almost every choice (q1, . . . , qN) ∈ (1, ∞)N. +Sequences of Beurling primes of the type considered in Theorem 1.1 previously appeared in +[16]. +Our main result requires more careful analysis. +Theorem 1.2. Let q = {qn}n≥1 be an increasing sequence of primes such that q1 > 1 and +σc(ζq) < ∞. Then, for every A > 0 there exists a sequence of Beurling primes ˜q = {˜qn}n≥1 for +which Bohr’s condition (1) holds and +|qn − ˜qn| ≤ q−A +n +, +n ∈ N. +Theorem 1.2 and the work of Diamond, Montgomery, Vorhauer [8] and Zhang [21], allows +us to construct a system of Beurling primes that satisfies Bohr’s theorem and the “Riemann +Hypothesis”. +Corollary 1.3. There exists a system of Beurling primes q = {qn}n≥1 such that: + +A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS +3 +(i) The Riemann zeta function ζq(s) has an analytic extension to Re s > 1 +2, except for a simple +pole at s = 1. +(ii) The Riemann zeta function has no zeros in the half–plane C 1 +2 = {Re s > 1/2}. +(iii) The prime counting πq(x) satisfies +πq(x) = li(x) + O +�√x +� +, +where li(x) = +x´ +2 +(log u)−1 du. +(iv) Bohr’s condition holds for the associated class of generalized Dirichlet series. +We remark that as a direct consequence of (iii), we have the prime number theorem (5) as +well as Nq(x) ∼ ax, where a is the residue of ζq(s) at s = 1, see for example [7, 15]. +The proofs of our results investigate how well “irrational numbers” may be approximated by +fractions of Beurling integers. We will comment further on this kind of Diophantine approxi- +mation problems in Section 3. +In Section 3 we will also return to the original motivation for our work. There has been an +interest in studying Hardy spaces of generalized Dirichlet series since the 60s [6, 11, 12]. However, +other than the ordinary integers, no examples of Beurling integers exist which simultaneously +satisfy the prime number theorem (iii) and Bohr’s theorem, despite the fact that Bohr’s theorem +is crucial. Furthermore, since many aspects of the function theory of the Hardy space do not +depend on the choice q of Beurling primes, the idea behind Corollary 1.3 was to find a Beurling +system Nq in which we can assume the Riemann hypothesis. As a function theoretic application, +we construct an outer function f which has a zero, +(6) +f(s) = +1 +ζq(s + 1/2 + ε), +0 < ε < 1/2. +Originally, we intended to make use of (6) in order to disprove a conjecture of Helson [13] +about outer functions in Hardy spaces formed from frequencies satisfying the Bohr condition. +Unfortunately, we did not quite succeed, since we are unable to demonstrate the convergence +of (6) for Res > 0; the typical proof of this relies on the Lindel¨of hypothesis. Finding a system +which in addition to the properties of Corollary 1.3 satisfies the Lindel¨of hypothesis seems to +demand a deeper interplay between our tools and the probabilistic methods of [8]. +Notation. Throughout the article, we will be using the convention that C denotes a positive +constant which may vary from line to line. We will write that C = C(Ω) when the constant +depends on the parameter Ω. +2. Proof of the main results +Lemma 2.1. Suppose that {qn}n≥1 is a Beurling system such that dn := νn+1 − νn ≫ ν−C +n+1. +Then, for almost every q′ > 1 and every ε > 0, the Beurling system {qn}n≥1∪{q′} has a distance +function satisfying +(7) +d′ +n = ν′ +n+1 − ν′ +n ≫ ν−C +′ +n+1 , +n ∈ N, +where C′(q′, q) = max (C, 2σc(ζq) − 1 + ε). +Proof. Let x0 > 1. First we will prove that the set M of all numbers q′ ≥ x0 such that there +exist infinitely many triples (j, n, m) ∈ N3 with +����(q′)j − νn +νm +���� ≤ ν−C0 +n +ν−C0 +m +, +C0 = σ(ζq) + ε, + +4 +ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT +has measure zero. Since +�����q′ − +� νn +νm +� 1 +j ����� ≤ C(x0)x−j +0 +����(q′)j − νn +νm +���� . +we have that M ⊂ lim supm,n,j Ωm,n,j, where +Ωm,n,j = +�� νn +νm +� 1 +j +− C(x0)x−j +0 ν−C0 +n +ν−C0 +m +, +� νn +νm +� 1 +j ++ C(x0)x−j +0 ν−C0 +n +ν−C0 +m +� +, +j, n, m ≥ 1. +The Borel–Cantelli lemma thus shows that |M| = 0, since +� +m≥1 +� +n≥1 +� +j≥1 +|Ωm,n,j| ≤ C(x0)ζq (C0)2 < ∞. +Fix a number q′ ∈ [x0, ∞) \ M such that log q′ is not in the (countable) set spanQ{log qn}. +Note that the set of such numbers has full measure in [x0, ∞), and that x0 > 1 is arbitrary. By +construction, there are finitely many triples (j, n, m) such that +(8) +����(q′)j − νn +νm +���� ≤ ν−C0 +n +ν−C0 +m +. +For these exceptional triples, the left-hand side is at least positive, since log q′ /∈ spanQ{log qn}. +Therefore +����(q′)j − νn +νm +���� ≫ ν−C0 +n +ν−C0 +m +for all (j, n, m) ∈ N3. +Now we consider two arbitrary consecutive Beurling integers generated by the prime system +{qn}n≥1 ∪ {q′}, +ν′ +n+1 = (q′)aνm, +ν′ +n = (q′)bνl. +If a = b, then l = m − 1 and +ν′ +n+1 − ν′ +n ≫ ν−C +m +≥ +� +ν′ +n+1 +�−C , +by the hypothesis on the distances dn for the original Beurling system. Otherwise, if, say, b < a, +then +��ν′ +n+1 − ν′ +n +�� = (q′)bνm +����(q′)a−b − νl +νm +���� ≫ ν−C0 +l +ν−C0+1 +m +(q′)b ≫ +� +ν′ +n+1 +�−C +′ +. +where C +′ = 2σc(ζq) − 1 + ε. +□ +Proof of Theorem 1.1. The proof is a direct consequence of Lemma 2.1. +□ +In order to prove Bohr’s theorem for more general Beurling systems, we need to control the +constant in the distance estimate (7), which comes from the exceptional triples satisfying (8). +Proof of Theorem 1.2. Fix a small ε > 0 and x0 ∈ (1+ε/2, 1+ε). Consider first any Beurling +system Nρ = {νn}n≥1 generated by Beurling primes such that ρ1 > 1 + ε and σc(ζρ) < ∞. For +a number σ∞ > max(2, A) to be chosen in a moment, let +N = +� +m≥2 +� +n≥2 +� +j≥1 +Ωm,n,j, +where Ωm,n,j is defined as in the proof of Lemma 2.1, +Ωm,n,j = +�� νn +νm +� 1 +j +− C(x0)x−j +0 ν−σ∞ +n +ν−σ∞ +m +, +� νn +νm +� 1 +j ++ C(x0)x−j +0 ν−σ∞ +n +ν−σ∞ +m +� +. + +A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS +5 +Then |N| ≤ C(ε) (ζρ(σ∞) − 1)2 . Furthermore, for x > 2, let +Ix = [x − x− σ∞ +2 , x + x− σ∞ +2 ]. +Note that if σ∞ is sufficiently large, σ∞ ≥ C(ε), then Ix∩Ωm,n,j ̸= ∅ only if νn ≥ x/2. Therefore +|Ix ∩ N| ≤ +� +m≥2 +j≥1 +νn≥ x +2 +|Ωm,n,j| ≤ C(ε) (ζρ(σ∞) − 1) +� +νn≥ x +2 +ν−σ∞ +n +≤ C(ε) (ζρ(σ∞) − 1) ζρ +�σ∞ +4 +� +x− 3σ∞ +4 . +We will construct a sequence of Beurling systems such that +(9) +(ζρ(σ∞) − 1) ζρ +�σ∞ +4 +� +≤ 1 +for the number σ∞ > 0, still to be chosen later. Therefore +(10) +|Ix ∩ N| ≤ C(ε)x− σ∞ +4 |Ix|, +We conclude that whenever x is sufficiently large, Ix ̸⊂ N. +To include triples where νn or νm equals one in our considerations, we increase the power +σ∞. The inequality +(11) +����xj − νn +νm +���� ≤ ν−3σ∞ +n +ν−3σ∞ +m +implies, whenever x ≥ x0, that +�����x − +� νn +νm +� 1 +j ����� ≤ C(x0)x−j +0 +����xj − ν2 +nνm +ν2mνn +���� ≤ C(x0)x−j +0 +� +ν2 +mνn +�−σ∞ � +ν2 +nνm +�−σ∞ . +Therefore M ⊂ N, where M this time denotes the set of all x ≥ x0 for which there exists an +exceptional triple (j, n, m) ∈ N3 such that (11) holds. +Now let q be a sequence of primes in the statement of Theorem 1.2, assuming that ε < q1 −1. +As described, we will only be able to effectively apply (10) when x is sufficiently large, say, +x ≥ B = B(ε) = C(ε)4 + 2, where C(ε) in this instance refers to the same constant as in (10). +Let N be such that {q1, . . . , qN} = (1, B) ∩ q. Then, as a corollary of Theorem 1.1, we already +know that there exists an increasing finite sequence of primes {˜q1, . . . ˜qN}, ˜q1 > 1, such that +|qj − ˜qj| ≤ q−A +j +, j = 1, . . . , N, and such that Bohr’s condition holds for {ν(N) +n +}n≥1 = N{˜q1,...˜qN }. +Further, we choose σ∞ so large that +���ν(N) +n+1 − ν(N) +n +��� ≥ +� +ν(N) +n+1 +�−6σ∞ +, +n ∈ N, +and +(12) +(ζq′(σ∞) − 1) ζq′ +�σ∞ +4 +� +≤ 1, +q′ = {˜q1, . . . ˜qN, qN+1 − 1, qN+2 − 1, qN+3 − 1, . . .}. +This is made possible by the hypothesis that σc(ζq) < ∞, since +ζq′(σ) ≤ +� +j≥1 +1 +1 − (q′ +j)−σ ≤ ζq +� σ +C +� +, +σ > 0, +C ≥ sup +n≥1 +log(qn) +log(q′n). +From here we proceed by induction. Suppose that ˜q1, . . . ˜qk have been chosen, where k ≥ N, +with corresponding Beurling integers {ν(k) +n }n≥1 = N{˜qn}k +n=1 satisfying that +���ν(k) +n+1 − ν(k) +n +��� ≥ +� +ν(N) +n+1 +�−6σ∞ . + +6 +ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT +We apply the preceding discussion to the Beurling primes ρ = {˜q1, . . . ˜qk} and x = qk+1, con- +cluding that there exists a number ˜qk+1 ∈ Iqk+1 such that +�����˜qj +k+1 − ν(k) +n +ν(k) +m +����� ≥ +� +ν(k) +n +�−3σ∞ � +ν(k) +m +�−3σ∞ +, +(j, n, m) ∈ N3. +By the same argument as in the last paragraph of the proof of Theorem 1.1 the Beurling system +{ν(k+1) +n +}n≥1 = N{˜qn}k+1 +n=1, then satisfies that +���ν(k+1) +n+1 +− ν(k+1) +n +��� ≥ +� +ν(k+1) +n+1 +�−6σ∞ +, +n ∈ N. +At each step of the construction, (12) ensures that (9) holds. We hence obtain a sequence +˜q = {˜qn}n≥1, satisfying that |˜qn − qn| ≤ q +− σ∞ +2 +n +as well as Bohr’s condition (1), specifically, +|˜νn+1 − ˜νn| ≥ (˜νn+1)−6σ∞ , +n ∈ N. +where {˜νn}n≥1 = N˜q. +□ +Proof of Corollary 1.3. By [21, Theorem 1] there exists a Beurling system qRH that satisfies +(i)–(iii). Theorem 1.2 then implies that there exist primes q, asymptotically approaching qRH, +satisfying Bohr’s theorem, and such that +(13) +πq(x) = li(x) + O +�√x +� +. +We observe that +log ζq(s) = − +∞ +ˆ +1 +log +� +1 − u−s� +dπq(u), +Re s > 1, +see [8, Lemma 10]. With f(u) = +u−1 +u log u, u > 1, consider the function +Z(s) = exp + + +∞ +ˆ +1 +u−sf(u) du + + = +s +s − 1, +Re s > 1. +We then have that +(14) +log ζq(s) +Z(s) = − +∞ +ˆ +1 +� +log +� +1 − u−s� ++ u−s� +dπq(u) + +∞ +ˆ +1 +u−s (dπq(u) − f(u) du) . +The quantity under the first integral sign satisfies +log +� +1 − u−s� ++ u−s = O +� +u−2σ� +. +Thus, the first integral in (14) is analytic and uniformly bounded for Re s > 1 +2 + ε, ε > 0. By +(13), the second integral in (14) is also analytic in the half-plane C 1 +2 . Therefore, ζq has an +analytic continuation to the half–plane C 1 +2 , except for a simple pole at s = 1. Furthermore, ζq +cannot have any zeros in this half-plane. +□ +3. Further discussion +Diophantine approximation and Beurling integers. Using the Borel–Cantelli theorem to +study the irrationality of real numbers is a standard technique of Diophantine approximation. +The irrationality measure µ(x) of a real number x ∈ R is defined as the infimum of the set +Rx = +� +r > 0 : +���x − m +n +��� < 1 +nr for at most finitely many pairs (m, n) ∈ N × N +� +. + +A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS +7 +For a Beurling system Nq = {νn}n≥1, we may also introduce the irrationality measure µq(x) of +a real number x ∈ R as the infimum of the set +Rx = +� +r > 0 : +����x − νm +νn +���� < 1 +νrn +for at most finitely many pairs (m, n) ∈ N × N +� +. +Then, by slightly modifying the proof of Lemma 2.1, we obtain the following proposition. +Proposition 3.1. Let q = {qn}n≥1 be a sequence of Beurling primes with σc(ζq) < ∞. Then, +for almost every x ∈ R, it holds that +µq(x) ≤ 2σc(ζq). +In the classical case, Dirichlet’s approximation theorem therefore implies that µ(x) = 2 for +almost every x ∈ R. We also recall Roth’s theorem [5], which states that µ(x) = 2 for every +algebraic irrational number. It would be very interesting to develop corresponding results in the +context of Beurling integers. +Hardy spaces of Dirichlet series and a conjecture of Helson. For a sequence q of Beurling +primes, we introduce the Hardy space H2 +q as +H2 +q = + + +f(s) = +� +n≥1 +anν−s +n +: ∥f∥2 +H2q = +� +n≥1 +|an|2 < ∞ + + + . +More generally, for 1 ≤ p < ∞, we define Hp +q as the completion of polynomials (finite sums +� anν−s +n ) under the Besicovitch norm +∥P∥Hp +q := + + lim +T →∞ +1 +2T +T +ˆ +−T +|P(it)|p dt + + +1 +p +. +The function theory of these spaces originated with Helson [12], and was, in the distuingished +case where q is the sequence of ordinary primes, continued in very influential papers of Bayart +[1] and Hedenmalm, Lindqvist, and Seip [11]. More generally, there is a developing theory of +Hardy spaces of Dirichlet series � ane−λns whose frequencies are related to other groups than +T∞, but we shall restrict our attention to frequencies given by Beurling primes. A cornerstone +of the theory is that there is a natural multiplicative linear isometric isomorphism between Hp +q +and the Hardy space Hp +q (T∞) of the infinite torus [6, 13]. However, more is needed in order to +identify H∞(T∞) with H∞ +q , the space of Dirichlet series � anν−s +n +which converge to a bounded +function in C0. In fact, Bohr’s condition is typically used in order to establish this isomorphism +[19]. +In identifying Hp +q with Hp +q (T∞) one is naturally led to consider twisted Dirichlet series +fχ(s) = +� +n≥1 +anχ(νn)ν−s +n , +where a point χ ∈ T∞ is interpreted as the completely multiplicative character χ: Nq → T +such that χ(qn) = χn. Helson [13] proved that if f ∈ H2 +q and the associated frequencies satisfy +Bohr’s condition, then fχ(s) converges in C0 for almost every χ ∈ T∞. Helson went on to +make a conjecture, which we state only in the special case that the frequencies correspond to a +Beurling system. +Recall that f ∈ H2 +q is said to be outer if +� +fg : g ∈ H∞ +q +� +is dense in H2 +q. +Conjecture. If Nq is a Beurling system that satisfies Bohr’s condition and f is outer in H2 +q, +then fχ never has any zeros in its half-plane of convergence. + +8 +ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT +Suppose now that the Beurling primes q are chosen as in Corollary 1.3, so that we have the +“Riemann hypothesis” at our disposal, and consider the Dirichlet series +f(s) = +1 +ζq(s + 1/2 + ε), +for some 0 < ε < 1/2. +Through a routine calculation with coefficients, one checks that +f, f 2, 1/f, 1/f 2 ∈ H2 +q. Therefore, there are polynomials pn which converge to 1/f in H4 +q, so +that +∥1 − pnf∥H2q ≤ ∥f∥H4q∥1/f − pn∥H4q → 0, +n → ∞. +Thus f is outer. On the other hand, it has a zero at s = 1/2 − ε. +Corollary 1.3 ensures that f has an analytic extension to C0. The problem is that we do not +know if f actually converges there, as required by Helson’s conjecture. Since we have Bohr’s +condition, a standard argument [20, Section 14.25] with the Perron formula shows that the +Lindel¨of hypothesis for ζq implies the convergence of f. However, in the Beurling prime setting, +“Riemann implies Lindel¨of” is only true when ζq has finite order in C 1 +2 . +This crux further +highlights one of the main issues of [14, 15]: how can we ensure that ζq is zero-free and has finite +order in some half-plane Re s > 1 − η? Or, equivalently, when do we have that +πq(x) = li(x) + O(xθ1), +Nq(x) = kx + O(xθ2), +for some θ1, θ2 < 1? The best estimate we are able to add to Corollary 1.3 comes from the +general result [15, Theorem 2.2]: there exists a number c > 0 such that +Nq(x) = ax + O +� +x exp(−c +� +log x log log x) +� +. +References +[1] Fr´ed´eric Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136 +(2002), no. 3, 203–236. +[2] Fr´ed´eric Bayart, Convergence and almost sure properties in Hardy spaces of Dirichlet series, Math. Ann. +382 (2022), no. 3-4, 1485–1515. +[3] Arne Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers g´en´eralis´es. I, Acta +Math. 68 (1937), no. 1, 255–291. +[4] Harald Bohr, ¨Uber die gleichm¨aßige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math. 143 (1913), +203–211. +[5] H. Davenport and K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 160– +167. +[6] Andreas Defant and Ingo Schoolmann, Hp-theory of general Dirichlet series, J. Fourier Anal. Appl. 25 +(2019), no. 6, 3220–3258. +[7] Harold G. Diamond, When do Beurling generalized integers have a density?, J. Reine Angew. Math. 295 +(1977), 22–39. +[8] Harold G. Diamond, Hugh L. Montgomery, and Ulrike M. A. Vorhauer, Beurling primes with large oscilla- +tion, Math. Ann. 334 (2006), no. 1, 1–36. +[9] Andrew Granville, The lattice points of an n-dimensional tetrahedron, Aequationes Math. 41 (1991), no. 2-3, +234–241. +[10] G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series, Cambridge Tracts in Mathematics and +Mathematical Physics, No. 18, Stechert-Hafner, Inc., New York, 1964. +[11] H˚akan Hedenmalm, Peter Lindqvist, and Kristian Seip, A Hilbert space of Dirichlet series and systems of +dilated functions in L2(0, 1), Duke Math. J. 86 (1997), no. 1, 1–37. +[12] Henry Helson, Compact groups with ordered duals, Proc. London Math. Soc. (3) 14a (1965), 144–156. +[13] Henry Helson, Compact groups and Dirichlet series, Ark. Mat. 8 (1969), 139–143. +[14] Titus W. Hilberdink, Well-behaved Beurling primes and integers, J. Number Theory 112 (2005), no. 2, +332–344. +[15] Titus W. Hilberdink and Michel L. Lapidus, Beurling zeta functions, generalised primes, and fractal mem- +branes, Acta Appl. Math. 94 (2006), no. 1, 21–48. +[16] Athanasios Kouroupis, Composition operators and generalized primes, Proc. Amer. Math. Soc., to appear, +https://doi.org/10.1090/proc/16395. + +A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS +9 +[17] Edmund Landau, ¨Uber die gleichm¨aßige Konvergenz Dirichletscher Reihen, Math. Z. 11 (1921), no. 3-4, +317–318. +[18] Herv´e Queffelec and Martine Queffelec, Diophantine approximation and Dirichlet series, Texts and Readings +in Mathematics, vol. 80, Hindustan Book Agency, New Delhi; Springer, Singapore, [2020] ©2020, Second +edition [of 3099268]. +[19] I. Schoolmann, On Bohr’s theorem for general Dirichlet series, Math. Nachr. 293 (2020), no. 8, 1591–1612. +[20] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press, Oxford +University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown. +[21] Wen-Bin Zhang, Beurling primes with RH and Beurling primes with large oscillation, Math. Ann. 337 +(2007), no. 3, 671–704. +Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), +7491 Trondheim, Norway +Email address: athanasios.kouroupis@ntnu.no +Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), +7491 Trondheim, Norway +Email address: karl-mikael.perfekt@ntnu.no + diff --git a/69FKT4oBgHgl3EQfUC2Z/content/tmp_files/load_file.txt b/69FKT4oBgHgl3EQfUC2Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..63cc34f4f6bfa6b5fc1b50ab8c85b143291eb900 --- /dev/null +++ b/69FKT4oBgHgl3EQfUC2Z/content/tmp_files/load_file.txt @@ -0,0 +1,311 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf,len=310 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='11782v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='NT] 27 Jan 2023 A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Given a sequence of frequencies {λn}n≥1, a corresponding generalized Dirichlet series is of the form f(s) = � n≥1 ane−λns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We are interested in multiplicatively generated systems, where each number eλn arises as a finite product of some given numbers {qn}n≥1, 1 < qn → ∞, referred to as Beurling primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' In the classical case, where λn = log n, Bohr’s theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane {Re s > θ}, then it actually converges uniformly in every half-plane {Re s > θ +ε}, ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr’s condition, and therefore the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Applying our result in conjunction with work of Diamond–Montgomery–Vorhauer and Zhang, we find a system of Beurling primes for which both Bohr’s theorem and the Riemann hypothesis are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We discuss the connections between our work and Diophantine approximation with Beurling integers, as well as with a conjecture of Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Introduction For an increasing sequence of positive frequencies λ = {λn}n≥1, and a generalized Dirichlet series f(s) = � n≥1 ane−λns, the abscissas σc, σu, and σa of point-wise, uniform, and absolute convergence are defined as in the classical theory of Dirichlet series [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' In this article we wish to find sets of frequencies such that the analogue of a theorem of Bohr [4] holds: if σc(f) < ∞ and f has a bounded analytic extension to a half-plane {Re s > θ}, then σu(f) ≤ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The problem of finding frequencies for which the abscissas of bounded and uniform convergence always coincide, which originated with Bohr and Landau [17], has recently been revisited [2, 19] with the context of Hardy spaces of Dirichlet series in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Indeed, Bohr’s theorem is essentially a necessity for a satisfactory Hardy space theory, see [18, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' An important class of frequencies were introduced by Beurling [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Given an arbitrary in- creasing sequence q = {qn}n≥1, 1 < qn → ∞, such that {log qn}n≥1 is linearly independent over Q, we will denote by Nq = {νn}n≥1 the set of numbers that can be written (uniquely) as finite products with factors from q, ordered in an increasing manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The numbers qn are known as Beurling primes, and the numbers νn are Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The corresponding generalized Dirichlet series are of the form f(s) = � n≥1 anν−s n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' There are a number of criteria to guarantee the validity of Bohr’s theorem for frequencies {λn}n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Bohr’s original condition asks for the existence of c1, c2 > 0 such that (1) λn+1 − λn ≥ c1e−c2λn+1, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Landau relaxed the condition somewhat: for every δ > 0 there should be a c > 0 such that (2) λn+1 − λn ≥ ce−eδλn+1 , n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 1 2 ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT Landau’s condition was recently relaxed further by Bayart [2]: for every δ > 0 there should be a C > 0 such that (3) log �λm + λn λm − λn � + (m − n) ≤ Ceδλn, m > n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' All of these conditions are usually very difficult to check for any given Beurling system, since they involve the distances between the corresponding Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Furthermore, while it is easy to construct Beurling integers for which conditions (1)-(3) fail, it appears to be rather difficult to construct Beurling systems which do satisfy the Bohr condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' This is especially true if one wants to retain properties of the ordinary integers, such as the asympotic behaviour of the counting function Nq(x) = � νn≤x 1, see for example [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' One motivation for considering Beurling integers is to investigate the properties of the q-zeta function ζq(s) = � n≥1 ν−s n = � n≥1 1 1 − q−s n , and their interplay with the counting functions Nq(x) = � νn≤x 1, πq(x) = � qn≤x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' As an example, Beurling [3] himself showed that the condition (4) Nq(x) = ax + O( x (log x)γ ), for some γ > 3 2, implies the analogue of the prime number theorem, (5) πq(x) := � qn≤x 1 ∼ x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We refer to [15] for an overview of further developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' In Section 2 we begin with a preparatory result which is interesting in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' It states that starting with the classical set of primes numbers we can add almost any finite sequence of Beurling primes while retaining the validity of Bohr’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Let {pn}n≥1 be the sequence of ordinary prime numbers and let N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Then Bohr’s condition (1) holds for the Beurling integers generated by the primes q = {pn}n≥1 � {qj}N j=1, for almost every choice (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' , qN) ∈ (1, ∞)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Sequences of Beurling primes of the type considered in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1 previously appeared in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Our main result requires more careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Let q = {qn}n≥1 be an increasing sequence of primes such that q1 > 1 and σc(ζq) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Then, for every A > 0 there exists a sequence of Beurling primes ˜q = {˜qn}n≥1 for which Bohr’s condition (1) holds and |qn − ˜qn| ≤ q−A n , n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='2 and the work of Diamond, Montgomery, Vorhauer [8] and Zhang [21], allows us to construct a system of Beurling primes that satisfies Bohr’s theorem and the “Riemann Hypothesis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' There exists a system of Beurling primes q = {qn}n≥1 such that: A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS 3 (i) The Riemann zeta function ζq(s) has an analytic extension to Re s > 1 2, except for a simple pole at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' (ii) The Riemann zeta function has no zeros in the half–plane C 1 2 = {Re s > 1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' (iii) The prime counting πq(x) satisfies πq(x) = li(x) + O �√x � , where li(x) = x´ 2 (log u)−1 du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' (iv) Bohr’s condition holds for the associated class of generalized Dirichlet series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We remark that as a direct consequence of (iii), we have the prime number theorem (5) as well as Nq(x) ∼ ax, where a is the residue of ζq(s) at s = 1, see for example [7, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The proofs of our results investigate how well “irrational numbers” may be approximated by fractions of Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We will comment further on this kind of Diophantine approxi- mation problems in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' In Section 3 we will also return to the original motivation for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' There has been an interest in studying Hardy spaces of generalized Dirichlet series since the 60s [6, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' However, other than the ordinary integers, no examples of Beurling integers exist which simultaneously satisfy the prime number theorem (iii) and Bohr’s theorem, despite the fact that Bohr’s theorem is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Furthermore, since many aspects of the function theory of the Hardy space do not depend on the choice q of Beurling primes, the idea behind Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='3 was to find a Beurling system Nq in which we can assume the Riemann hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' As a function theoretic application, we construct an outer function f which has a zero, (6) f(s) = 1 ζq(s + 1/2 + ε), 0 < ε < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Originally, we intended to make use of (6) in order to disprove a conjecture of Helson [13] about outer functions in Hardy spaces formed from frequencies satisfying the Bohr condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Unfortunately, we did not quite succeed, since we are unable to demonstrate the convergence of (6) for Res > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' the typical proof of this relies on the Lindel¨of hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Finding a system which in addition to the properties of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='3 satisfies the Lindel¨of hypothesis seems to demand a deeper interplay between our tools and the probabilistic methods of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Throughout the article, we will be using the convention that C denotes a positive constant which may vary from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We will write that C = C(Ω) when the constant depends on the parameter Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Proof of the main results Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Suppose that {qn}n≥1 is a Beurling system such that dn := νn+1 − νn ≫ ν−C n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Then, for almost every q′ > 1 and every ε > 0, the Beurling system {qn}n≥1∪{q′} has a distance function satisfying (7) d′ n = ν′ n+1 − ν′ n ≫ ν−C ′ n+1 , n ∈ N, where C′(q′, q) = max (C, 2σc(ζq) − 1 + ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Let x0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' First we will prove that the set M of all numbers q′ ≥ x0 such that there exist infinitely many triples (j, n, m) ∈ N3 with ����(q′)j − νn νm ���� ≤ ν−C0 n ν−C0 m , C0 = σ(ζq) + ε, 4 ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT has measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Since �����q′ − � νn νm � 1 j ����� ≤ C(x0)x−j 0 ����(q′)j − νn νm ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' we have that M ⊂ lim supm,n,j Ωm,n,j, where Ωm,n,j = �� νn νm � 1 j − C(x0)x−j 0 ν−C0 n ν−C0 m , � νn νm � 1 j + C(x0)x−j 0 ν−C0 n ν−C0 m � , j, n, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The Borel–Cantelli lemma thus shows that |M| = 0, since � m≥1 � n≥1 � j≥1 |Ωm,n,j| ≤ C(x0)ζq (C0)2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Fix a number q′ ∈ [x0, ∞) \\ M such that log q′ is not in the (countable) set spanQ{log qn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Note that the set of such numbers has full measure in [x0, ∞), and that x0 > 1 is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' By construction, there are finitely many triples (j, n, m) such that (8) ����(q′)j − νn νm ���� ≤ ν−C0 n ν−C0 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' For these exceptional triples, the left-hand side is at least positive, since log q′ /∈ spanQ{log qn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Therefore ����(q′)j − νn νm ���� ≫ ν−C0 n ν−C0 m for all (j, n, m) ∈ N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Now we consider two arbitrary consecutive Beurling integers generated by the prime system {qn}n≥1 ∪ {q′}, ν′ n+1 = (q′)aνm, ν′ n = (q′)bνl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' If a = b, then l = m − 1 and ν′ n+1 − ν′ n ≫ ν−C m ≥ � ν′ n+1 �−C , by the hypothesis on the distances dn for the original Beurling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Otherwise, if, say, b < a, then ��ν′ n+1 − ν′ n �� = (q′)bνm ����(q′)a−b − νl νm ���� ≫ ν−C0 l ν−C0+1 m (q′)b ≫ � ν′ n+1 �−C ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' where C ′ = 2σc(ζq) − 1 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The proof is a direct consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' □ In order to prove Bohr’s theorem for more general Beurling systems, we need to control the constant in the distance estimate (7), which comes from the exceptional triples satisfying (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Fix a small ε > 0 and x0 ∈ (1+ε/2, 1+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Consider first any Beurling system Nρ = {νn}n≥1 generated by Beurling primes such that ρ1 > 1 + ε and σc(ζρ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' For a number σ∞ > max(2, A) to be chosen in a moment, let N = � m≥2 � n≥2 � j≥1 Ωm,n,j, where Ωm,n,j is defined as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1, Ωm,n,j = �� νn νm � 1 j − C(x0)x−j 0 ν−σ∞ n ν−σ∞ m , � νn νm � 1 j + C(x0)x−j 0 ν−σ∞ n ν−σ∞ m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS 5 Then |N| ≤ C(ε) (ζρ(σ∞) − 1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Furthermore, for x > 2, let Ix = [x − x− σ∞ 2 , x + x− σ∞ 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Note that if σ∞ is sufficiently large, σ∞ ≥ C(ε), then Ix∩Ωm,n,j ̸= ∅ only if νn ≥ x/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Therefore |Ix ∩ N| ≤ � m≥2 j≥1 νn≥ x 2 |Ωm,n,j| ≤ C(ε) (ζρ(σ∞) − 1) � νn≥ x 2 ν−σ∞ n ≤ C(ε) (ζρ(σ∞) − 1) ζρ �σ∞ 4 � x− 3σ∞ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We will construct a sequence of Beurling systems such that (9) (ζρ(σ∞) − 1) ζρ �σ∞ 4 � ≤ 1 for the number σ∞ > 0, still to be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Therefore (10) |Ix ∩ N| ≤ C(ε)x− σ∞ 4 |Ix|, We conclude that whenever x is sufficiently large, Ix ̸⊂ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' To include triples where νn or νm equals one in our considerations, we increase the power σ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The inequality (11) ����xj − νn νm ���� ≤ ν−3σ∞ n ν−3σ∞ m implies, whenever x ≥ x0, that �����x − � νn νm � 1 j ����� ≤ C(x0)x−j 0 ����xj − ν2 nνm ν2mνn ���� ≤ C(x0)x−j 0 � ν2 mνn �−σ∞ � ν2 nνm �−σ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Therefore M ⊂ N, where M this time denotes the set of all x ≥ x0 for which there exists an exceptional triple (j, n, m) ∈ N3 such that (11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Now let q be a sequence of primes in the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='2, assuming that ε < q1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' As described, we will only be able to effectively apply (10) when x is sufficiently large, say, x ≥ B = B(ε) = C(ε)4 + 2, where C(ε) in this instance refers to the same constant as in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Let N be such that {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' , qN} = (1, B) ∩ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Then, as a corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1, we already know that there exists an increasing finite sequence of primes {˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' ˜qN}, ˜q1 > 1, such that |qj − ˜qj| ≤ q−A j , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' , N, and such that Bohr’s condition holds for {ν(N) n }n≥1 = N{˜q1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='˜qN }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Further, we choose σ∞ so large that ���ν(N) n+1 − ν(N) n ��� ≥ � ν(N) n+1 �−6σ∞ , n ∈ N, and (12) (ζq′(σ∞) − 1) ζq′ �σ∞ 4 � ≤ 1, q′ = {˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' ˜qN, qN+1 − 1, qN+2 − 1, qN+3 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' This is made possible by the hypothesis that σc(ζq) < ∞, since ζq′(σ) ≤ � j≥1 1 1 − (q′ j)−σ ≤ ζq � σ C � , σ > 0, C ≥ sup n≥1 log(qn) log(q′n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' From here we proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Suppose that ˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' ˜qk have been chosen, where k ≥ N, with corresponding Beurling integers {ν(k) n }n≥1 = N{˜qn}k n=1 satisfying that ���ν(k) n+1 − ν(k) n ��� ≥ � ν(N) n+1 �−6σ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 6 ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT We apply the preceding discussion to the Beurling primes ρ = {˜q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' ˜qk} and x = qk+1, con- cluding that there exists a number ˜qk+1 ∈ Iqk+1 such that �����˜qj k+1 − ν(k) n ν(k) m ����� ≥ � ν(k) n �−3σ∞ � ν(k) m �−3σ∞ , (j, n, m) ∈ N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' By the same argument as in the last paragraph of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1 the Beurling system {ν(k+1) n }n≥1 = N{˜qn}k+1 n=1, then satisfies that ���ν(k+1) n+1 − ν(k+1) n ��� ≥ � ν(k+1) n+1 �−6σ∞ , n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' At each step of the construction, (12) ensures that (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We hence obtain a sequence ˜q = {˜qn}n≥1, satisfying that |˜qn − qn| ≤ q − σ∞ 2 n as well as Bohr’s condition (1), specifically, |˜νn+1 − ˜νn| ≥ (˜νn+1)−6σ∞ , n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' where {˜νn}n≥1 = N˜q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' □ Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' By [21, Theorem 1] there exists a Beurling system qRH that satisfies (i)–(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='2 then implies that there exist primes q, asymptotically approaching qRH, satisfying Bohr’s theorem, and such that (13) πq(x) = li(x) + O �√x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We observe that log ζq(s) = − ∞ ˆ 1 log � 1 − u−s� dπq(u), Re s > 1, see [8, Lemma 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' With f(u) = u−1 u log u, u > 1, consider the function Z(s) = exp \uf8eb \uf8ed ∞ ˆ 1 u−sf(u) du \uf8f6 \uf8f8 = s s − 1, Re s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We then have that (14) log ζq(s) Z(s) = − ∞ ˆ 1 � log � 1 − u−s� + u−s� dπq(u) + ∞ ˆ 1 u−s (dπq(u) − f(u) du) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The quantity under the first integral sign satisfies log � 1 − u−s� + u−s = O � u−2σ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Thus, the first integral in (14) is analytic and uniformly bounded for Re s > 1 2 + ε, ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' By (13), the second integral in (14) is also analytic in the half-plane C 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Therefore, ζq has an analytic continuation to the half–plane C 1 2 , except for a simple pole at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Furthermore, ζq cannot have any zeros in this half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Further discussion Diophantine approximation and Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Using the Borel–Cantelli theorem to study the irrationality of real numbers is a standard technique of Diophantine approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The irrationality measure µ(x) of a real number x ∈ R is defined as the infimum of the set Rx = � r > 0 : ���x − m n ��� < 1 nr for at most finitely many pairs (m, n) ∈ N × N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' A NOTE ON BOHR’S THEOREM FOR BEURLING INTEGER SYSTEMS 7 For a Beurling system Nq = {νn}n≥1, we may also introduce the irrationality measure µq(x) of a real number x ∈ R as the infimum of the set Rx = � r > 0 : ����x − νm νn ���� < 1 νrn for at most finitely many pairs (m, n) ∈ N × N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Then, by slightly modifying the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1, we obtain the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Let q = {qn}n≥1 be a sequence of Beurling primes with σc(ζq) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Then, for almost every x ∈ R, it holds that µq(x) ≤ 2σc(ζq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' In the classical case, Dirichlet’s approximation theorem therefore implies that µ(x) = 2 for almost every x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' We also recall Roth’s theorem [5], which states that µ(x) = 2 for every algebraic irrational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' It would be very interesting to develop corresponding results in the context of Beurling integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Hardy spaces of Dirichlet series and a conjecture of Helson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' For a sequence q of Beurling primes, we introduce the Hardy space H2 q as H2 q = \uf8f1 \uf8f2 \uf8f3f(s) = � n≥1 anν−s n : ∥f∥2 H2q = � n≥1 |an|2 < ∞ \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' More generally, for 1 ≤ p < ∞, we define Hp q as the completion of polynomials (finite sums � anν−s n ) under the Besicovitch norm ∥P∥Hp q := \uf8eb \uf8ed lim T →∞ 1 2T T ˆ −T |P(it)|p dt \uf8f6 \uf8f8 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The function theory of these spaces originated with Helson [12], and was, in the distuingished case where q is the sequence of ordinary primes, continued in very influential papers of Bayart [1] and Hedenmalm, Lindqvist, and Seip [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' More generally, there is a developing theory of Hardy spaces of Dirichlet series � ane−λns whose frequencies are related to other groups than T∞, but we shall restrict our attention to frequencies given by Beurling primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' A cornerstone of the theory is that there is a natural multiplicative linear isometric isomorphism between Hp q and the Hardy space Hp q (T∞) of the infinite torus [6, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' However, more is needed in order to identify H∞(T∞) with H∞ q , the space of Dirichlet series � anν−s n which converge to a bounded function in C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' In fact, Bohr’s condition is typically used in order to establish this isomorphism [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' In identifying Hp q with Hp q (T∞) one is naturally led to consider twisted Dirichlet series fχ(s) = � n≥1 anχ(νn)ν−s n , where a point χ ∈ T∞ is interpreted as the completely multiplicative character χ: Nq → T such that χ(qn) = χn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Helson [13] proved that if f ∈ H2 q and the associated frequencies satisfy Bohr’s condition, then fχ(s) converges in C0 for almost every χ ∈ T∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Helson went on to make a conjecture, which we state only in the special case that the frequencies correspond to a Beurling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Recall that f ∈ H2 q is said to be outer if � fg : g ∈ H∞ q � is dense in H2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' If Nq is a Beurling system that satisfies Bohr’s condition and f is outer in H2 q, then fχ never has any zeros in its half-plane of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 8 ATHANASIOS KOUROUPIS AND KARL-MIKAEL PERFEKT Suppose now that the Beurling primes q are chosen as in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='3, so that we have the “Riemann hypothesis” at our disposal, and consider the Dirichlet series f(s) = 1 ζq(s + 1/2 + ε), for some 0 < ε < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Through a routine calculation with coefficients, one checks that f, f 2, 1/f, 1/f 2 ∈ H2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Therefore, there are polynomials pn which converge to 1/f in H4 q, so that ∥1 − pnf∥H2q ≤ ∥f∥H4q∥1/f − pn∥H4q → 0, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Thus f is outer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' On the other hand, it has a zero at s = 1/2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='3 ensures that f has an analytic extension to C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The problem is that we do not know if f actually converges there, as required by Helson’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Since we have Bohr’s condition, a standard argument [20, Section 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='25] with the Perron formula shows that the Lindel¨of hypothesis for ζq implies the convergence of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' However, in the Beurling prime setting, “Riemann implies Lindel¨of” is only true when ζq has finite order in C 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' This crux further highlights one of the main issues of [14, 15]: how can we ensure that ζq is zero-free and has finite order in some half-plane Re s > 1 − η?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Or, equivalently, when do we have that πq(x) = li(x) + O(xθ1), Nq(x) = kx + O(xθ2), for some θ1, θ2 < 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' The best estimate we are able to add to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='3 comes from the general result [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='2]: there exists a number c > 0 such that Nq(x) = ax + O � x exp(−c � log x log log x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' References [1] Fr´ed´eric Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 136 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 3, 203–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' [2] Fr´ed´eric Bayart, Convergence and almost sure properties in Hardy spaces of Dirichlet series, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 382 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content=' 3-4, 1485–1515.' metadata={'source': 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Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway Email address: karl-mikael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='perfekt@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} +page_content='no' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQfUC2Z/content/2301.11782v1.pdf'} diff --git a/6NE2T4oBgHgl3EQfOwb7/content/tmp_files/2301.03753v1.pdf.txt b/6NE2T4oBgHgl3EQfOwb7/content/tmp_files/2301.03753v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1289876685e0b56cadad28b94e607fd05efb3571 --- /dev/null +++ b/6NE2T4oBgHgl3EQfOwb7/content/tmp_files/2301.03753v1.pdf.txt @@ -0,0 +1,776 @@ +arXiv:2301.03753v1 [math.NA] 10 Jan 2023 +ENFORCING NEUMANN BOUNDARY CONDITIONS WITH POLYNOMIAL +EXTENSION OPERATORS TO ACHEIVE OPTIMAL CONVERGENCE RATES ON +POLYTOPIAL MESHES IN THE FINITE ELEMENT METHOD +JAMES CHEUNG +Millennium Space Systems, A Boeing Company. 2265 E. El Segundo Blvd, El Segundo, CA. +Abstract. In [4], the authors presented two finite element methods for approximating second order bound- +ary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear map- +pings. This was done by enforcing the boundary conditions through judiciously chosen polynomial extension +operators. The H1 error estimates were proven to be optimal for the solutions of both the Dirichlet and +Neumann boundary value problems. It was also proven that the Dirichlet problem approximation converges +optimally in L2. However, optimality of the Neumann approximation in the L2 norm was left as an open +problem. In this work, we seek to close this problem by presenting new analysis that proves optimal error +estimates for the Neumann approximation in the W 1 +∞ and L2 norms. +1. Introduction +The purpose of this note is to derive optimal error estimates for the polynomial extension finite element +method (PE-FEM) described in [4] for approximating elliptic boundary value problems with Neumann +conditions. This manuscript is very much an appendix to our previous work. As such, we highly suggest +that the reader refer to that work, especially since we will not redefine our notation here for the sake of +brevity. Our analysis will be structured in the following manner: We present new technical lemmas involving +the Averaged Taylor series in §2, then in §3 we move on to prove well-posedness and derive optimal error +estimates for the numerical solution in the W 1 +∞(Ω) norm, using this new result we are finally able to derive +optimal error estimates for the numerical solution in the L2(Ω) norm in §4. We then discuss our results and +propose additional future research in §5. +2. Additional Results for Averaged Taylor Polynomials +We begin our analysis by deriving some technical results for the Averaged Taylor polynomials in the +W m +∞(Ωh) setting. +Lemma 1. Let v ∈ L∞(Ωh), then ���T k +h (v) +�� +η(ξ) +��� +L∞(Ωh) ≤ C∥v∥L∞(Ωh). +Proof. Using a scaling argument on [2, Corollary 4.1.15], we immediately have that +���T k +h (v) +�� +η(ξ) +��� +L∞(Ei∩Si,ℓ) ≤ Cδ−d +h ∥v∥L1(σi,ℓ) +≤ C∥v∥L∞(σi,ℓ) +We conclude by seeing that���T k +h (v) +�� +η(ξ) +��� +L∞(Γh) = max +i,ℓ +���T k +h (v) +�� +η(ξ) +��� +L∞(Ei∩Si,ℓ) . +□ +Lemma 2. Let v ∈ W k+1 +∞ +(Ωh). Then for an integer m < k + 1, the following is satisfied +���T k +h (v) +�� +η(ξ) − v +��� +W m +∞(Γh) ≤ Cδk+1−m +h +|v|W k+1 +∞ +(Ωh). +1 + +Proof. Using [2, Proposition 4.3.2], we have that +���T k +h (v) +�� +η(ξ) − v +��� +W m +∞(Ei∩Si,ℓ) = +��T k +h (v) − v +�� +W m +∞(η(Ei)∩Si,ℓ) +≤ +��T k +h (v) − v +�� +W m +∞(Si,ℓ) +≤ Cδk+1−m +h +|v|W k+1 +∞ +(Si,ℓ). +Taking the maximum over all i, ℓ ∈ N concludes the proof. +□ +Lemma 3. Let v ∈ V +k +h, then +����T k′,k +h +(v) +��� +η(ξ) +���� +L∞(Γh) +≤ C +k +� +α|=1 +h−|α|δ|α| +h ∥v∥L∞(Ωh) +Proof. From the definition of T k′,k +h +(v), we see directly that +����T k′,k +h +(v) +��� +η(ξ) +���� +L∞(Γh) +≤ +k +� +|α|=1 +δ|α| +h +|α|! ∥Dαv∥L∞(Γh) +≤ +k +� +|α|=1 +δ|α| +h +|α|! ∥Dαv∥L∞(Ωh) +≤ +k +� +|α|=1 +δ|α| +h h−|α| +|α|! +∥v∥L∞(Ωh) , +after applying the inverse inequality. +□ +With these technical lemmas derived, we are now ready to prove that the solution of the Neumann +approximation presented in [4] is well-posed and optimal in W 1 +∞(Ω). +3. Well-Posedness and Error Estimates in W 1 +∞(Ωh) +In this section, we determine that the solution uh ∈ V k +h is bounded in W 1 +∞(Ωh). +Additionally, we +demonstrate that the error estimate is optimal in the same norm. The analysis presented here is remarkably +standard since the perturbations incurred by the extensions in the discrete bilinear form are not large enough +to cause stability loss. Additionally, Strang’s Lemma arguments are also used to handle the error incurred +by the nonconforming perturbations in the bilinear form. +Theorem 1. Assume that δh ∼ O(h2) and that h ∈ R+ is sufficiently small. Further assume that �p, �q ∈ +Ck+1(Ωh). Then there exist positive constants γ, M ∈ R+ such that +(1) +sup +vh∈V k +h ∩W 1 +1 (Ωh) +Bh,N(wh, vh) +∥vh∥W 1 +1 (Ωh) +≥ γ∥wh∥W 1 +∞(Ωh) +and +(2) +Bh,N(wh, vh) ≤ M∥wh∥W 1 +∞(Ωh)∥vh∥W 1 +1 (Ωh) +for all uh, vh ∈ V k +h . +Proof. We begin by recalling that +Bh,N(wh, vh) = +� +Ωh +(�p∇wh · ∇vh + ˜qwhvh) dx + +� +�p ◦ η(ξ) Tk−1 +h +(∇wh) +�� +η(ξ) · n − �p(ξ)∇wh · nh, vh +� +Γh +. +We then see that +Bh,N(wh, vh) ≥ +� +Ωh +(�p∇wh · ∇vh + �qwhvh) dx − Ch|u|W 1 +∞(Ωh)∥vh∥W 1 +1 (Ωh), +2 + +after applying Lemma 3 and the trace inequality [2, Theorem 1.6.6]. Dividing both sides with ∥vh∥W 1 +1 (Ωh) +allows us to see that +sup +vh∈V k +h ∩W 1 +1 (Ωh) +Bh,N(wh, vh) +∥vh∥W 1 +1 (Ωh) +≥ +sup +vh∈V k +h ∩W 1 +1 (Ωh) +� +Ωh (�p∇wh · ∇vh + �qwhvh) dx +∥vh∥W 1 +1 (Ωh) +− Ch|u|W 1 +∞(Ωh). +Now, choosing vh = wh allows us to see that supvh∈V k +h ∩W 1 +1 (Ωh) +� +Ωh(�p∇wh·∇vh+�qwhvh)dx +∥vh∥W 1 +1 (Ωh) +> 0. As such, we can +choose a constant Cp,q > 0 such that +sup +vh∈V k +h ∩W 1 +1 (Ωh) +� +Ωh (�p∇wh · ∇vh + �qwhvh) dx +∥vh∥W 1 +1 (Ωh) +≥ Cp,q +sup +vh∈V k +h ∩W 1 +1 (Ωh) +� +Ωh (∇wh · ∇vh + whvh) dx +∥vh∥W 1 +1 (Ωh) += Cp,q∥wh∥W 1 +∞(Ωh). +Therefore, +sup +vh∈V k +h ∩W 1 +1 (Ωh) +Bh,N(wh, vh) +∥vh∥W 1 +1 (Ωh) +≥ Cp,q∥wh∥W 1 +∞Ωh − Ch∥uh∥W 1 +∞(Ωh). +And thus, we have that (1) is satisfied. +Using Lemma 1, H¨older’s inequality, and the trace inequality allows us to derive (2). +□ +Theorem 2. Let uh ∈ V k +h satisfy [4, Equation 26]. Assume that u ∈ W k+1 +∞ +(Ω) and that f ∈ W k−1 +∞ +(Ω). +Furthermore, let �u ∈ W k+1 +∞ +(Ωh) and �f, �f ∈ W k−1 +∞ +(Ωh) be extensions of u and f respectively from Ω to Ωh. +We then have that +∥�u − uh∥W 1 +∞(Ωh) ≤ Chk � +|u|W k+1 +∞ +(Ω) + ∥f∥W k−1 +∞ +(Ω) +� +under the conditions specified in Theorem 1. +Proof. Let uI ∈ V k +h be the piecewise polynomial interpolant of �u ∈ W k+1 +∞ +(Ωh) defined on Ωh. From [2, +Theorem 4.2.20] and the Stein extension theorem [1], we have that +(3) +∥�u − uI∥W 1 +∞(Ωh) ≤ Chk|u|W k+1 +∞ +(Ω). +We begin the analysis by seeing that +∥uI − uh∥W 1 +∞(Ωh) ≤ +sup +vh∈V k +h ∩W 1 +1 (Ωh) +Bh,N(uI − uh, vh) +∥vh∥W 1 +1 (Ωh) += +sup +vh∈V k +h ∩W 1 +1 (Ωh) +Bh,N(uI − �u, vh) + Bh,N(�u − uh, vh) +∥vh∥W 1 +1 (Ωh) +≤ M∥�u − uh∥W 1 +∞(Ωh) + +sup +vh∈V k +h ∩W 1 +1 (Ωh) +� +�f − �f, vh +� +Ωh +− +� +�p ◦ η(ξ)Rk−1 (∇�u)|η(ξ) · n, vh +� +Γh +∥vh∥W 1 +1 (Ωh) +≤ Chk|u|W k+1 +∞ +(Ω) + Cδk−1 +h +|f|W k−1 +∞ +(Ω) + Cδk−1 +h +|u|W k+1 +∞ +(Ω) +≤ Chk|u|W k+1 +∞ +(Ω) +after utilizing (3), Lemma 2, and seeing that +Bh,N(�u, vh) = +� +�f, vh +� +Ωh ++ +� +gN ◦ η(ξ) − �p ◦ η(ξ)Rk−1 (∇u)|η(ξ) · n, vh +� +Γh +. +The proof is completed by seeing that +∥�u − uh∥W 1 +∞(Ωh) ≤ ∥u − uI∥W 1 +∞(Ωh) + ∥uI − uh∥W 1 +∞(Ωh), +and applying the above bound along with (3). +□ +3 + +4. Error Estimates in L2(Ωh) +We are now ready to derive the optimal error estimates for the solution of the Neumann approximation +in the L2(Ω) norm. The analysis begins by estimating the nonconformity error. This nonconformity error +will then be used in the following duality argument to bound the terms in the discrete problem that are not +orthogonal in the Galerkin sense with respect to the continuous bilinear form. +4.1. Nonconformity Error. Let us begin the derivation of the L2(Ωh) error bound by analyzing the +nonconformity induced by Bh,N(·, ·) := H1(Ωh) × V k +h → R+. +Lemma 4. Assume that all the conditions in Theorem 2 hold, then the following is satisfied +Nh(�u − uh, vh) ≤ Chk+1 � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +(Ω) +� +∥vh∥1,Ωh +for all V k +h ∩ W 1 +1 (Ωh). +Proof. Notice that +Bh,N(�u, vh) = +� +�f, vh +� +Ωh ++ +� +gN ◦ η(ξ) − �p ◦ η(ξ) Rk−1 +h +(∇�u) +�� +η(ξ) · n, vh +� +Γh +∀v ∈ H1(Ωh). +Taking the difference with [4, Equation (24)] yields +Bh,N(�u − uh, vh) = +� +�f − �f, vh +� +Ωh +− +� +�p ◦ η(ξ) Rk−1 +h +(∇�u) +�� +η(ξ) · n, vh +� +Γh +∀vh ∈ V k +h . +Let us define eh := �u − uh, then from the definition of Bh,N(·, ·) (See [4, Equation (25)]), we have that +(4) +Nh(eh, vh) = +� +�f − �f, vh +� +Ωh +− +� +�p ◦ η(ξ) Rk−1 +h +(∇�u) +�� +η(ξ) · n, vh +� +Γh +− +� +�p ◦ η(ξ) Tk−1 +h +(∇eh) +�� +η(ξ) · n, vh +� +Γh ++ ⟨�p(ξ)∇eh · nh, vh⟩Γh += +� +�f − �f, vh +� +Ωh +− +� +�p ◦ η(ξ) Rk−1 +h +(∇�u) +�� +η(ξ) · n, vh +� +Γh ++ ⟨�p ◦ η(ξ)∇eh · (n − nh), vh⟩Γh + +� +�p(ξ) T1,k−1 +h +(∇eh) +��� +η(ξ) · n, vh +� +Γh ++ +� +(�p(ξ) − �p ◦ η(ξ)) Tk−1 +h +(∇eh) +�� +η(ξ) · n, vh +� +Γh +. +We will now analyze each of the terms on the right hand side of (4) seperately. +Let Ωh +diff := Ωh \ (Ω ∩ Ωh), then we have by the definition of the extension that +� +�f − �f, vh +� +Ωh += +� +�f − �f, vh +� +Ωh +diff +≤ +� +max +i,ℓ ∥ �f − T k−1 +h +f∥L∞(Si,ℓ) + max +i,ℓ ∥ �f − T k−1 +h +f∥L∞(Si,ℓ) +� � +Ωh +diff +1 · |v|dx +≤ Cδk−1 +h +|Ωh +diff| +1 +2 |f|W k−1 +∞ +(Ωh)∥vh∥0,Ωh +≤ Cδ +k− 1 +2 +h +|f|W k−1 +∞ +(Ωh)∥vh∥1,Ωh, +after applying [2, Lemma 4.3.8], H¨older’s inequality, and seeing that |Ωh +diff| ∼ O(δh). Since we have assumed +that δh ∼ O(h2), we have that +(5) +� +�f − �f, vh +� +Ωh +≤ Ch2k−1|f|W k−1 +∞ +(Ωh)∥vh∥1,Ωh. +Next, using Lemma 2, we have that +(6) +� +�p ◦ η(ξ) Rk−1 +h +(∇�u) +�� +η(ξ) · n, vh +� +Γh +≤ p +���Rk−1 +h +∇�u +�� +η(ξ) +��� +L∞(Γh) ∥vh∥L1(Γh) +≤ Cδk +h|∇�u|W k +∞(Ω)∥vh∥0,Γh +≤ Ch2k|u|W k+1 +∞ +(Ω)∥vh∥1,Ωh, +4 + +after applying the trace theorem and the assumption that δh ∼ O(h2). +Additionally, we have that +(7) +⟨�p ◦ η(ξ)∇eh · (n − nh), vh⟩Γh ≤ p +� +max +ξ∈Γh ∥n ◦ η(ξ) − nh(ξ)∥Rd +� +∥∇eh∥L∞(Γh)∥vh∥L1(Γh) +≤ Ch∥eh∥W 1 +∞(Ωh)∥vh∥1,Ωh +≤ Chk+1 � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +(Ω) +� +∥vh∥1,Ωh, +after applying Theorem 2. +Then, we have that +(8) +� +�p(ξ) T1,k−1 +h +(∇eh) +��� +η(ξ) · n, vh +� +Γh +≤ Cp∥vh∥1,1Ωh +k−1 +� +|α|=1 +δ|α| +h ∥Dα∇eh∥L∞(Γh) +≤ Cp∥vh∥1,1Ωh +k−1 +� +|α|=1 +δ|α| +h +� +∥Dα∇(�u − uI)∥L∞(Ωh) + ∥Dα∇(uI − uh)∥L∞(Ωh) +� +≤ Cp∥vh∥1,1Ωh +k−1 +� +|α|=1 +δ|α| +h +� +hk−|α||u|W k+1 +∞ +(Ω) + Ch−|α| � +∥∇(eh)∥L∞(Ωh) − ∥∇(�u − uI)∥L∞(Ωh) +�� +≤ Cp∥vh∥1,1Ωh +k−1 +� +|α|=1 +δ|α| +h hk−|α| � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +(Ω) +� +≤ Chk+1 � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +(Ω) +� +∥vh∥1,Ωh +after applying Theorem 2 and the interpolation estimate [2, Theorem 4.2.20]. +We finally move on to the last term, where we see that +(9) +� +(�p(ξ) − �p ◦ η(ξ)) Tk−1 +h +(∇eh) +�� +η(ξ) · n, vh +� +Γh +≤ Cδh +���Tk−1 +h +(∇eh) +�� +η(ξ) +��� +L∞(Γh) ∥vh∥L1(Γh) +≤ Cδh∥∇eh∥L∞(Ωh)∥vh∥1,Ωh +≤ Chk+2 � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +(Ω) +� +∥vh∥1,Ωh, +where we used Lemma 1 and Theorem 2. +Inserting (5), (6), (7), (8), and (9) into (4) gives us +Nh(�u − uh, vh) ≤ Chk+1 � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +(Ω) +� +∥vh∥1,Ωh +∀vh ∈ V k +h +This concludes this proof. +□ +Now that we have established that the nonconformity in the bilinear form is bounded above by O(hk+1), +we are now ready to analyze the L2(Ω) error of the numerical solution. +4.2. The Dual Problem. The dual problem we are interested in utilizing is to seek a ψ ∈ H1(Ωh) such +that +Nh(v, ψ) = ⟨�u − uh, v⟩Ωh +∀v ∈ H1(Ωh). +The corresponding finite element approximation to the dual problem is to seek a ψh ∈ V k +h such that +Nh(v, ψh) = ⟨�u − uh, v⟩Ωh +∀vh ∈ V k +h . +In general, φ ∈ H1(Ωh) does not full H2(Ωh) regularity due to the presence of interior angles in Γh. From +the literature, [2, 5] it is known that +(10) +∥ψ∥1+s,Ωh ≤ C∥u − uh∥0,Ωh +5 + +and +(11) +∥φ∥1,Ωh ≤ C∥u − uh∥−1,Ωh ≤ C∥u − uh∥0,Ωh. +Furthermore, we have that +(12) +∥ψ − ψh∥1,Ωh ≤ Chs|φ|1+s,Ωh, +where s ∈ +� 1 +2, 1 +� +depends on the magnitude of the largest interior angle of Γh. See [4, Remark 7]. +4.3. Analysis of L2(Ωh) Convergence. With the previous results proven, we are now ready to derive op- +timal L2(Ωh) error estimates for the PE-FEM approximation of elliptic Neumann boundary value problems. +Theorem 3. Assume that all the conditions in Theorem 2 are satisfied. Then we have that +∥�u − uh∥0,Ωh ≤ Chk+s � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +Ω +� +, +where s ∈ +� 1 +2, 1 +� +depends on the largest interior angle of Γh. If Γh is convex, then s = 1. +Proof. From the definition of the dual problem, we have that +∥�u − uh∥2 +0,Ωh = Nh(�u − uh, φ) += Nh(�u − uh, φ − φh) + Nh(�u − uh, φh) +≤ C∥�u − uh∥1,Ωh∥φ − φh∥1,Ωh + Nh(�u − uh, φh) +([4, Theorem 2]) +≤ Chk+s (|u|k+1,Ωh + |f|k−1,Ωh) |φ|2,Ωh + Nh(�u − uh, φh) +(12) and [4, Theorem 5] +≤ Chk+s (|u|k+1,Ωh + |f|k−1,Ωh) |φ|2,Ωh ++ Chk+1 � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +Ω +� +∥φh∥1,Ωh +Lemma 4 +≤ Chk+s � +|u|W k+1 +∞ +(Ω) + |f|W k−1 +∞ +Ω +� +∥�u − uh∥0,Ωh +(10) and (11). +This concludes this proof. +□ +5. Conclusion +In this manuscript, we have presented new results that indicate that the solution of the Neumann ap- +proximation presented in [4] is optimal in W 1 +∞(Ω) and L2(Ω). Our analysis implies hat the solution to the +continuous Neumann problem must be pointwise bounded a.e. in its (k + 1)-th derivative in order for us to +achieve additional accuracy in the L2(Ω) norm, otherwise the L2(Ω) error is bounded by the O(hk) H1(Ω) +error. +In our future work, we plan on utilizing these results in the derivation of optimal L2(Ω) × L2(Ω) error +estimates for the interface coupling method presented in [3]. We will then aim to establish a generalized +theory for the well-posedness and approximation properties of extension boundary methods. We believe that +this approach can be generalized across many types of partial differential equations since the work presented +in this manuscript and in [4] indicates that the averaged Taylor series approximation only generates an O(h) +perturbation to the discretized variational operator that vanishes as h → 0. +References +[1] Robert A Adams and John JF Fournier. Sobolev spaces. Elsevier, 2003. +[2] Susanne C Brenner, L Ridgway Scott, and L Ridgway Scott. The mathematical theory of finite element methods, volume 3. +Springer, 2008. +[3] James Cheung, Max Gunzburger, Pavel Bochev, and Mauro Perego. An optimally convergent higher-order finite element +coupling method for interface and domain decomposition problems. Results in Applied Mathematics, 6:100094, 2020. +[4] James Cheung, Mauro Perego, Pavel Bochev, and Max Gunzburger. Optimally accurate higher-order finite element methods +for polytopial approximations of domains with smooth boundaries. Mathematics of Computation, 88(319):2187–2219, 2019. +[5] Philippe G Ciarlet. The finite element method for elliptic problems. SIAM, 2002. +6 + diff --git a/6NE2T4oBgHgl3EQfOwb7/content/tmp_files/load_file.txt b/6NE2T4oBgHgl3EQfOwb7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b49c2978f69516c5b765b4c61445e922a84927a --- /dev/null +++ b/6NE2T4oBgHgl3EQfOwb7/content/tmp_files/load_file.txt @@ -0,0 +1,184 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf,len=183 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='03753v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='NA] 10 Jan 2023 ENFORCING NEUMANN BOUNDARY CONDITIONS WITH POLYNOMIAL EXTENSION OPERATORS TO ACHEIVE OPTIMAL CONVERGENCE RATES ON POLYTOPIAL MESHES IN THE FINITE ELEMENT METHOD JAMES CHEUNG Millennium Space Systems, A Boeing Company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 2265 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' El Segundo Blvd, El Segundo, CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' In [4], the authors presented two finite element methods for approximating second order bound- ary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear map- pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' This was done by enforcing the boundary conditions through judiciously chosen polynomial extension operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The H1 error estimates were proven to be optimal for the solutions of both the Dirichlet and Neumann boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' It was also proven that the Dirichlet problem approximation converges optimally in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' However, optimality of the Neumann approximation in the L2 norm was left as an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' In this work, we seek to close this problem by presenting new analysis that proves optimal error estimates for the Neumann approximation in the W 1 ∞ and L2 norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Introduction The purpose of this note is to derive optimal error estimates for the polynomial extension finite element method (PE-FEM) described in [4] for approximating elliptic boundary value problems with Neumann conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' This manuscript is very much an appendix to our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' As such, we highly suggest that the reader refer to that work, especially since we will not redefine our notation here for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Our analysis will be structured in the following manner: We present new technical lemmas involving the Averaged Taylor series in §2, then in §3 we move on to prove well-posedness and derive optimal error estimates for the numerical solution in the W 1 ∞(Ω) norm, using this new result we are finally able to derive optimal error estimates for the numerical solution in the L2(Ω) norm in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We then discuss our results and propose additional future research in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Additional Results for Averaged Taylor Polynomials We begin our analysis by deriving some technical results for the Averaged Taylor polynomials in the W m ∞(Ωh) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Let v ∈ L∞(Ωh), then ���T k h (v) �� η(ξ) ��� L∞(Ωh) ≤ C∥v∥L∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Using a scaling argument on [2, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='15], we immediately have that ���T k h (v) �� η(ξ) ��� L∞(Ei∩Si,ℓ) ≤ Cδ−d h ∥v∥L1(σi,ℓ) ≤ C∥v∥L∞(σi,ℓ) We conclude by seeing that���T k h (v) �� η(ξ) ��� L∞(Γh) = max i,ℓ ���T k h (v) �� η(ξ) ��� L∞(Ei∩Si,ℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Let v ∈ W k+1 ∞ (Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Then for an integer m < k + 1, the following is satisfied ���T k h (v) �� η(ξ) − v ��� W m ∞(Γh) ≤ Cδk+1−m h |v|W k+1 ∞ (Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Using [2, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='2], we have that ���T k h (v) �� η(ξ) − v ��� W m ∞(Ei∩Si,ℓ) = ��T k h (v) − v �� W m ∞(η(Ei)∩Si,ℓ) ≤ ��T k h (v) − v �� W m ∞(Si,ℓ) ≤ Cδk+1−m h |v|W k+1 ∞ (Si,ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Taking the maximum over all i, ℓ ∈ N concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Let v ∈ V k h, then ����T k′,k h (v) ��� η(ξ) ���� L∞(Γh) ≤ C k � α|=1 h−|α|δ|α| h ∥v∥L∞(Ωh) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' From the definition of T k′,k h (v), we see directly that ����T k′,k h (v) ��� η(ξ) ���� L∞(Γh) ≤ k � |α|=1 δ|α| h |α|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' ∥Dαv∥L∞(Γh) ≤ k � |α|=1 δ|α| h |α|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' ∥Dαv∥L∞(Ωh) ≤ k � |α|=1 δ|α| h h−|α| |α|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' ∥v∥L∞(Ωh) , after applying the inverse inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' □ With these technical lemmas derived, we are now ready to prove that the solution of the Neumann approximation presented in [4] is well-posed and optimal in W 1 ∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Well-Posedness and Error Estimates in W 1 ∞(Ωh) In this section, we determine that the solution uh ∈ V k h is bounded in W 1 ∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Additionally, we demonstrate that the error estimate is optimal in the same norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The analysis presented here is remarkably standard since the perturbations incurred by the extensions in the discrete bilinear form are not large enough to cause stability loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Additionally, Strang’s Lemma arguments are also used to handle the error incurred by the nonconforming perturbations in the bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Assume that δh ∼ O(h2) and that h ∈ R+ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Further assume that �p, �q ∈ Ck+1(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Then there exist positive constants γ, M ∈ R+ such that (1) sup vh∈V k h ∩W 1 1 (Ωh) Bh,N(wh, vh) ∥vh∥W 1 1 (Ωh) ≥ γ∥wh∥W 1 ∞(Ωh) and (2) Bh,N(wh, vh) ≤ M∥wh∥W 1 ∞(Ωh)∥vh∥W 1 1 (Ωh) for all uh, vh ∈ V k h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We begin by recalling that Bh,N(wh, vh) = � Ωh (�p∇wh · ∇vh + ˜qwhvh) dx + � �p ◦ η(ξ) Tk−1 h (∇wh) �� η(ξ) · n − �p(ξ)∇wh · nh, vh � Γh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We then see that Bh,N(wh, vh) ≥ � Ωh (�p∇wh · ∇vh + �qwhvh) dx − Ch|u|W 1 ∞(Ωh)∥vh∥W 1 1 (Ωh), 2 after applying Lemma 3 and the trace inequality [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Dividing both sides with ∥vh∥W 1 1 (Ωh) allows us to see that sup vh∈V k h ∩W 1 1 (Ωh) Bh,N(wh, vh) ∥vh∥W 1 1 (Ωh) ≥ sup vh∈V k h ∩W 1 1 (Ωh) � Ωh (�p∇wh · ∇vh + �qwhvh) dx ∥vh∥W 1 1 (Ωh) − Ch|u|W 1 ∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Now, choosing vh = wh allows us to see that supvh∈V k h ∩W 1 1 (Ωh) � Ωh(�p∇wh·∇vh+�qwhvh)dx ∥vh∥W 1 1 (Ωh) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' As such, we can choose a constant Cp,q > 0 such that sup vh∈V k h ∩W 1 1 (Ωh) � Ωh (�p∇wh · ∇vh + �qwhvh) dx ∥vh∥W 1 1 (Ωh) ≥ Cp,q sup vh∈V k h ∩W 1 1 (Ωh) � Ωh (∇wh · ∇vh + whvh) dx ∥vh∥W 1 1 (Ωh) = Cp,q∥wh∥W 1 ∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Therefore, sup vh∈V k h ∩W 1 1 (Ωh) Bh,N(wh, vh) ∥vh∥W 1 1 (Ωh) ≥ Cp,q∥wh∥W 1 ∞Ωh − Ch∥uh∥W 1 ∞(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' And thus, we have that (1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Using Lemma 1, H¨older’s inequality, and the trace inequality allows us to derive (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Let uh ∈ V k h satisfy [4, Equation 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Assume that u ∈ W k+1 ∞ (Ω) and that f ∈ W k−1 ∞ (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Furthermore, let �u ∈ W k+1 ∞ (Ωh) and �f, �f ∈ W k−1 ∞ (Ωh) be extensions of u and f respectively from Ω to Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We then have that ∥�u − uh∥W 1 ∞(Ωh) ≤ Chk � |u|W k+1 ∞ (Ω) + ∥f∥W k−1 ∞ (Ω) � under the conditions specified in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Let uI ∈ V k h be the piecewise polynomial interpolant of �u ∈ W k+1 ∞ (Ωh) defined on Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' From [2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='20] and the Stein extension theorem [1], we have that (3) ∥�u − uI∥W 1 ∞(Ωh) ≤ Chk|u|W k+1 ∞ (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We begin the analysis by seeing that ∥uI − uh∥W 1 ∞(Ωh) ≤ sup vh∈V k h ∩W 1 1 (Ωh) Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='N(uI − uh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh) ∥vh∥W 1 1 (Ωh) = sup vh∈V k h ∩W 1 1 (Ωh) Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='N(uI − �u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh) + Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='N(�u − uh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh) ∥vh∥W 1 1 (Ωh) ≤ M∥�u − uh∥W 1 ∞(Ωh) + sup vh∈V k h ∩W 1 1 (Ωh) � �f − �f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Ωh − � �p ◦ η(ξ)Rk−1 (∇�u)|η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Γh ∥vh∥W 1 1 (Ωh) ≤ Chk|u|W k+1 ∞ (Ω) + Cδk−1 h |f|W k−1 ∞ (Ω) + Cδk−1 h |u|W k+1 ∞ (Ω) ≤ Chk|u|W k+1 ∞ (Ω) after utilizing (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Lemma 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' and seeing that Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='N(�u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh) = � �f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Ωh + � gN ◦ η(ξ) − �p ◦ η(ξ)Rk−1 (∇u)|η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Γh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The proof is completed by seeing that ∥�u − uh∥W 1 ∞(Ωh) ≤ ∥u − uI∥W 1 ∞(Ωh) + ∥uI − uh∥W 1 ∞(Ωh), and applying the above bound along with (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' □ 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Error Estimates in L2(Ωh) We are now ready to derive the optimal error estimates for the solution of the Neumann approximation in the L2(Ω) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The analysis begins by estimating the nonconformity error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' This nonconformity error will then be used in the following duality argument to bound the terms in the discrete problem that are not orthogonal in the Galerkin sense with respect to the continuous bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Nonconformity Error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Let us begin the derivation of the L2(Ωh) error bound by analyzing the nonconformity induced by Bh,N(·, ·) := H1(Ωh) × V k h → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Assume that all the conditions in Theorem 2 hold, then the following is satisfied Nh(�u − uh, vh) ≤ Chk+1 � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ (Ω) � ∥vh∥1,Ωh for all V k h ∩ W 1 1 (Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Notice that Bh,N(�u, vh) = � �f, vh � Ωh + � gN ◦ η(ξ) − �p ◦ η(ξ) Rk−1 h (∇�u) �� η(ξ) · n, vh � Γh ∀v ∈ H1(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Taking the difference with [4, Equation (24)] yields Bh,N(�u − uh, vh) = � �f − �f, vh � Ωh − � �p ◦ η(ξ) Rk−1 h (∇�u) �� η(ξ) · n, vh � Γh ∀vh ∈ V k h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Let us define eh := �u − uh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' then from the definition of Bh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='N(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' ·) (See [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Equation (25)]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' we have that (4) Nh(eh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh) = � �f − �f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Ωh − � �p ◦ η(ξ) Rk−1 h (∇�u) �� η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Γh − � �p ◦ η(ξ) Tk−1 h (∇eh) �� η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Γh + ⟨�p(ξ)∇eh · nh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh⟩Γh = � �f − �f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Ωh − � �p ◦ η(ξ) Rk−1 h (∇�u) �� η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Γh + ⟨�p ◦ η(ξ)∇eh · (n − nh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh⟩Γh + � �p(ξ) T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='k−1 h (∇eh) ��� η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Γh + � (�p(ξ) − �p ◦ η(ξ)) Tk−1 h (∇eh) �� η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Γh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We will now analyze each of the terms on the right hand side of (4) seperately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Let Ωh diff := Ωh \\ (Ω ∩ Ωh), then we have by the definition of the extension that � �f − �f, vh � Ωh = � �f − �f, vh � Ωh diff ≤ � max i,ℓ ∥ �f − T k−1 h f∥L∞(Si,ℓ) + max i,ℓ ∥ �f − T k−1 h f∥L∞(Si,ℓ) � � Ωh diff 1 · |v|dx ≤ Cδk−1 h |Ωh diff| 1 2 |f|W k−1 ∞ (Ωh)∥vh∥0,Ωh ≤ Cδ k− 1 2 h |f|W k−1 ∞ (Ωh)∥vh∥1,Ωh, after applying [2, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='8], H¨older’s inequality, and seeing that |Ωh diff| ∼ O(δh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Since we have assumed that δh ∼ O(h2), we have that (5) � �f − �f, vh � Ωh ≤ Ch2k−1|f|W k−1 ∞ (Ωh)∥vh∥1,Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Next, using Lemma 2, we have that (6) � �p ◦ η(ξ) Rk−1 h (∇�u) �� η(ξ) · n, vh � Γh ≤ p ���Rk−1 h ∇�u �� η(ξ) ��� L∞(Γh) ∥vh∥L1(Γh) ≤ Cδk h|∇�u|W k ∞(Ω)∥vh∥0,Γh ≤ Ch2k|u|W k+1 ∞ (Ω)∥vh∥1,Ωh, 4 after applying the trace theorem and the assumption that δh ∼ O(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Additionally, we have that (7) ⟨�p ◦ η(ξ)∇eh · (n − nh), vh⟩Γh ≤ p � max ξ∈Γh ∥n ◦ η(ξ) − nh(ξ)∥Rd � ∥∇eh∥L∞(Γh)∥vh∥L1(Γh) ≤ Ch∥eh∥W 1 ∞(Ωh)∥vh∥1,Ωh ≤ Chk+1 � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ (Ω) � ∥vh∥1,Ωh, after applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' we have that (8) � �p(ξ) T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='k−1 h (∇eh) ��� η(ξ) · n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' vh � Γh ≤ Cp∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='1Ωh k−1 � |α|=1 δ|α| h ∥Dα∇eh∥L∞(Γh) ≤ Cp∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='1Ωh k−1 � |α|=1 δ|α| h � ∥Dα∇(�u − uI)∥L∞(Ωh) + ∥Dα∇(uI − uh)∥L∞(Ωh) � ≤ Cp∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='1Ωh k−1 � |α|=1 δ|α| h � hk−|α||u|W k+1 ∞ (Ω) + Ch−|α| � ∥∇(eh)∥L∞(Ωh) − ∥∇(�u − uI)∥L∞(Ωh) �� ≤ Cp∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='1Ωh k−1 � |α|=1 δ|α| h hk−|α| � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ (Ω) � ≤ Chk+1 � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ (Ω) � ∥vh∥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='Ωh after applying Theorem 2 and the interpolation estimate [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We finally move on to the last term, where we see that (9) � (�p(ξ) − �p ◦ η(ξ)) Tk−1 h (∇eh) �� η(ξ) · n, vh � Γh ≤ Cδh ���Tk−1 h (∇eh) �� η(ξ) ��� L∞(Γh) ∥vh∥L1(Γh) ≤ Cδh∥∇eh∥L∞(Ωh)∥vh∥1,Ωh ≤ Chk+2 � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ (Ω) � ∥vh∥1,Ωh, where we used Lemma 1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Inserting (5), (6), (7), (8), and (9) into (4) gives us Nh(�u − uh, vh) ≤ Chk+1 � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ (Ω) � ∥vh∥1,Ωh ∀vh ∈ V k h This concludes this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' □ Now that we have established that the nonconformity in the bilinear form is bounded above by O(hk+1), we are now ready to analyze the L2(Ω) error of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The Dual Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The dual problem we are interested in utilizing is to seek a ψ ∈ H1(Ωh) such that Nh(v, ψ) = ⟨�u − uh, v⟩Ωh ∀v ∈ H1(Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The corresponding finite element approximation to the dual problem is to seek a ψh ∈ V k h such that Nh(v, ψh) = ⟨�u − uh, v⟩Ωh ∀vh ∈ V k h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' In general, φ ∈ H1(Ωh) does not full H2(Ωh) regularity due to the presence of interior angles in Γh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' From the literature, [2, 5] it is known that (10) ∥ψ∥1+s,Ωh ≤ C∥u − uh∥0,Ωh 5 and (11) ∥φ∥1,Ωh ≤ C∥u − uh∥−1,Ωh ≤ C∥u − uh∥0,Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Furthermore, we have that (12) ∥ψ − ψh∥1,Ωh ≤ Chs|φ|1+s,Ωh, where s ∈ � 1 2, 1 � depends on the magnitude of the largest interior angle of Γh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' See [4, Remark 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Analysis of L2(Ωh) Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' With the previous results proven, we are now ready to derive op- timal L2(Ωh) error estimates for the PE-FEM approximation of elliptic Neumann boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Assume that all the conditions in Theorem 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Then we have that ∥�u − uh∥0,Ωh ≤ Chk+s � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ Ω � , where s ∈ � 1 2, 1 � depends on the largest interior angle of Γh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' If Γh is convex, then s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' From the definition of the dual problem, we have that ∥�u − uh∥2 0,Ωh = Nh(�u − uh, φ) = Nh(�u − uh, φ − φh) + Nh(�u − uh, φh) ≤ C∥�u − uh∥1,Ωh∥φ − φh∥1,Ωh + Nh(�u − uh, φh) ([4, Theorem 2]) ≤ Chk+s (|u|k+1,Ωh + |f|k−1,Ωh) |φ|2,Ωh + Nh(�u − uh, φh) (12) and [4, Theorem 5] ≤ Chk+s (|u|k+1,Ωh + |f|k−1,Ωh) |φ|2,Ωh + Chk+1 � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ Ω � ∥φh∥1,Ωh Lemma 4 ≤ Chk+s � |u|W k+1 ∞ (Ω) + |f|W k−1 ∞ Ω � ∥�u − uh∥0,Ωh (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' This concludes this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Conclusion In this manuscript, we have presented new results that indicate that the solution of the Neumann ap- proximation presented in [4] is optimal in W 1 ∞(Ω) and L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Our analysis implies hat the solution to the continuous Neumann problem must be pointwise bounded a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' in its (k + 1)-th derivative in order for us to achieve additional accuracy in the L2(Ω) norm, otherwise the L2(Ω) error is bounded by the O(hk) H1(Ω) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' In our future work, we plan on utilizing these results in the derivation of optimal L2(Ω) × L2(Ω) error estimates for the interface coupling method presented in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We will then aim to establish a generalized theory for the well-posedness and approximation properties of extension boundary methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' We believe that this approach can be generalized across many types of partial differential equations since the work presented in this manuscript and in [4] indicates that the averaged Taylor series approximation only generates an O(h) perturbation to the discretized variational operator that vanishes as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' References [1] Robert A Adams and John JF Fournier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Elsevier, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' [2] Susanne C Brenner, L Ridgway Scott, and L Ridgway Scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The mathematical theory of finite element methods, volume 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' [3] James Cheung, Max Gunzburger, Pavel Bochev, and Mauro Perego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' An optimally convergent higher-order finite element coupling method for interface and domain decomposition problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Results in Applied Mathematics, 6:100094, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' [4] James Cheung, Mauro Perego, Pavel Bochev, and Max Gunzburger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Optimally accurate higher-order finite element methods for polytopial approximations of domains with smooth boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' Mathematics of Computation, 88(319):2187–2219, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' [5] Philippe G Ciarlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' The finite element method for elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' SIAM, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE2T4oBgHgl3EQfOwb7/content/2301.03753v1.pdf'} diff --git a/7dAzT4oBgHgl3EQfEvrf/content/tmp_files/2301.01000v1.pdf.txt b/7dAzT4oBgHgl3EQfEvrf/content/tmp_files/2301.01000v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..770f94391b4c78f40d0f8af0328d2b2e270707a1 --- /dev/null +++ b/7dAzT4oBgHgl3EQfEvrf/content/tmp_files/2301.01000v1.pdf.txt @@ -0,0 +1,1816 @@ +Draft version January 4, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Deep Synoptic Array science: a 50 Mpc fast radio burst constrains the mass of the Milky Way +circumgalactic medium +Vikram Ravi,1, 2 Morgan Catha,2 Ge Chen,1 Liam Connor,1 James M. Cordes,3, 4 Jakob T. Faber,1 +James W. Lamb,2 Gregg Hallinan,1, 2 Charlie Harnach,2 Greg Hellbourg,1, 2 Rick Hobbs,2 David Hodge,1 +Mark Hodges,2 Casey Law,1, 2 Paul Rasmussen,2 Kritti Sharma,1 Myles B. Sherman,1 Jun Shi,1 Dana Simard,1 +Jean J. Somalwar,1 Reynier Squillace,1 Sander Weinreb,1 David P. Woody,2 Nitika Yadlapalli,1 +(The Deep Synoptic Array team) +1Cahill Center for Astronomy and Astrophysics, MC 249-17 California Institute of Technology, Pasadena CA 91125, USA. +2Owens Valley Radio Observatory, California Institute of Technology, Big Pine CA 93513, USA. +3Department of Astronomy, Cornell University, Ithaca, NY 14853, USA +4Cornell Center for Astrophysics and Planetary Science and Department of Astronomy, Cornell University, Ithaca, NY 14853, USA +ABSTRACT +We present the Deep Synoptic Array (DSA-110) discovery and interferometric localization of the so +far non-repeating FRB 20220319D. The FRB originates in a young, rapidly star-forming barred spiral +galaxy, IRAS 02044+7048, at just 50 Mpc. Although the NE2001 and YMW16 models for the Galactic +interstellar-medium (ISM) contribution to the DM of FRB 20220319D exceed its total observed DM, +we show that uncertainties in these models accommodate an extragalactic origin for the burst. We +derive a conservative upper limit on the DM contributed by the circumgalactic medium (CGM) of +the Milky Way: the limit is either 28.7 pc cm−3 and 47.3 pc cm−3, depending on which of two pulsars +nearby on the sky to FRB 20220319D is used to estimate the ISM DM. These limits both imply that +the total Galactic CGM mass is < 1011M⊙, and that the baryonic mass of the Milky Way is ≲ 60% of +the cosmological average given the total halo mass. More stringent albeit less conservative constraints +are possible when the DMs of pulsars in the distant globular cluster M53 are additionally considered. +Although our constraints are sensitive to possible anisotropy in the CGM and to the assumed form of +the radial-density profile, they are not subject to uncertainties in the chemical and thermal properties of +the CGM. Our results strongly support scenarios commonly predicted by galaxy-formation simulations +wherein feedback processes expel baryonic matter from the halos of galaxies like the Milky Way. +Keywords: Barred spiral galaxies — circumgalactic medium — radio interferometers — radio transient +sources — star formation — warm ionized medium +1. INTRODUCTION +Galaxies like the Milky Way are embedded in a multi- +phase (∼ 104 − 107 K), highly ionized (hydrogen neu- +tral fractions ≪ 0.01%), kinematically complex, spa- +tially clumpy and anisotropic circumgalactic medium +(CGM; Tumlinson et al. 2017). Likely extending beyond +dark-matter halo virial radii rvir, the CGM may repre- +sent the dominant baryon component by mass within +Corresponding author: Vikram Ravi +vikram@caltech.edu +halos. +The physical properties of the CGM of exter- +nal galaxies, including density, temperature, kinematic +structure, and chemical composition have long been +probed by absorption-line measurements towards back- +ground objects. +More recently, detections of thermal +bremsstrahlung (e.g., Li et al. 2018) and the Sunyaev- +Zeldovich effect around nearby galaxies (e.g., Bregman +et al. 2022) provide independent constraints on the +CGM density and temperature structure, potentially +confirming the presence of extended coronae around +galaxies. +Evidence for extended gas reservoirs associ- +ated with nearby galaxies, either in the CGM or the +intra-group medium, has also been observed in the dis- +arXiv:2301.01000v1 [astro-ph.GA] 3 Jan 2023 + +2 +Ravi et al. +persion measures (DMs) of background fast radio bursts +(FRBs; Connor & Ravi 2022; Wu & McQuinn 2022). +Galaxy-formation simulations highlight the dependence +of CGM properties on feedback from stellar winds, su- +pernovae, and AGN, as well as on the galaxy merger his- +tories (e.g., Wijers et al. 2020; Hafen et al. 2020; Zheng +et al. 2020; Fielding et al. 2020; Appleby et al. 2021; +Ramesh et al. 2022). The simulations predict varying +degrees of anisotropy in the CGM of individual galax- +ies, and different total CGM masses for galaxies with +comparable global characteristics. A ubiquitous predic- +tion of simulations is that feedback processes determine +the integrated CGM mass, MCGM. +In general, CGM +baryon fractions fCGM = MCGMΩM +MtotΩb +≲ 0.5 are predicted, +where Mtot is the total (dark matter and baryonic) halo +mass. +A halo of ∼ 106 K gas at densities ≲ 10−3 cm−3 sur- +rounding the Milky Way was proposed by Spitzer (1956) +as a solution to the problem of confining distant cold- +gas clouds at high Galactic latitudes. +Tentative evi- +dence for such coronal gas had previously been found +in observations of the diffuse radio-synchrotron back- +ground (e.g., Baldwin 1956). +A theoretical basis for +the origins of such a hot CGM around galaxies was de- +veloped in the 1970s (Rees & Ostriker 1977; White & +Rees 1978), with the gas infalling onto dark-matter ha- +los heated to virial temperatures of ≳ 106 K, possibly +in shocks at the halo virial radii (see also Birnboim & +Dekel 2003). It would, however, be nearly two decades +before further observational evidence for a hot Milky +Way CGM was identified using observations of shadows +in the soft X-ray background from high-latitude (Snow- +den et al. 1991) and high-velocity (Herbstmeier et al. +1995) cold clouds, and in the spectral decomposition of +the background into absorbed and unabsorbed compo- +nents (Garmire et al. 1992). +Today the hot CGM of +the Milky Way is best traced through X-ray observa- +tions of OVII and OVIII diffuse emission, and absorp- +tion towards bright background sources. Although high +Galactic latitude data are well fit by a spheroidal com- +ponent with a mass of a few ×1010M⊙ (e.g., Bregman +& Lloyd-Davies 2007; Gupta et al. 2012), more recent +models include a substantially less massive (∼ 0.1%), +but denser disk-like component to explain the line ob- +servations (Li & Bregman 2017; Nakashima et al. 2018; +Kaaret et al. 2020; Ueda et al. 2022). Structure in the +hot Milky Way CGM is also observed in the form of +the Fermi (Su et al. 2010), Parkes (Carretti et al. 2013) +and eROSITA (Predehl et al. 2020) bubbles. The neu- +tral and warm (≲ 105 K) components of the Milky Way +CGM, as primarily traced through observations of high- +velocity clouds, are likely sub-dominant in mass to the +hot component (few ×108M⊙; Putman et al. 2012), but +highlight the complexities of the disk-halo interface (e.g., +Koo et al. 1992; Werk et al. 2019; Clark et al. 2022). +The mass of the Galactic CGM is uncertain. Here we +assume Mtot = 1.5 × 1012M⊙ (following Prochaska & +Zheng 2019) and a total Galactic mass in stars and cold +gas of 6 × 1010M⊙ (Draine 2011). Setting fCGM = 1 +and Ωb/ΩM = 0.188 (Planck Collaboration et al. 2020) +implies MCGM = 2.82 × 1011M⊙. Recent models for the +distribution of OVII and OVIII emission and absorp- +tion result in MCGM values between 3 − 4 × 1010M⊙ +(Li & Bregman 2017), 5.5 − 8.6 × 1010M⊙ (Kaaret et +al. 2020), and ∼ 1.2 × 1011M⊙ (Yamasaki & Totani +2020). Faerman et al. (2017) argue that a portion of +the hot CGM is also traced by OVI absorption, as is +observed in external galaxies (Tumlinson et al. 2011), +and find MCGM = 1.2 × 1011M⊙. +Systematic effects +that make the estimation of MCGM difficult include the +model uncertainties in foregrounds and in the spatial +distribution and clumpiness of the gas, and the poorly +constrained gas metallicity. An alternate measurement +of MCGM was found by Salem et al. (2015) by model- +ing the ram-pressure stripping of the Large Magellanic +Cloud (LMC); a low value of (2.7 ± 1.4) × 1010M⊙ was +found. +The DMs of FRBs and pulsars within the Milky Way +halo offer a unique probe of the content of the Galactic +CGM (e.g., Anderson & Bregman 2010; Platts et al. +2020). FRBs are transient radio emissions from distant +extragalactic sources that, like the emissions from radio +pulsars, are so short in duration that the dispersion in +intervening plasma is evident as an arrival-time delay at +lower frequencies: +∆t(ν) = +q2 +e +2πmecν2 +� D +0 +ne(l)dl, +(1) +where qe is the electron charge, me is the electron mass, +ν is the frequency, D is the source distance, and ne(l) is +the electron number density along the sightline. Under +physical conditions of CGM gas, DMs quantify the col- +umn densities along the source sightlines. In all cases, +however, the observed DMs of FRBs, DMobs, are com- +posed of a selection of additive components: +DMobs = DMISM + DMCGM + DMIGM + DMhost, (2) +where in this formulation DMISM includes the warm ion- +ized medium (WIM) in the Milky Way disk (e.g., Cordes +& Lazio 2002; Yao et al. 2017), DMCGM is the DM con- +tributed by the Galactic CGM, DMhost is the DM as- +sociated with the FRB host-galaxy interstellar medium +(ISM) and their halos, and DMIGM includes contribu- +tions from the intergalactic medium (IGM) and any in- +tervening galaxy systems. For pulsars within the Milky + +Mass of the Milky Way CGM +3 +Way halo, such as in the Magellanic Clouds and in sev- +eral globular clusters, DMIGM = 0 pc cm−3. These ex- +pressions apply to the redshift ∼ 0 regime relevant to +this paper. If the components of DMobs can be accu- +rately modeled or bounded, constraints can be placed +on DMCGM. +Modeling of the DMs of pulsars in the +LMC led Anderson & Bregman (2010) to find a very low +MCGM ∼ 1.5×1010M⊙ assuming a Navarro et al. (1997) +profile (an NFW profile); it is likely that an overly large +DMISM was assumed, and the NFW profile is no longer +favored for the Galactic CGM. Although most FRBs are +too distant to allow for accurate modeling of the compo- +nents of DMobs, FRBs from nearby galaxies (Bhardwaj +et al. 2021; Kirsten et al. 2022) have begun to be used to +place constraints on DMCGM, assuming specific models +for DMISM, that are in tension with some models for the +Milky Way CGM. +In this paper we present observations with the +Deep +Synoptic +Array +(DSA-110) +of +a +new +FRB +(FRB 20220319D) that motivates stringent new con- +straints on the mass of the Milky Way CGM. Remark- +ably, FRB 20220319D was observed to have DMobs < +DMISM for leading models of the warm-gas distribution +in the Galactic disk, despite the unambiguous associ- +ation we find with a galaxy at a distance of 50 Mpc. +We detail observations of the FRB in §2. +We con- +sider the association with its host galaxy in §3, wherein +we show that the DM of FRB 20220319D is consis- +tent with an extragalactic origin given uncertainties in +models for DMISM, using observations of the DMs of +pulsars in globular clusters with accurate parallax dis- +tances. +We present an analysis of the host environ- +ment of FRB 20220319D in §4. We then place conser- +vative constraints on DMCGM, and hence MCGM, using +FRB 20220319D and the distant high-latitude globular +cluster M53, in §5. We discuss the implications of our +results in §6, and conclude in §7. +2. DSA-110 OBSERVATION OF FRB 20220319D +The DSA-1101 is a radio interferometer hosted at +the Owens Valley Radio Observatory (OVRO), pur- +pose built for the discovery and arcsecond-localization +of FRBs. A full description of the instrument will be +presented in Ravi et al. (in prep.). During the observa- +tions discussed herein, the DSA-110 was configured as +described in Ravi et al. (2022). +Of particular note is +that when FRB 20220319D was observed, early in DSA- +110 commissioning, all candidates at DMs in excess of +1 https://deepsynoptic.org +100 +0 +100 +degrees +Intrinsic PPA +0 +10 +20 +30 +S/N +Intensity (I) +Linear Polarization (L) +Circular Polarization (V) +-0.8 +-0.5 +-0.2 +0.0 +0.2 +0.5 +0.8 +Time (ms) +1325 +1350 +1375 +1400 +1425 +1450 +1475 +Frequency (MHz) +Figure 1. Dedispersed temporal profile and dynamic spec- +trum of FRB 20220319D. A value of DMobs = 110.95 pc cm−3 +was used. The top panel shows the estimated absolute po- +sition angle based on an approximate parallactic angle, the +∼ 20◦ error due to RM uncertainty is not included in the +error bars. The middle panel shows temporal profiles in to- +tal intensity, linearly polarized intensity, and Stokes V, as la- +beled. The bottom panel shows the dynamic spectrum of the +total-intensity data. The time resolution in the time-series +and dynamic-spectrum plots is 32.768 µs, and the temporal +profile is in units of signal to noise ratio. The reference time +is MJD 59657.93275696795, which was the burst arrival time +at OVRO at 1530 MHz. The polarized data on the burst was +corrected for the measured RM of 50 rad m−2. +80 pc cm−3 were saved for further inspection, regardless +of the expected DM through the Galactic disk. +FRB 20220319D was detected during standard com- +missioning observations, with an arrival time at OVRO +at 1530 MHz of MJD 59657.93275696795. During these +observations, the DSA-110 was parked at a pointing- +center declination of +71.6◦. +The DSA-110 refer- +ence position is −118.283◦ longitude, +37.2334◦ lat- +itude. +FRB 20220319D was detected with a DM of +110.96 pc cm−3, and a signal to noise ratio (S/N) of 41.7. +To derive optimized burst parameters and polarization +properties, we coherently combined voltage data from + +4 +Ravi et al. +the 48 core antennas towards the detection-beam direc- +tion. The resulting temporal profile and calibrated dy- +namic spectrum, produced with incoherent dedispersion +at the native time and frequency resolutions, are shown +in Figure 1, and optimized parameters are given in Ta- +ble 1. The polarization analysis was done following pro- +cedures described in Ravi et al. (2022). The matched- +filter total-intensity S/N estimate is 79. +The burst exhibits a narrow temporal profile with +a strongly modulated spectrum. +The DM smearing +timescale ranges between 8–13 µs within the DSA-110 +band, and so the data resolve the temporal structure of +the burst. Scaling the system-equivalent flux density of +the DSA-110 by the primary-beam attenuation at the +burst position, we derive a fluence of 11.2 ± 0.2 Jy ms. +Following techniques described in Connor et al. (2020), +we derive a spectral decorrelation bandwidth, νd = +3.6 ± 0.2 MHz, defined as the 1/e scale of the autocor- +relation function of the spectrum. The burst spectro- +temporal morphology is consistent with the population +of so far non-repeating FRBs (Pleunis et al. 2021). We +derive a low fractional polarization of just 16% (Fig- +ure 1), with no significant circular polarization. A Fara- +day rotation measure (RM) of 50 ± 15 rad m−2 was de- +tected. The error was estimated by simulating the re- +covery of RMs with the same linear-polarization S/N as +FRB 20220319D (Sherman et al., in prep.). +The interferometric localization of FRB 20220319D +was derived following procedures largely described in +Ravi et al. (2022). Correlation products for all baselines +formed from the 63 functioning antennas of the DSA-110 +at the time of detection were integrated over 262.144µs +centered on the burst arrival time. Bandpass calibra- +tion was done using a 10-min observation of 3C309.1 ob- +served 11 hr prior to the burst detection. Complex gain +calibration at the time of the burst detection was derived +using an NRAO VLA Sky Survey (NVSS; Condon et al. +1998) model and 5 min of visibility data. We show im- +ages of the point-spread function (PSF; a flat spectrum +was assumed), and the pre- and post-deconvolution im- +ages of FRB 20220319D in Figure 2. Briggs weighting +with a ‘robust’ parameter of 0.5 was used to suppress +sidelobes, given the high S/N of FRB 20220319D, yield- +ing a PSF FWHM of 26.9′′ × 15.6′′. +The position of +FRB 20220319D, given in Table 1, was derived by fit- +ting an elliptical Gaussian to the deconvolved image of +FRB 20220319D. +We derived the uncertainty in the FRB position us- +ing data on nine compact bright sources from the Radio +Fundamental Catalog (RFC; rfc 2022c), obtained within +2 hr of the burst observation. A fit to a linear trend in +R.A. and Decl. was used to derive arcsecond-level astro- +Table 1. Properties of FRB 20220319D. +Parameter +Value +Arrival time (MJD)a +59657.93275696795 +DM (pc cm−3) +110.95(1) +Full-width half-maximum (ms) +0.16(1) +Fluence (Jy ms) +11.2(2) +Spectral energy (erg Hz−1) +3.3(1)×1028 +L/Ib +0.16(3) +|V/I|b +0.04(3) +RM (rad m−2) +50(15) +νd (MHz)c +3.6(2) +R.A. (J2000) +02:08:42.7(1) +Decl. (J2000) ++71:02:06.9(6) +aArrival time at OVRO at 1530 MHz. +b L/I is the fraction of linearly polarized fluence, and +|V/I| is the absolute value of the fraction of circularly +polarized fluence. +c νd is the characteristic spectral decorrelation band- +width. +metric corrections, and the corresponding uncertainties. +These uncertainties (0.52′′ in R.A. and 0.5′′ in Decl.) +were then added in quadrature to the statistical uncer- +tainty in the fitted burst position (0.29′′ in R.A. and +0.17′′ in Decl.) to derive a final 90% confidence contain- +ment ellipse with semi-axes of 1.25′′ in R.A. and 1.18′′ in +Decl. This ellipse is shown in the bottom-left panel of +Figure 2, together with the detrended position offsets of +the RFC sources. We also assessed the quality of the +in-field calibration by checking the positions of bright +(> 50 mJy) compact (< 20′′) NVSS sources within the +primary-beam FWHM against NVSS catalog positions. +These offsets (with the RFC corrections applied) are also +shown in Figure 2. A more detailed analysis of the DSA- +110 localization accuracy will be presented in Ravi et al. +(in prep.). +3. ASSOCIATION WITH IRAS 02044+7048, AND +UNCERTAINTIES IN DMISM +The +90% +confidence +localization +region +of +FRB 20220319D is shown in Figure 3 overlaid on a +Pan-STARRS1 (PS1) i-band image. +The galaxy co- +incident with the FRB location is cataloged in the +NASA Extragalactic Database as IRAS 02044+7048, +with a spectroscopic redshift of 0.011 indicating a dis- +tance of approximately 50 Mpc. +Using observations +described below, we derive a spectroscopic redshift of + +Mass of the Milky Way CGM +5 +100 +50 +0 +50 +100 +100 +50 +0 +50 +100 +PSF +100 +50 +0 +50 +100 +100 +50 +0 +50 +100 +FRB 20220319 - dirty +100 +50 +0 +50 +100 +100 +50 +0 +50 +100 +FRB 20220319 - clean +2 +0 +2 +2 +1 +0 +1 +2 +Offsets +NVSS offsets +VLBI sources +Right Ascension offset (arcsec) +Declination offset (arcsec) +Figure 2. DSA-110 localization of FRB 20220319D. Top left: Point-spread function (PSF) of the DSA-110 for the observation +of FRB 20220319D, assuming a flat-spectrum source. Top right: Dirty image of FRB 20220319D, with no deconvolution applied. +Bottom left: Deconvolved image of FRB 20220319D. The synthesized beam is represented by an ellipse with half-power diameters +of 26.9′′×15.6′′, at a position angle of 80◦. Briggs weighting was used with a ‘robust’ parameter of 0.5, in order to partially +suppress PSF sidelobes. Baselines shorter than 200λ were excluded from the analysis because they were affected by spurious +cross-coupling. Contours in the preceding three panels are at −0.4, −0.2 (dashed), 0.2, 0.4, 0.6, 0.8 and 0.9 (solid) of the peak +intensity. The images are centered on the coordinates (R.A. J2000, decl. J2000) = (02:08:42.7, +71:02:06.9). Bottom right: +Offsets of known astronomical sources from their true positions as measured by the DSA-110. The blue disks show offsets +of sources from cataloged positions in the NRAO VLA Sky Survey (NVSS; Condon et al. 1998) in a 5 min DSA-110 image +formed from data taken at a time centered on the burst detection. Only sources with cataloged flux densities > 50 mJy and +measured major-axis diameters < 20′′, within 2 deg of the pointing center, were considered. The symbol size is proportional to +flux density. The red crosses show measured offsets of nine RFC calibrator sources (see text for details) observed immediately +preceding and after the burst. The ellipse indicates the 90% confidence containment region for FRB 20220319D, derived from +the RFC calibrators. + +6 +Ravi et al. +2h08m45s +42s +39s +36s +71°02'30" +15" +00" +01'45" +Right Ascension (J2000) +Declination (J2000) +Figure 3. Pan-STARRS i-band image of IRAS 02044+7048, +with the 3.7 × 2.2 arcsec 90% confidence localization region +of FRB 20220319D indicated as a white ellipse. +0.0111 ± 0.0004, indicating a luminosity distance of +49.6 Mpc. An isophotal fit to the PS1 i-band image in- +dicates an effective radius of 2.7 ± 0.2 kpc, and the FRB +is offset by 2.3 ± 0.3 kpc. The isophotal analysis also +indicates an inclination of 23 ± 3 deg. A barred spiral +morphology, with a classification of SBa, is evident from +the image. +Under +the +assumption +that +FRB 20220319D +is +extragalactic, +the +spatial +association +between +FRB 20220319D and IRAS 02044+7048 is secure. It is +unlikely that FRB 20220319D lies significantly beyond +IRAS 02044+7048; the morphology of the galaxy indi- +cates the presence of a warm-ISM phase, and scattering +in this ISM would likely cause temporal broadening +of several hundred milliseconds at our observing fre- +quencies (Cordes et al. 2022; Ocker et al. 2022). The +Gravitational Wave Galaxy Catalog (GWGC; White +et al. 2011) lists 15869 galaxies at distances less than +50 Mpc, subtending a total of 4 deg2. Thus the chance- +association probability of FRB 20220319D and a galaxy +at < 50 Mpc is ≲ 10−4. +The extragalactic nature of FRB 20220319D is called +into question by its DM of 110.95 pc cm−3. +The pre- +dicted DM for extragalactic objects along its l = 129.2◦, +b = +9.1◦ sightline is 132.9 pc cm−3 according to the +NE2001 model for the Galactic ionized ISM distribu- +tion (Cordes & Lazio 2002), and 187.7 pc cm−3 accord- +ing to the YMW16 model (Yao et al. 2017). The RM +detection for FRB 20220319D of 50±15 rad m−2 is, how- +ever, marginally consistent with being in excess of the +expected Milky Way contribution along the burst sight- +line. A recent model for the Galactic RM foreground +along this sightline (Hutschenreuter et al. 2022) indi- +cates an expected Galactic RM of −14 ± 18 rad m−2 +towards FRB 20220319D. We cannot draw conclusions +from the measured spectral decorrelation bandwidth of +νd = 3.6±0.2 MHz, which is far in excess of the 0.2 MHz +decorrelation bandwidth predicted by NE2001 due to +scattering in the Milky Way ISM, because the origins of +the spectral structure cannot easily be determined. +We now analyze the performance of the NE2001 and +YMW16 models with the aim of determining whether +the model uncertainties can accommodate an extra- +galactic origin for FRB 20220319D. Both model the dis- +tribution of the WIM (Draine 2011) in the Galactic +disk using observations of radio-wave propagation, with +a dominant thick-disk component encapsulating several +overdensities and voids motivated by multiwavelength +data. The YMW16 model is fit to pulsars with indepen- +dent DM and distance estimates, whereas the NE2001 +model is additionally fit to pulsar and extragalactic +radio-source scattering measurements. For low-latitude +extragalactic sightlines, the radial extent of the thick +disk is critical in determining DMISM. +In NE2001, a +cutoff at a radius of 20 kpc is motivated by observations +of HII regions in other galaxies and scattering of extra- +galactic sources. Surveys of Galactic HII regions on the +other hand motivate the cutoff of 15 kpc in YMW16. +Neither cutoff is estimated in the NE2001 or YMW16 +fits; rather they are fixed model choices that do not im- +pact the match between the models and the data un- +der consideration. The functional forms for the WIM +distribution in the thick disk also differ, with NE2001 +incorporating an oblate spheroid, and YMW16 incorpo- +rating a plane-parallel slab that results in significant de- +viations from NE2001 at low latitudes in the second and +third quadrants (e.g., Price et al. 2021). High-latitude +extragalactic sightlines are likely to be better modeled +because the scale height of the thick disk is a critical fit- +ted parameter. However, density fluctuations caused by +turbulence in the WIM, together with clumps and voids, +complicate our picture even at high latitudes (Ocker et +al. 2020). Large-scale WIM inhomogeneities at the disk- +halo interface such as Galactic chimneys (e.g., Koo et al. +1992; Normandeau et al. 1996) can result in significant +departures from spatially smooth models for DMISM. +We first note that the recent PSRπ sample of pul- +sar parallax measurements (Deller et al. 2019) reveals +discrepancies between the NE2001 and YMW16 models +and measured pulsar distances at low latitudes in the +second quadrant, where FRB 20220319D appears on the +sky. In general, known errors in NE2001 and YMW16 + +Mass of the Milky Way CGM +7 +exhibit significant spatial correlation (e.g., Price et al. +2021), motivating the consideration of sources along +sightlines similar to FRB 20220319D. Table 2 lists the +DMs of the three nearest pulsars in the sample to +FRB 20220319D, together with the predicted DMs given +the measured distances. +The YMW16 model signifi- +cantly overpredicts all three DMs, and although NE2001 +is more accurate for the modestly distant pulsars likely +in the Cygnus-Orion arm, it also overpredicts the DM +of the more distant PSR J0406+6138. +A more detailed analysis of the performance of the +models for sightlines towards distant objects is enabled +by the recent convergence in the distances to globu- +lar clusters hosting pulsars. +We obtain accurate dis- +tance estimates (less than few-percent errors) to all +21 globular clusters with associated pulsars (Manch- +ester et al. 2005) at |b| > 5◦ from the compilation +of Baumgardt & Vasiliev (2021), primarily based on +Gaia parallaxes (Vasiliev & Baumgardt 2021). +Clus- +ters at lower latitudes are not considered because their +DMs are influenced by the thin disk and other compo- +nents in the Galactic plane. For each cluster we com- +pare the difference between the DMs predicted by the +NE2001 and YMW16 models for the cluster distances, +and the true cluster DMs defined as the mean DMs of +the associated pulsars. The results are shown in Fig- +ure 4; FRB 20220319D is also included with the pre- +dicted DM set to DMISM. +The errors in NE2001 at +low latitudes can clearly accommodate an extragalactic +origin for FRB 20220319D. YMW16 generally performs +better than NE2001 for the low-latitude clusters, but +these are nearly all in the first and fourth quadrants +(with NGC 1851 as the only exception), and so perhaps +not representative of errors along sightlines closer to the +Galactic anticenter. +As an additional check of the Galactic sightline to- +wards FRB 20220319D, we also consider independent +measures of DMISM. First, we use the relation between +the neutral hydrogen column density and pulsar DMs +derived by He et al. (2013) to estimate the DMs to- +wards FRB 20220319D and each globular cluster, as- +suming no HI gas beyond the clusters. We obtain HI +column densities from the HI4PI data (HI4PI Collabo- +ration et al. 2016). The results are shown in Figure 4, +including errors representing intrinsic scatter in the rela- +tion. The HI column densities result in underestimates +of the DMs towards low-latitude clusters, possibly re- +flecting the increased ionization of HI gas towards the +central regions of the Galaxy (e.g., Madsen & Reynolds +2005). +The HI-predicted DMISM for FRB 20220319D +is consistent with an extragalactic origin, although the +uncertainties are large. Second, we estimate the DMs +towards FRB 20220319D and each globular cluster us- +ing Hα emission measures (EMs) derived from the Wis- +consin Hα Mapper (WHAM) all sky maps (Haffner et +al. 2003). +We follow the analysis of Berkhuijsen et +al. (2006), wherein the photon fluxes, FHα (in units of +Rayleighs), integrated over the range of Galactic veloci- +ties are converted to DMs using the following relations: +EM = 2.25FHαe2.2E(B−V ) cm−6 pc +(3) +DM = (EM × D × F)1/2. +(4) +F = +f +ζ(1 + ϵ2) +(5) +The expression for F (e.g., Ocker et al. 2020, and +references therein) encodes an ionized cloudlet model, +wherein the WIM is composed of cloudlets with a volume +filling factor f, ζ = ⟨n2 +e⟩/⟨ne⟩/2 ∼ 2 captures the cloud +to cloud variation, and ϵ2 ≲ 1 is the density variance +internal to the cloudlets. Further, E(B − V ) is a mea- +sure of the interstellar redenning along a given sightline, +and D is the distance through the WIM. We have as- +sumed a uniform WIM electron temperature of 8000 K, +and equivalence between the line-of-sight and volume +filling factors. We find that for the cluster sightlines the +Hα-based DM estimates perform remarkably well for a +nominal value F = 0.1 (i.e., f ≳ 0.4), comparably to +YMW16 and better than NE2001 for low-latitude sight- +lines. This is surprising given previous results for pulsar +DMs (e.g., Schnitzeler 2012), but may be explained by +globular clusters generally lying beyond the outer extent +of the WIM, obviating the need for distance corrections +to the WHAM EMs. +For the FRB 20220319D sight- +line, the Hα flux implies DMISM = 89 pc cm−3, which is +22 pc cm−3 lower than the measured DM. +We +conclude +that +models +for +the +DM +towards +FRB 20220319D contributed by the Galactic WIM are +consistent with an extragalactic origin. +This se- +cures the association of the burst with the galaxy +IRAS 02044+7048. Consistent with the original presen- +tations of the NE2001 and YMW16 models for the WIM +distribution, we confirm that there are significant model +uncertainties at low Galactic latitudes. We used a large +sample of globular-cluster parallax distances, which have +DM estimates from associated pulsars, to demonstrate +that an alternative predictor of DMISM based on the to- +tal Galactic Hα flux can provide a useful check on the +aforementioned models. +In order to estimate DMISM +towards FRB 20220319D we consider pulsars that are +nearby on the sky and at a similar Galactic latitude. The +two nearest pulsars within 1 deg of the burst in Galactic +latitude, PSR J0231+7026 (total separation of 1.9 deg) + +8 +Ravi et al. +Table 2. Measured and predicted DMs of PSRπ pulsars near FRB 20220319D. +Pulsar +l (deg.) +b (deg.) +FRB offset (deg) +Distance (kpc) +DMa +NE2001a +YMW16a +PSR J0147+5922 +130.1 +−2.7 +12 +2.02+0.46 +−0.16 +40.1 +31.8 +79.0 +PSR J0157+6212 +130.6 +0.3 +9 +1.80+0.08 +−0.12 +30.2 +31.5 +59.3 +PSR J0406+6138 +144.0 +7.0 +15 +4.58+1.63 +−0.87 +65.3 +123.7 +141.7 +aAll DMs are given in units of pc cm−3. +NGC6517 +NGC6539 +M62 +M22 +FRB20220319 +NGC6652 +NGC6397 +NGC5986 +OmegaCen +M4 +M10 +NGC6752 +M3 +M15 +M14 +NGC1851 +M2 +M13 +47Tuc +M5 +M30 +M53 +200 +150 +100 +50 +0 +50 +100 +150 +(Predicted DM) - (True DM) (pc/cc) +6.7 +6.8 +7.3 +7.6 +9.8 +11.4 +12.0 +13.3 +14.9 +16.0 +23.1 +25.6 +26.1 +27.3 +27.7 +35.0 +35.8 +40.9 +44.9 +46.8 +46.8 +79.8 +|b| (deg) +NE2001 +YMW16 +Halpha (ff=0.1) +NH +Figure 4. Difference between the predicted DM and true DM of all globular clusters that host pulsars, sorted by increasing +|b|. Predictions are made for each globular cluster given its distance and location on the sky, using four different models for the +warm ISM of the Milky Way. The two literature models are NE2001 (blue solid thick curve) and YMW16 (orange solid thin +curve). We also convert the measured Hα emission measures (green dashed curve) and HI columns (red dots and shaded region) +along each sightline to DM predictions using methods described in the text. We also include predictions for FRB 20220319D +assuming a location beyond the Galactic disk. +and PSR J0325+6744 (total separation of 7.4 deg), have +DMs of 46.7 pc cm−3 and 65.3 pc cm−3, and we consider +these as conservative and less conservative estimates re- +spectively of DMISM along the burst sightline. We note +that the predicted NE2001 and YMW16 DMISM values +along the sightlines towards these pulsars are lower than +along the burst sightline. +4. THE HOST ENVIRONMENT OF FRB 20220319D +We +obtained +long-slit +optical +spectroscopy +of +IRAS 02044+7048 +using +the +Double +Spectrograph +(DBSP; Oke & Gunn 1982) mounted on the Palo- +mar 200-inch Hale telescope on 2022 June 01 (UT). +The observations were undertaken under conditions of +1.2′′ seeing, at an airmass of 1.9, and used a 1.5′′ slit +positioned to include both the FRB location and the +galaxy nucleus at a position angle of 104◦. Two 600 s +exposures were obtained with the D55 dichroic and the +316/7500 grating on the red arm; we only considered +data from the red arm of the spectrograph given the +significant extinction along this sightline (AV = 2.209; +Schlafly & Finkbeiner 2011). The two-dimensional spec- +tral data were reduced according to standard procedures +using a PypeIt-based pipeline (Prochaska et al. 2020). +We then defined the trace function using observations +of the standard star Feige 34, and extracted spectra in +1.5′′ windows along the slit. +Feige 34 was also used + +Mass of the Milky Way CGM +9 +Table 3. Observed and derived parame- +ters of the host galaxy of FRB 20220319D, +IRAS 02044+7048. +1σ errors in the last +significant figures are given in parentheses. +Parameter +Value +Redshift +0.0111(4) +Luminosity distance (Mpc) +49.6 +Effective radius (kpc) +2.7(2) +FRB projected offset (kpc) +2.3(3) +log M∗ (M⊙) +9.93(7) +log Za +0.1(0.2) +Internal AV b +0.2+0.2 +−0.1 +SFR (M⊙ yr−1)c +1.8(0.7) +Inclination (deg) +23(3) +aMetallicity with respect to solar. +b V -band extinction corresponding to a +uniform dust slab. +c Averaged over the past 100 Myr. +6 +4 +2 +0 +2 +Offset along slit (kpc) +0 +10 +20 +30 +40 +50 +H flux (1040 erg s +1) +FRB +Nucleus +Figure +5. +Total Hα flux observed at different lo- +cations within our longslit spectroscopic observations of +IRAS 02044+7048. Measurements were obtained at 1.5′′ in- +tervals along a 1.5′′ longslit oriented at a position angle of +104◦, which covered both the FRB position and the cen- +ter of the galaxy. The locations of the galaxy nucleus and +FRB 20220319D are indicated in the figure. +for flux calibration, and all spectra were corrected for +extinction according to the Fitzpatrick & Massa (2007) +extinction curve. +We detected Hα emission at several locations along +the slit, as shown in Figure 5. The strongest Hα emission +is observed at the galaxy core, from which we also detect +lines corresponding to [NII] and [SII]. These lines were +together used to derive a redshift of 0.0111 ± 0.0004 for +IRAS 02044+7048. Although we do not detect Hβ or +[OIII], the ratio log [NII]/Hα = −0.23 ± 0.02 indicates a +< 10% chance of the nuclear ionization being purely due +to star formation activity (Ho et al. 1997; Kewley et al. +2006). There is evidence of Hα emission from spiral-arm +features on either side of the nucleus; these features are +too extended to represent HII regions. The FRB appears +associated with the eastern arm, although not at the +center of its Hα radial profile. No additional evidence +for or against an arm association can be gleaned from +the residuals of a fit of the IRAS 02044+7048 image in +Figure 3 to a two-dimensional Sersic profile. +In +order +to +determine +global +properties +of +IRAS 02044+7048, we derived and modeled its spec- +tral energy distribution (SED). We executed aperture +photometry on archival images from the Panoramic +Survey Telescope and Rapid Response System (Pan- +STARRS; Chambers et al. 2016), Two Micron All Sky +Survey (2MASS; Skrutskie et al. 2006), and ALLWISE +(Cutri et al. 2021) surveys. We identified an elliptical +aperture that captured the i-band extent of the galaxy +in Pan-STARRS data, and masked likely foreground +objects. We then used this aperture, convolved with the +PSF of each survey, to measure extinction-corrected AB +magnitudes in all filters used by Pan-STARRS, 2MASS, +and WISE. We modeled this SED using the Prospector +stellar population synthesis modeling code (Johnson et +al. 2021). We ran Prospector using recommended tech- +niques and priors and a non-parametric star-formation +history (SFH) model (Leja et al. 2019), and sampled +from the posterior using emcee (Foreman-Mackey et al. +2013). +Non-parametric models for the SFH result in +less bias in both stellar-mass (M∗) and star-formation +rate (SFR) estimates, because specific SFH models are +not excluded a priori. We included a model for dust re- +radiation in the likelihood function. The spectral energy +distribution of IRAS 02044+7048 is shown in Figure 6, +together with the results from the Prospector analysis, +and derived maximum aposteriori probability parame- +ters are listed in Table 3. The SFR of 1.8 M⊙ yr−1 was +derived by integrating the SFH over the past 100 Myr. +IRAS 02044+7048, +and +the +location +of +FRB 20220319D within it, are largely consistent with +previous results on FRB host galaxies and environ- +ments (Bhandari et al. 2020; Heintz et al. 2020; Boch- +enek et al. 2021; Mannings et al. 2021; Bhandari et al. +2022). Most FRB host galaxies have detectable ongoing +star formation, and some exhibit signatures of addi- +tional nuclear ionization sources. +The stellar mass of +IRAS 02044+7048 is entirely consistent with the distri- + +10 +Ravi et al. +Figure 6. +Left: +Prospector fit to the spectral energy distribution (SED) of the host galaxy of FRB 20220319D, +IRAS 02044+7048. The SED measurements from archival PanSTARRS, 2MASS, and WISE data are shown in red, together +with representative 10% errors. Filter transmission curves are shown in grey. The maximum aposteriori probability (MAP) +model photometry is shown in green, together with the MAP spectrum. 100 spectra generated using random draws from the +posterior distributions of the model parameters are shown in black. Right: Measured SFH of IRAS 02044+7048 (green). The +results from 100 random draws from the posterior distributions are shown in black. +bution of masses found for both repeating and so far +non-repeating FRB hosts. The potential association of +FRB 20220319D with a spiral-arm feature is consistent +with the results of Mannings et al. (2021) based on +Hubble Space Telescope imaging of eight FRB hosts. +In the absence of very long baseline interferometry, our +data do not have sufficient angular resolution to deter- +mine whether or not the FRB originates from within an +association of young stars (e.g., Tendulkar et al. 2021). +The location of FRB 20220319D at an offset of ∼ 85% of +the effective radius from the galaxy nucleus is consistent +with the range of previously observed FRB offsets. +However, the SFR of IRAS 02044+7048 is rather high +for its stellar mass, with respect to the observed sam- +ple of hosts of so far non-repeating FRBs (Bhandari +et al. 2022). +Only one so far non-repeating FRB +(20190102C, at z = 0.2912) has a comparably high +specific SFR (sSFR), of log sSFR = −9.74. Although +IRAS 02044+7048 is consistent with a location on the +star-forming main sequence of galaxies (Speagle et al. +2014), non-repeating FRB hosts as a population are con- +sistent with originating from below this sequence. +It +is possible that FRBs from more actively star-forming +hosts are selected against in existing radio surveys due to +propagation effects in the host ISM (Seebeck et al. 2021), +in which case it is not surprising that IRAS 02044+7048 +has a face-on orientation. +5. THE MASS OF THE MILKY WAY CGM +The constraints on the extragalactic DM contribution +along the sightline towards FRB 20220319D are severe. +Adopting the two estimates discussed above for DMISM +from pulsars nearby to the FRB on the sky, we find +possible upper limits on the extragalactic DM contribu- +tion (DMCGM+DMIGM+DMhost) of either 45.7 pc cm−3 +or 64.3 pc cm−3. Even with our conservative estimates +for DMISM, FRB 20220319D has the lowest extragalactic +DM yet measured for an FRB localized to a host galaxy. +We thus have the opportunity to stringently bound the +heretofore poorly constrained value of DMCGM, and thus +directly bound the mass of the Galactic CGM. The IGM +likely contributes 7 pc cm−3 towards IRAS 02044+7048, +assuming an IGM baryon fraction of 0.7 (e.g., Macquart +et al. 2020). +The detection of extended Hα emission +from the galaxy itself, and in particular at the location +of FRB 20220319D, indicates that the WIM component +of the ISM is present, although we do not know where +within the ISM column the FRB source is located.2 We +simply assume a nominal value of DMhost = 10 pc cm−3 +for our analysis below. +Thus, we find that for the +FRB 20220319D sightline, two possible upper limits on +DMCGM are 28.7 pc cm−3 and 47.3 pc cm−3, depending +on the assumption for DMISM. These upper limits are +comparable to previous results from the closest known +FRB source (FRB 20200120E), in a globular cluster as- +sociated with M81, which provide possible upper limits +on DMCGM of 32 pc cm−3 and 42 pc cm−3] depending on +the choice of NE2001 or YMW16 for DMISM (Kirsten et +al. 2022). The constraints from FRB 20200120E are af- +fected by a reliance on the NE2001/YMW16 models for +2 With a measurement of scattering in the host ISM, it is possible +to estimate DMhost (Cordes et al. 2022). However, upper limits +on the temporal broadening of FRB 20220319D of ∼ 0.1 ms are +not usefully constraining. + +Model spectrum (MAP) +Model photometry (MAP) +Observed photometry +Flux Density [maggies] +10 +10 +Wavelength [A]5 +4 - +SFR [Mo/yr] +3 +2 +0 +.6 +18 +0 +2 +4 +10 +12 +Lookback Time [Gyr]Mass of the Milky Way CGM +11 +DMISM, and by stronger assumptions for the halo DM +of M81. +A recent synthesis of models for the Galactic CGM +DM by Keating & Pen (2020) highlighted the wide range +of extant predictions. In this work, we use a representa- +tive set of models to demonstrate the implications of the +constraint on DMCGM from FRB 20220319D. The mod- +els are illustrated in Figure 7. We also consider measure- +ments of the DM contributed by sightlines through the +halo from pulsars in the LMC (Ridley et al. 2013) and +in the distant (18.5 kpc; Baumgardt & Vasiliev 2021) +globular cluster M53 (Kulkarni et al. 1991; Pan et al. +2021). Although the LMC is at b ≈ −33◦ and M53 is at +b ≈ +80◦, we conservatively subtract the lowest DMISM +to be found in either the NE2001 or YMW16 models of +18 pc cm−3 to estimate the halo DM contributions along +their sightlines. In the top two panels of Figure 7, we +show the resulting upper limits on DMCGM towards M53 +and the LMC, with ranges corresponding to the range of +associated pulsar DMs. The bottom-left panel shows the +upper limit on DMCGM from FRB 20220319D, placed at +the virial radius of the Milky Way halo. In all cases we +show model predictions for DMCGM assuming a spher- +ically symmetric galactocentric baryon halo, but with +the DM evaluated along sightlines from the position of +the Earth (GRAVITY Collaboration et al. 2019). We +evaluate the modified NFW (mNFW) and Maller & Bul- +lock (2004) profiles following Prochaska & Zheng (2019), +with a total Milky Way mass of Mtot = 1.5 × 1012M⊙, +and a halo baryon fraction of fCGM = 0.75. The Pen +(1999) model is evaluated for the same Mtot and with +the core radius set to the halo virial radius. The semi- +empirical Miller & Bregman (2013) and Faerman et +al. (2017) models and the semi-analytic Faerman et al. +(2020) model are all evaluated as given. Finally, in the +bottom-right panel of Figure 7, we also show the model +predictions for the halo baryon density along the LMC +sightline, together with constraints from ram-pressure +stripping analyses of dwarf galaxies (Salem et al. 2015; +Putman et al. 2021) and modeling of OVII and OVIII +emission measurements (Kaaret et al. 2020). +The data are in favor of models that predict lower to- +tal CGM DMs and lower inner densities (e.g., Pen 1999; +Faerman et al. 2020; Miller & Bregman 2013). The DMs +of M53 pulsars and FRB 20220319D deliver consistent +constraints on the range of models, albeit on very dif- +ferent radial distance scales. The constraints from LMC +pulsars are less impactful, but nonetheless also exclude +the three models (Maller & Bullock 2004; Faerman et +al. 2017; Prochaska & Zheng 2019) considered here that +predict larger values of DMCGM. We note that our treat- +ment of the CGM contribution to the DMs of LMC pul- +sars is far more conservative than that of Anderson & +Bregman (2010); this is motivated by the uncertainties +discussed above in estimating DMISM. +The DM con- +straints are consistent with the most robust density es- +timates that we consider (Salem et al. 2015; Kaaret et +al. 2020). +The density estimates are however subject +to the uncertainties discussed in §1, and we proceed by +considering only the constraints on DMCGM. +We now derive constraints on the mass of the Galac- +tic CGM, MCGM, by finding β models that are consis- +tent with the FRB 20220319D and M53 DMs. +When +converting the electron column density to a mass col- +umn density, we assume 1.18 proton masses per elec- +tron following Yamasaki & Totani (2020), which corre- +sponds to roughly solar metallicity gas. The β profile is +widely adopted in the field to convert (column-)density +estimates to halo masses (e.g., Miller & Bregman 2013; +Salem et al. 2015; Kaaret et al. 2020), as it is empirically +motivated. The electron number density at a radius r is +given by +n(r) = n0 +� +1 + +� r +rc +�2�−3β/2 +, +(6) +where n0 is a central density, and rc is a core radius that +we fix to 0.47 kpc following Miller & Bregman (2013). +The exact value of the core radius is unimportant for +our conclusions regarding MCGM. +Typical values of +β found using various CGM tracers are in the range +0.4 ≲ β ≲ 0.5 (e.g., Miller & Bregman 2013, 2015; Salem +et al. 2015; Kaaret et al. 2020). We consider three con- +straints on DMCGM: the lower and upper constraints +from FRB 20220319D, and the lower constraint from the +M53 pulsars. For each constraint, we derive correspond- +ing values of MCGM from an integrated β profile for dif- +ferent values of β. A useful joint constraint on MCGM +and β is not possible with the data in hand. The results +are shown in Figure 8. +For typical values of β of between 0.4 and 0.5, +the DM of FRB 20220319D implies an upper limit on +log MCGM of between 11.0 and 10.8. +The M53 pul- +sars are even more constraining, limiting log MCGM to +below a value of between 10.7 to 10.5. +We consider +the FRB 20220319D constraints to be more conserva- +tive because the M53 estimate of DMCGM is very sen- +sitive to the assumed DMISM along its sightline. The +mass constraints all favor values of fCGM < 0.5 (re- +call fCGM = MCGMΩM +MtotΩb ) for our fiducial value of Mtot = +1.5×1012M⊙, with M53 potentially accommodating val- +ues of fCGM ≲ 0.2 for reasonable values of β. + +12 +Ravi et al. +0 +50 +100 +150 +200 +250 +Distance (kpc) +100 +101 +102 +DM (pc cm +3) +M53 +mNFW +MB04 +MB13 +Pen99 +Faerman1 +Faerman2 +0 +50 +100 +150 +200 +250 +Distance (kpc) +100 +101 +102 +DM (pc cm +3) +LMC +mNFW +MB04 +MB13 +Pen99 +Faerman1 +Faerman2 +0 +50 +100 +150 +200 +250 +Distance (kpc) +100 +101 +DM (pc cm +3) +FRB 20220319 +mNFW +MB04 +MB13 +Pen99 +Faerman1 +Faerman2 +0 +50 +100 +150 +200 +250 +Distance (kpc) +10 +5 +10 +4 +10 +3 +Baryon number + density (cm +3) +Density (LMC direction) +mNFW +MB04 +MB13 +Pen99 +Faerman1 +Faerman2 +Figure 7. Illustration of the constraining power of different measurements of the CGM DM and density on a range of models. +In all panels, curves show predictions of the default modified NFW (mNFW) model of Prochaska & Zheng (2019) (blue), the +model of Maller & Bullock (2004) (orange; MB04), the model of Miller & Bregman (2013) (green; MB13), a Pen (1999) model +(Pen99) assuming a core radius set to the virial radius (red), and the models of Faerman et al. (2017) (purple; Faerman1) and +Faerman et al. (2020) (brown; Faerman2). Top panels: upper limit on the CGM DM towards M53 (left) and the LMC (right). +In both cases, the minimum DM (18 pc cm−3) across the sky in either the NE2001 or YMW16 models has been subtracted. The +ranges indicate the range of DMs of pulsars associated with each object. Bottom left: upper limit on the CGM DM towards +FRB 20220319D. The range indicates different assumptions for the ISM DM that correspond to the two closest pulsars on the +sky. Bottom right: constraints on the CGM density compared with the models. The dark disks indicate measurements using +ram-pressure stripping modeling of the LMC (Salem et al. 2015) and modeling of diffuse X-ray line emission (Kaaret et al. +2020). The lower limits displayed as triangles indicate estimates using ram-pressure stripping modeling of a large selection of +Milky Way satellites (Putman et al. 2021). +6. DISCUSSION +Our results demonstrate that the total Galactic +baryon mass is likely significantly lower than the cos- +mological average for a halo as massive as the Milky +Way. +The mass constraints we derive are consistent +with some previous observational results from analyses +of OVII and OVIII emission and absorption (e.g., Li +& Bregman 2017; Kaaret et al. 2020) and ram-pressure +stripping of the LMC (Salem et al. 2015), but inconsis- +tent with other results that posit higher CGM masses +(e.g., Faerman et al. 2017; Yamasaki & Totani 2020). +Analyses of the dynamics of Milky Way satellites im- +ply a total virial mass of Mtot = (1.4 ± 0.3) × 1012M⊙ +(Watkins et al. 2010), a recent Gaia-based analysis +of the dynamics of Milky Way globular clusters yields +Mtot = (1.3 ± 0.3) × 1012M⊙ (Posti & Helmi 2019), and +an analysis of the dynamics of the Magellanic Stream +yields Mtot = (1.5 ± 0.3) × 1012M⊙ (Craig et al. 2022). +It is possible that a full accounting for the orbit of the +LMC would reduce the estimates of Mtot by ∼ 15% +(Correa Magnus & Vasiliev 2022). +However, in all +cases fCGM ≲ 0.4 is implied by FRB 20220319D, and +fCGM ≲ 0.2 is implied by the M53 DM. Our constraints +are affected by different systematic effects to previous +results, and do not suffer from uncertainties in model- +ing the chemical and thermal states of the CGM, nor +from uncertainties in modeling the interaction of satel- +lite galaxies with the CGM. These values are consistent +with several simulations of the CGM contents of galaxies +like the Milky Way that account for the effects of kinetic +and thermal feedback on reducing the halo baryon con- +tent. +The observational constraints on MCGM are affected +by uncertainties in identifying the CGM DM contri- +bution along the sightlines of interest. +For the low- +latitude FRB 20220319D sightline, we adopted conser- +vative estimates of DMISM, and there are uncertainties +at the level of a few pc cm−3 in the DMIGM and DMhost + +Mass of the Milky Way CGM +13 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +10.4 +10.6 +10.8 +11.0 +11.2 +11.4 +11.6 +log MCGM (M ) +fCGM = 0.5 +fCGM = 0.2 +fCGM = 0.1 +FRB lower +FRB upper +M53 +Figure 8. +Upper limits on the CGM mass for different +assumptions for the radial density profile. +We adopt a β +model (see text for details), and show limits corresponding +to the upper and lower ends of the range of constraints from +FRB 20220319D (dashed and solid lines respectively), and +from the most constraining pulsar in M53 (red solid line). +Typical values of β found in the literature are ≳ 0.4. For +these values, the inferred fCGM values are typically below +0.5, as indicated by blue horizontal lines. +terms. We therefore consider the constraints on MCGM +based on FRB 20220319D to be robust to uncertainties +in DMCGM. The impact of uncertainty in DMISM for the +M53 sightline is greater given the low values of DMCGM +under consideration. Although we attempt to be con- +servative, even a few-unit increase in DMCGM would sig- +nificantly raise the derived upper limits on MCGM. Our +analysis is also sensitive to how we model the distribu- +tion of baryons in the CGM. First, we do not account +for sightline to sightline variance, which simulations sug- +gest may be at the level of ≳ 10% (e.g., Zheng et al. +2020; Ramesh et al. 2022). Second, it may be that a β +profile is not the correct model to adopt, and our con- +straints are sensitive to the specific value of β. Guidance +on these points from simulations for future analyses of +DMCGM will be important. Finally, although we assume +a specific helium abundance in translating the DM con- +straints to a total CGM mass, our results are sensitive +to variations in this assumption only at the < 10% level. +The discovery of FRB 20220319D has major implica- +tions for the search for FRBs in the local universe, and +for future FRB studies of the CGM of the Milky Way +and other galaxies. First, we have shown that for low +Galactic latitudes (|b| ≲ 15◦; see Figure 4) there can +be significant uncertainties in the widely used NE2001 +and YMW16 models for DMISM, which may lead to +nearby extragalactic FRBs being missed in surveys. In +all cases we recommend that FRB surveys save candi- +dates at DMs below the model values of DMISM. The +possibility of missing FRBs with low extragalactic DMs +also needs to be considered when using FRB samples to +directly infer a characteristic DMCGM (e.g., Platts et al. +2020). These studies are also likely to be impacted by +the uncertainties in deriving DMCGM that in most cases +will be on the order of the values of DMCGM allowed +by our analysis (see Figure 7). We recommend that in- +ferences of DMCGM with FRBs focus on high-latitude +sightlines towards FRBs interferometrically localized to +nearby galaxies, wherein uncertainties in DMIGM and +DMhost can be minimized. Nonetheless, if the charac- +teristic DMCGM is indeed on the order of 10 pc cm−3, +a very large number of local FRBs will be required to +suppress variance between halo sightlines, and uncer- +tainties in other DM contributions. These samples may +be furnished by future coherent all-sky radio monitors +(e.g., Lin et al. 2022). Finally, consistent with previ- +ous studies (Keating & Pen 2020; Kirsten et al. 2022), +our constraints on fCGM firmly exclude the fiducial pa- +rameterizations of the widely used modified-NFW and +Maller & Bullock (2004) models for the CGM DM as +described by Prochaska & Zheng (2019). This impacts +studies that have assumed the correspondingly large val- +ues of DMCGM (e.g., Ravi 2019), as well as forecasts for +the contributions of the CGM of intervening galaxies to +FRB DMs. +The host of FRB 20220319D, IRAS 02044+7048, ap- +pears to have an unusually large specific SFR. This is +despite FRB 20220319D being the closest so far non- +repeating FRB yet discovered, and the strong evolution +towards higher SFRs of the star-forming main sequence +of galaxies with redshift (Speagle et al. 2014). Our anal- +ysis of the SFH of the IRAS 02044+7048 host galaxy +suggests that most stars were formed in a burst in the +last ≲ 2 Gyr. Thus it is likely that the progenitor of +FRB 20220319D does not have a delay time (i.e., age +from its formation epoch) in excess of this timescale. +Although several progenitor scenarios remain consistent +with this constraint, the additional possible association +of the FRB source with a spiral arm may imply a pro- +genitor age that is a fraction of a ∼ 100 Myr dynamical +timescale. We note that it is not straightforward to com- +pare our inferred SFR, which is averaged over the past +100 Myr of the SFH, to the Hα-based SFRs common in +the FRB literature (e.g., Bhandari et al. 2022), which +are sensitive to the last ∼ 10 Myr (Kennicutt 1998). A +direct comparison of non-parametric SFH estimates may +lead to a more complete view of the formation channels +of FRB progenitors. +7. SUMMARY AND CONCLUSIONS +We present the DSA-110 discovery and interfero- +metric localization of the nearest so far non-repeating + +14 +Ravi et al. +FRB (20220319). The burst was observed with a high +S/N of 79, a DM of 110.95 pc cm−3, and has a lin- +ear polarization fraction of 16 ± 3% and an RM of +50 ± 15 rad m−2 (Figure 1, and Table 1). +The FRB +originated from the position (R.A. J2000, decl. J2000) += (02:08:42.7(1), +71:02:06.9(6)), where uncertainties +in the last significant figures are given in parenthe- +ses (Figure 2). +We associate FRB 20220319D with a +spiral-arm feature of a face-on star-forming galaxy at a +distance of 50 Mpc (IRAS 02044+7048; Figures 3 and +5; Table 3). +IRAS 02044+7048 is moderately mas- +sive (log M∗ = 9.93 ± 0.07), with an approximately +solar metallicity and an SFR averaged over the past +100 Myr of 1.8 ± 0.7 M⊙ yr−1 (Figure 6). The low DM +of FRB 20220319D motivates a sensitive new constraint +on the baryon mass of the Milky Way CGM. Our con- +clusions are summarized as follows. +• The spatial coincidence of FRB 20220319D and +IRAS 02044+7048 +(false +alarm +probability +of +< +10−4), +and a potential RM detection for +FRB 20220319D that is in excess of the Galac- +tic foreground, both imply a secure association. +However, the DM of FRB 20220319D is somewhat +lower than the predicted Galactic ISM DM along +its sightline from leading models for the WIM dis- +tribution (NE2001 and YMW16; Cordes & Lazio +2002; Yao et al. 2017). We show through an anal- +ysis of the DMs towards pulsar-hosting globular +clusters with Gaia parallax distances (Figure 4) +that there are significant uncertainties in NE2001 +predictions for DMISM for |b| ≲ 15◦, and that +YMW16 is particularly uncertain at low latitudes +in the second quadrant, where FRB 20220319D is +located (see also Table 2). We find that estimates +for DMISM based on Hα EMs perform compara- +bly well to the models. We conclude that models +for DMISM towards FRB 20220319D can accom- +modate an extragalactic origin, and urge FRB sur- +veys to consider candidate bursts at DMs below +standard model predictions for DMISM. +• The SFH of IRAS 02044+7048 is consistent with +most stars being formed in a burst within the +past two Gigayears. This, together with the pos- +sible location of FRB 20220319D within a spi- +ral arm of IRAS 02044+7048, suggests a mod- +erately low delay time for the burst progenitor. +Despite its nearby distance, the specific SFR of +IRAS 02044+7048 derived from SED modeling is +larger than all but one so far non-repeating FRB, +although the comparison with literature samples +of host galaxies would be aided by the widespread +use of non-parametric SFH analyses. +• We find conservative upper limits on the Galactic +CGM contribution to the DM of FRB 20220319D +of either 28.7 pc cm−3 or 47.3 pc cm−3: the two val- +ues are based on DMISM-estimates from the DMs +of two pulsars nearby on the sky. These limits ex- +clude some literature models (Maller & Bullock +2004; Faerman et al. 2017; Prochaska & Zheng +2019) for the baryon distribution in the CGM (Fig- +ure 7). +We find that consistent results are ob- +tained from estimates of the CGM DM contribu- +tions towards pulsars in the LMC and in the dis- +tant (18.5 kpc) high-latitude globular cluster M53. +The M53 pulsars in particular provide interest- +ing constraints in the inner regions of the Milky +Way halo that complement the FRB constraint, +although the M53 DMCGM estimate is sensitive to +the assumed ISM contribution. +• We derive an upper limit on the mass of baryons in +the Galactic CGM of log MCGM ≲ 10.8 − 11 from +FRB 20220319D, and log MCGM ≲ 10.5−10.7 from +M53 (Figure 8). +The range corresponds to the +range of assumed indices (0.4–0.5) of the β pro- +file for the baryon halo density distribution. For +a fiducial total mass (baryons and dark matter) +of the Milky Way of 1.5 × 1012M⊙, and an as- +sumed baryon-disk mass of 6×1010M⊙, the Milky +Way contains ≪ 60% of the cosmological-average +baryon mass. +This is consistent with a baryon +census, predicted by galaxy-formation simulations, +wherein these “missing” baryons are expelled from +halos like the Milky Way into the IGM through ki- +netic and thermal feedback from AGN, supernovae +and massive stars. +Although our analysis relies +on just a few sightlines, the conservative upper +bounds on MCGM assuming spherical symmetry +are robust to variations in the density distribution +of halo baryons, and to assumptions on the chem- +ical and thermal state of the halo. +Studies of unseen baryons with FRB propagation sig- +natures, like DMs, promise to transform our understand- +ing of the distribution of baryons around and in be- +tween galaxies (Ravi et al. 2019). As more FRBs like +FRB 20220319D are localized to nearby galaxies (see +also Kirsten et al. 2022), constraints on the CGM con- +tent of the Milky Way will continue to improve, and +direct measurements of MCGM may be possible. These +measurements are likely to be important in assembling +an in-situ picture of the processes whereby a galaxy +grows from and impacts its environment. + +Mass of the Milky Way CGM +15 +The authors thank staff members of the Owens Valley +Radio Observatory and the Caltech radio group, includ- +ing Kristen Bernasconi, Stephanie Cha-Ramos, Sarah +Harnach, Tom Klinefelter, Lori McGraw, Corey Posner, +Andres Rizo, Michael Virgin, Scott White, and Thomas +Zentmyer. Their tireless efforts were instrumental to the +success of the DSA-110. The DSA-110 is supported by +the National Science Foundation Mid-Scale Innovations +Program in Astronomical Sciences (MSIP) under grant +AST-1836018. 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W., et al. 2020, ApJ, +896, 143. doi:10.3847/1538-4357/ab960a + diff --git a/7dAzT4oBgHgl3EQfEvrf/content/tmp_files/load_file.txt b/7dAzT4oBgHgl3EQfEvrf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac5210d5f987c20caac655ee7b7b09eb345be421 --- /dev/null +++ b/7dAzT4oBgHgl3EQfEvrf/content/tmp_files/load_file.txt @@ -0,0 +1,1724 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf,len=1723 +page_content='Draft version January 4, 2023 Typeset using LATEX twocolumn style in AASTeX631 Deep Synoptic Array science: a 50 Mpc fast radio burst constrains the mass of the Milky Way circumgalactic medium Vikram Ravi,1, 2 Morgan Catha,2 Ge Chen,1 Liam Connor,1 James M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Cordes,3, 4 Jakob T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Faber,1 James W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Lamb,2 Gregg Hallinan,1, 2 Charlie Harnach,2 Greg Hellbourg,1, 2 Rick Hobbs,2 David Hodge,1 Mark Hodges,2 Casey Law,1, 2 Paul Rasmussen,2 Kritti Sharma,1 Myles B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Sherman,1 Jun Shi,1 Dana Simard,1 Jean J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Somalwar,1 Reynier Squillace,1 Sander Weinreb,1 David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Woody,2 Nitika Yadlapalli,1 (The Deep Synoptic Array team) 1Cahill Center for Astronomy and Astrophysics, MC 249-17 California Institute of Technology, Pasadena CA 91125, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2Owens Valley Radio Observatory, California Institute of Technology, Big Pine CA 93513, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 3Department of Astronomy, Cornell University, Ithaca, NY 14853, USA 4Cornell Center for Astrophysics and Planetary Science and Department of Astronomy, Cornell University, Ithaca, NY 14853, USA ABSTRACT We present the Deep Synoptic Array (DSA-110) discovery and interferometric localization of the so far non-repeating FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The FRB originates in a young, rapidly star-forming barred spiral galaxy, IRAS 02044+7048, at just 50 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although the NE2001 and YMW16 models for the Galactic interstellar-medium (ISM) contribution to the DM of FRB 20220319D exceed its total observed DM, we show that uncertainties in these models accommodate an extragalactic origin for the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We derive a conservative upper limit on the DM contributed by the circumgalactic medium (CGM) of the Milky Way: the limit is either 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 pc cm−3 and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 pc cm−3, depending on which of two pulsars nearby on the sky to FRB 20220319D is used to estimate the ISM DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These limits both imply that the total Galactic CGM mass is < 1011M⊙, and that the baryonic mass of the Milky Way is ≲ 60% of the cosmological average given the total halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' More stringent albeit less conservative constraints are possible when the DMs of pulsars in the distant globular cluster M53 are additionally considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although our constraints are sensitive to possible anisotropy in the CGM and to the assumed form of the radial-density profile, they are not subject to uncertainties in the chemical and thermal properties of the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Our results strongly support scenarios commonly predicted by galaxy-formation simulations wherein feedback processes expel baryonic matter from the halos of galaxies like the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Keywords: Barred spiral galaxies — circumgalactic medium — radio interferometers — radio transient sources — star formation — warm ionized medium 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' INTRODUCTION Galaxies like the Milky Way are embedded in a multi- phase (∼ 104 − 107 K), highly ionized (hydrogen neu- tral fractions ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='01%), kinematically complex, spa- tially clumpy and anisotropic circumgalactic medium (CGM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Tumlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Likely extending beyond dark-matter halo virial radii rvir, the CGM may repre- sent the dominant baryon component by mass within Corresponding author: Vikram Ravi vikram@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='edu halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The physical properties of the CGM of exter- nal galaxies, including density, temperature, kinematic structure, and chemical composition have long been probed by absorption-line measurements towards back- ground objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' More recently, detections of thermal bremsstrahlung (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2018) and the Sunyaev- Zeldovich effect around nearby galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Bregman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022) provide independent constraints on the CGM density and temperature structure, potentially confirming the presence of extended coronae around galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Evidence for extended gas reservoirs associ- ated with nearby galaxies, either in the CGM or the intra-group medium, has also been observed in the dis- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='01000v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='GA] 3 Jan 2023 2 Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' persion measures (DMs) of background fast radio bursts (FRBs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Connor & Ravi 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Wu & McQuinn 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Galaxy-formation simulations highlight the dependence of CGM properties on feedback from stellar winds, su- pernovae, and AGN, as well as on the galaxy merger his- tories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Wijers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Hafen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Fielding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The simulations predict varying degrees of anisotropy in the CGM of individual galax- ies, and different total CGM masses for galaxies with comparable global characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A ubiquitous predic- tion of simulations is that feedback processes determine the integrated CGM mass, MCGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In general, CGM baryon fractions fCGM = MCGMΩM MtotΩb ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 are predicted, where Mtot is the total (dark matter and baryonic) halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A halo of ∼ 106 K gas at densities ≲ 10−3 cm−3 sur- rounding the Milky Way was proposed by Spitzer (1956) as a solution to the problem of confining distant cold- gas clouds at high Galactic latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Tentative evi- dence for such coronal gas had previously been found in observations of the diffuse radio-synchrotron back- ground (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Baldwin 1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A theoretical basis for the origins of such a hot CGM around galaxies was de- veloped in the 1970s (Rees & Ostriker 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' White & Rees 1978), with the gas infalling onto dark-matter ha- los heated to virial temperatures of ≳ 106 K, possibly in shocks at the halo virial radii (see also Birnboim & Dekel 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' It would, however, be nearly two decades before further observational evidence for a hot Milky Way CGM was identified using observations of shadows in the soft X-ray background from high-latitude (Snow- den et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1991) and high-velocity (Herbstmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1995) cold clouds, and in the spectral decomposition of the background into absorbed and unabsorbed compo- nents (Garmire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Today the hot CGM of the Milky Way is best traced through X-ray observa- tions of OVII and OVIII diffuse emission, and absorp- tion towards bright background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although high Galactic latitude data are well fit by a spheroidal com- ponent with a mass of a few ×1010M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Bregman & Lloyd-Davies 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2012), more recent models include a substantially less massive (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1%), but denser disk-like component to explain the line ob- servations (Li & Bregman 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Nakashima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Structure in the hot Milky Way CGM is also observed in the form of the Fermi (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2010), Parkes (Carretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2013) and eROSITA (Predehl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020) bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The neu- tral and warm (≲ 105 K) components of the Milky Way CGM, as primarily traced through observations of high- velocity clouds, are likely sub-dominant in mass to the hot component (few ×108M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Putman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2012), but highlight the complexities of the disk-halo interface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Koo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Werk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The mass of the Galactic CGM is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Here we assume Mtot = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 × 1012M⊙ (following Prochaska & Zheng 2019) and a total Galactic mass in stars and cold gas of 6 × 1010M⊙ (Draine 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Setting fCGM = 1 and Ωb/ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='188 (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020) implies MCGM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='82 × 1011M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Recent models for the distribution of OVII and OVIII emission and absorp- tion result in MCGM values between 3 − 4 × 1010M⊙ (Li & Bregman 2017), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 × 1010M⊙ (Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020), and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 × 1011M⊙ (Yamasaki & Totani 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2017) argue that a portion of the hot CGM is also traced by OVI absorption, as is observed in external galaxies (Tumlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2011), and find MCGM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 × 1011M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Systematic effects that make the estimation of MCGM difficult include the model uncertainties in foregrounds and in the spatial distribution and clumpiness of the gas, and the poorly constrained gas metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' An alternate measurement of MCGM was found by Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2015) by model- ing the ram-pressure stripping of the Large Magellanic Cloud (LMC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' a low value of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4) × 1010M⊙ was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The DMs of FRBs and pulsars within the Milky Way halo offer a unique probe of the content of the Galactic CGM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Anderson & Bregman 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Platts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' FRBs are transient radio emissions from distant extragalactic sources that, like the emissions from radio pulsars, are so short in duration that the dispersion in intervening plasma is evident as an arrival-time delay at lower frequencies: ∆t(ν) = q2 e 2πmecν2 � D 0 ne(l)dl, (1) where qe is the electron charge, me is the electron mass, ν is the frequency, D is the source distance, and ne(l) is the electron number density along the sightline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Under physical conditions of CGM gas, DMs quantify the col- umn densities along the source sightlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In all cases, however, the observed DMs of FRBs, DMobs, are com- posed of a selection of additive components: DMobs = DMISM + DMCGM + DMIGM + DMhost, (2) where in this formulation DMISM includes the warm ion- ized medium (WIM) in the Milky Way disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Cordes & Lazio 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2017), DMCGM is the DM con- tributed by the Galactic CGM, DMhost is the DM as- sociated with the FRB host-galaxy interstellar medium (ISM) and their halos, and DMIGM includes contribu- tions from the intergalactic medium (IGM) and any in- tervening galaxy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For pulsars within the Milky Mass of the Milky Way CGM 3 Way halo, such as in the Magellanic Clouds and in sev- eral globular clusters, DMIGM = 0 pc cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These ex- pressions apply to the redshift ∼ 0 regime relevant to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' If the components of DMobs can be accu- rately modeled or bounded, constraints can be placed on DMCGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Modeling of the DMs of pulsars in the LMC led Anderson & Bregman (2010) to find a very low MCGM ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5×1010M⊙ assuming a Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (1997) profile (an NFW profile);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' it is likely that an overly large DMISM was assumed, and the NFW profile is no longer favored for the Galactic CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although most FRBs are too distant to allow for accurate modeling of the compo- nents of DMobs, FRBs from nearby galaxies (Bhardwaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Kirsten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022) have begun to be used to place constraints on DMCGM, assuming specific models for DMISM, that are in tension with some models for the Milky Way CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In this paper we present observations with the Deep Synoptic Array (DSA-110) of a new FRB (FRB 20220319D) that motivates stringent new con- straints on the mass of the Milky Way CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Remark- ably, FRB 20220319D was observed to have DMobs < DMISM for leading models of the warm-gas distribution in the Galactic disk, despite the unambiguous associ- ation we find with a galaxy at a distance of 50 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We detail observations of the FRB in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We con- sider the association with its host galaxy in §3, wherein we show that the DM of FRB 20220319D is consis- tent with an extragalactic origin given uncertainties in models for DMISM, using observations of the DMs of pulsars in globular clusters with accurate parallax dis- tances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We present an analysis of the host environ- ment of FRB 20220319D in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We then place conser- vative constraints on DMCGM, and hence MCGM, using FRB 20220319D and the distant high-latitude globular cluster M53, in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We discuss the implications of our results in §6, and conclude in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' DSA-110 OBSERVATION OF FRB 20220319D The DSA-1101 is a radio interferometer hosted at the Owens Valley Radio Observatory (OVRO), pur- pose built for the discovery and arcsecond-localization of FRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A full description of the instrument will be presented in Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' During the observa- tions discussed herein, the DSA-110 was configured as described in Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Of particular note is that when FRB 20220319D was observed, early in DSA- 110 commissioning, all candidates at DMs in excess of 1 https://deepsynoptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='org 100 0 100 degrees Intrinsic PPA 0 10 20 30 S/N Intensity (I) Linear Polarization (L) Circular Polarization (V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 Time (ms) 1325 1350 1375 1400 1425 1450 1475 Frequency (MHz) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Dedispersed temporal profile and dynamic spec- trum of FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A value of DMobs = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='95 pc cm−3 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The top panel shows the estimated absolute po- sition angle based on an approximate parallactic angle, the ∼ 20◦ error due to RM uncertainty is not included in the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The middle panel shows temporal profiles in to- tal intensity, linearly polarized intensity, and Stokes V, as la- beled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The bottom panel shows the dynamic spectrum of the total-intensity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The time resolution in the time-series and dynamic-spectrum plots is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='768 µs, and the temporal profile is in units of signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The reference time is MJD 59657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='93275696795, which was the burst arrival time at OVRO at 1530 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The polarized data on the burst was corrected for the measured RM of 50 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 80 pc cm−3 were saved for further inspection, regardless of the expected DM through the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' FRB 20220319D was detected during standard com- missioning observations, with an arrival time at OVRO at 1530 MHz of MJD 59657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='93275696795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' During these observations, the DSA-110 was parked at a pointing- center declination of +71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The DSA-110 refer- ence position is −118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='283◦ longitude, +37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2334◦ lat- itude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' FRB 20220319D was detected with a DM of 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='96 pc cm−3, and a signal to noise ratio (S/N) of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' To derive optimized burst parameters and polarization properties, we coherently combined voltage data from 4 Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' the 48 core antennas towards the detection-beam direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The resulting temporal profile and calibrated dy- namic spectrum, produced with incoherent dedispersion at the native time and frequency resolutions, are shown in Figure 1, and optimized parameters are given in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The polarization analysis was done following pro- cedures described in Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The matched- filter total-intensity S/N estimate is 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The burst exhibits a narrow temporal profile with a strongly modulated spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The DM smearing timescale ranges between 8–13 µs within the DSA-110 band, and so the data resolve the temporal structure of the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Scaling the system-equivalent flux density of the DSA-110 by the primary-beam attenuation at the burst position, we derive a fluence of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 Jy ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Following techniques described in Connor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2020), we derive a spectral decorrelation bandwidth, νd = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 MHz, defined as the 1/e scale of the autocor- relation function of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The burst spectro- temporal morphology is consistent with the population of so far non-repeating FRBs (Pleunis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We derive a low fractional polarization of just 16% (Fig- ure 1), with no significant circular polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A Fara- day rotation measure (RM) of 50 ± 15 rad m−2 was de- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The error was estimated by simulating the re- covery of RMs with the same linear-polarization S/N as FRB 20220319D (Sherman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The interferometric localization of FRB 20220319D was derived following procedures largely described in Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Correlation products for all baselines formed from the 63 functioning antennas of the DSA-110 at the time of detection were integrated over 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='144µs centered on the burst arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Bandpass calibra- tion was done using a 10-min observation of 3C309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 ob- served 11 hr prior to the burst detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Complex gain calibration at the time of the burst detection was derived using an NRAO VLA Sky Survey (NVSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1998) model and 5 min of visibility data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We show im- ages of the point-spread function (PSF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' a flat spectrum was assumed), and the pre- and post-deconvolution im- ages of FRB 20220319D in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Briggs weighting with a ‘robust’ parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 was used to suppress sidelobes, given the high S/N of FRB 20220319D, yield- ing a PSF FWHM of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9′′ × 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The position of FRB 20220319D, given in Table 1, was derived by fit- ting an elliptical Gaussian to the deconvolved image of FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We derived the uncertainty in the FRB position us- ing data on nine compact bright sources from the Radio Fundamental Catalog (RFC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' rfc 2022c), obtained within 2 hr of the burst observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A fit to a linear trend in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' and Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' was used to derive arcsecond-level astro- Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Properties of FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Parameter Value Arrival time (MJD)a 59657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='93275696795 DM (pc cm−3) 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='95(1) Full-width half-maximum (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='16(1) Fluence (Jy ms) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2(2) Spectral energy (erg Hz−1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3(1)×1028 L/Ib 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='16(3) |V/I|b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='04(3) RM (rad m−2) 50(15) νd (MHz)c 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6(2) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (J2000) 02:08:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7(1) Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (J2000) +71:02:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9(6) aArrival time at OVRO at 1530 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' b L/I is the fraction of linearly polarized fluence, and |V/I| is the absolute value of the fraction of circularly polarized fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' c νd is the characteristic spectral decorrelation band- width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' metric corrections, and the corresponding uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These uncertainties (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='52′′ in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5′′ in Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=') were then added in quadrature to the statistical uncer- tainty in the fitted burst position (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='29′′ in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='17′′ in Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=') to derive a final 90% confidence contain- ment ellipse with semi-axes of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='25′′ in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='18′′ in Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' This ellipse is shown in the bottom-left panel of Figure 2, together with the detrended position offsets of the RFC sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We also assessed the quality of the in-field calibration by checking the positions of bright (> 50 mJy) compact (< 20′′) NVSS sources within the primary-beam FWHM against NVSS catalog positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These offsets (with the RFC corrections applied) are also shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A more detailed analysis of the DSA- 110 localization accuracy will be presented in Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' ASSOCIATION WITH IRAS 02044+7048, AND UNCERTAINTIES IN DMISM The 90% confidence localization region of FRB 20220319D is shown in Figure 3 overlaid on a Pan-STARRS1 (PS1) i-band image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The galaxy co- incident with the FRB location is cataloged in the NASA Extragalactic Database as IRAS 02044+7048, with a spectroscopic redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='011 indicating a dis- tance of approximately 50 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Using observations described below, we derive a spectroscopic redshift of Mass of the Milky Way CGM 5 100 50 0 50 100 100 50 0 50 100 PSF 100 50 0 50 100 100 50 0 50 100 FRB 20220319 - dirty 100 50 0 50 100 100 50 0 50 100 FRB 20220319 - clean 2 0 2 2 1 0 1 2 Offsets NVSS offsets VLBI sources Right Ascension offset (arcsec) Declination offset (arcsec) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' DSA-110 localization of FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Top left: Point-spread function (PSF) of the DSA-110 for the observation of FRB 20220319D, assuming a flat-spectrum source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Top right: Dirty image of FRB 20220319D, with no deconvolution applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Bottom left: Deconvolved image of FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The synthesized beam is represented by an ellipse with half-power diameters of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9′′×15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6′′, at a position angle of 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Briggs weighting was used with a ‘robust’ parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5, in order to partially suppress PSF sidelobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Baselines shorter than 200λ were excluded from the analysis because they were affected by spurious cross-coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Contours in the preceding three panels are at −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 (dashed), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9 (solid) of the peak intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The images are centered on the coordinates (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' J2000, decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' J2000) = (02:08:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7, +71:02:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Bottom right: Offsets of known astronomical sources from their true positions as measured by the DSA-110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The blue disks show offsets of sources from cataloged positions in the NRAO VLA Sky Survey (NVSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1998) in a 5 min DSA-110 image formed from data taken at a time centered on the burst detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Only sources with cataloged flux densities > 50 mJy and measured major-axis diameters < 20′′, within 2 deg of the pointing center, were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The symbol size is proportional to flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The red crosses show measured offsets of nine RFC calibrator sources (see text for details) observed immediately preceding and after the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The ellipse indicates the 90% confidence containment region for FRB 20220319D, derived from the RFC calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 6 Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2h08m45s 42s 39s 36s 71°02\'30" 15" 00" 01\'45" Right Ascension (J2000) Declination (J2000) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Pan-STARRS i-band image of IRAS 02044+7048, with the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 arcsec 90% confidence localization region of FRB 20220319D indicated as a white ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0111 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0004, indicating a luminosity distance of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' An isophotal fit to the PS1 i-band image in- dicates an effective radius of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 kpc, and the FRB is offset by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The isophotal analysis also indicates an inclination of 23 ± 3 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A barred spiral morphology, with a classification of SBa, is evident from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Under the assumption that FRB 20220319D is extragalactic, the spatial association between FRB 20220319D and IRAS 02044+7048 is secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' It is unlikely that FRB 20220319D lies significantly beyond IRAS 02044+7048;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' the morphology of the galaxy indi- cates the presence of a warm-ISM phase, and scattering in this ISM would likely cause temporal broadening of several hundred milliseconds at our observing fre- quencies (Cordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Ocker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The Gravitational Wave Galaxy Catalog (GWGC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2011) lists 15869 galaxies at distances less than 50 Mpc, subtending a total of 4 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Thus the chance- association probability of FRB 20220319D and a galaxy at < 50 Mpc is ≲ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The extragalactic nature of FRB 20220319D is called into question by its DM of 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='95 pc cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The pre- dicted DM for extragalactic objects along its l = 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2◦, b = +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1◦ sightline is 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9 pc cm−3 according to the NE2001 model for the Galactic ionized ISM distribu- tion (Cordes & Lazio 2002), and 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 pc cm−3 accord- ing to the YMW16 model (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The RM detection for FRB 20220319D of 50±15 rad m−2 is, how- ever, marginally consistent with being in excess of the expected Milky Way contribution along the burst sight- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A recent model for the Galactic RM foreground along this sightline (Hutschenreuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022) indi- cates an expected Galactic RM of −14 ± 18 rad m−2 towards FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We cannot draw conclusions from the measured spectral decorrelation bandwidth of νd = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 MHz, which is far in excess of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 MHz decorrelation bandwidth predicted by NE2001 due to scattering in the Milky Way ISM, because the origins of the spectral structure cannot easily be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We now analyze the performance of the NE2001 and YMW16 models with the aim of determining whether the model uncertainties can accommodate an extra- galactic origin for FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Both model the dis- tribution of the WIM (Draine 2011) in the Galactic disk using observations of radio-wave propagation, with a dominant thick-disk component encapsulating several overdensities and voids motivated by multiwavelength data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The YMW16 model is fit to pulsars with indepen- dent DM and distance estimates, whereas the NE2001 model is additionally fit to pulsar and extragalactic radio-source scattering measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For low-latitude extragalactic sightlines, the radial extent of the thick disk is critical in determining DMISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In NE2001, a cutoff at a radius of 20 kpc is motivated by observations of HII regions in other galaxies and scattering of extra- galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Surveys of Galactic HII regions on the other hand motivate the cutoff of 15 kpc in YMW16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Neither cutoff is estimated in the NE2001 or YMW16 fits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' rather they are fixed model choices that do not im- pact the match between the models and the data un- der consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The functional forms for the WIM distribution in the thick disk also differ, with NE2001 incorporating an oblate spheroid, and YMW16 incorpo- rating a plane-parallel slab that results in significant de- viations from NE2001 at low latitudes in the second and third quadrants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' High-latitude extragalactic sightlines are likely to be better modeled because the scale height of the thick disk is a critical fit- ted parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' However, density fluctuations caused by turbulence in the WIM, together with clumps and voids, complicate our picture even at high latitudes (Ocker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Large-scale WIM inhomogeneities at the disk- halo interface such as Galactic chimneys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Koo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Normandeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1996) can result in significant departures from spatially smooth models for DMISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We first note that the recent PSRπ sample of pul- sar parallax measurements (Deller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2019) reveals discrepancies between the NE2001 and YMW16 models and measured pulsar distances at low latitudes in the second quadrant, where FRB 20220319D appears on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In general, known errors in NE2001 and YMW16 Mass of the Milky Way CGM 7 exhibit significant spatial correlation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021), motivating the consideration of sources along sightlines similar to FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Table 2 lists the DMs of the three nearest pulsars in the sample to FRB 20220319D, together with the predicted DMs given the measured distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The YMW16 model signifi- cantly overpredicts all three DMs, and although NE2001 is more accurate for the modestly distant pulsars likely in the Cygnus-Orion arm, it also overpredicts the DM of the more distant PSR J0406+6138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A more detailed analysis of the performance of the models for sightlines towards distant objects is enabled by the recent convergence in the distances to globu- lar clusters hosting pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We obtain accurate dis- tance estimates (less than few-percent errors) to all 21 globular clusters with associated pulsars (Manch- ester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2005) at |b| > 5◦ from the compilation of Baumgardt & Vasiliev (2021), primarily based on Gaia parallaxes (Vasiliev & Baumgardt 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Clus- ters at lower latitudes are not considered because their DMs are influenced by the thin disk and other compo- nents in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For each cluster we com- pare the difference between the DMs predicted by the NE2001 and YMW16 models for the cluster distances, and the true cluster DMs defined as the mean DMs of the associated pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The results are shown in Fig- ure 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' FRB 20220319D is also included with the pre- dicted DM set to DMISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The errors in NE2001 at low latitudes can clearly accommodate an extragalactic origin for FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' YMW16 generally performs better than NE2001 for the low-latitude clusters, but these are nearly all in the first and fourth quadrants (with NGC 1851 as the only exception), and so perhaps not representative of errors along sightlines closer to the Galactic anticenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' As an additional check of the Galactic sightline to- wards FRB 20220319D, we also consider independent measures of DMISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' First, we use the relation between the neutral hydrogen column density and pulsar DMs derived by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2013) to estimate the DMs to- wards FRB 20220319D and each globular cluster, as- suming no HI gas beyond the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We obtain HI column densities from the HI4PI data (HI4PI Collabo- ration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The results are shown in Figure 4, including errors representing intrinsic scatter in the rela- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The HI column densities result in underestimates of the DMs towards low-latitude clusters, possibly re- flecting the increased ionization of HI gas towards the central regions of the Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Madsen & Reynolds 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The HI-predicted DMISM for FRB 20220319D is consistent with an extragalactic origin, although the uncertainties are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Second, we estimate the DMs towards FRB 20220319D and each globular cluster us- ing Hα emission measures (EMs) derived from the Wis- consin Hα Mapper (WHAM) all sky maps (Haffner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We follow the analysis of Berkhuijsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2006), wherein the photon fluxes, FHα (in units of Rayleighs), integrated over the range of Galactic veloci- ties are converted to DMs using the following relations: EM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='25FHαe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2E(B−V ) cm−6 pc (3) DM = (EM × D × F)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (4) F = f ζ(1 + ϵ2) (5) The expression for F (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Ocker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020, and references therein) encodes an ionized cloudlet model, wherein the WIM is composed of cloudlets with a volume filling factor f, ζ = ⟨n2 e⟩/⟨ne⟩/2 ∼ 2 captures the cloud to cloud variation, and ϵ2 ≲ 1 is the density variance internal to the cloudlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Further, E(B − V ) is a mea- sure of the interstellar redenning along a given sightline, and D is the distance through the WIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We have as- sumed a uniform WIM electron temperature of 8000 K, and equivalence between the line-of-sight and volume filling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We find that for the cluster sightlines the Hα-based DM estimates perform remarkably well for a nominal value F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', f ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4), comparably to YMW16 and better than NE2001 for low-latitude sight- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' This is surprising given previous results for pulsar DMs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Schnitzeler 2012), but may be explained by globular clusters generally lying beyond the outer extent of the WIM, obviating the need for distance corrections to the WHAM EMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For the FRB 20220319D sight- line, the Hα flux implies DMISM = 89 pc cm−3, which is 22 pc cm−3 lower than the measured DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We conclude that models for the DM towards FRB 20220319D contributed by the Galactic WIM are consistent with an extragalactic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' This se- cures the association of the burst with the galaxy IRAS 02044+7048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Consistent with the original presen- tations of the NE2001 and YMW16 models for the WIM distribution, we confirm that there are significant model uncertainties at low Galactic latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We used a large sample of globular-cluster parallax distances, which have DM estimates from associated pulsars, to demonstrate that an alternative predictor of DMISM based on the to- tal Galactic Hα flux can provide a useful check on the aforementioned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In order to estimate DMISM towards FRB 20220319D we consider pulsars that are nearby on the sky and at a similar Galactic latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The two nearest pulsars within 1 deg of the burst in Galactic latitude, PSR J0231+7026 (total separation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9 deg) 8 Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Measured and predicted DMs of PSRπ pulsars near FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Pulsar l (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=') b (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=') FRB offset (deg) Distance (kpc) DMa NE2001a YMW16a PSR J0147+5922 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='16 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 PSR J0157+6212 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='80+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='12 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 PSR J0406+6138 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='58+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='87 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 aAll DMs are given in units of pc cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' NGC6517 NGC6539 M62 M22 FRB20220319 NGC6652 NGC6397 NGC5986 OmegaCen M4 M10 NGC6752 M3 M15 M14 NGC1851 M2 M13 47Tuc M5 M30 M53 200 150 100 50 0 50 100 150 (Predicted DM) - (True DM) (pc/cc) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 |b| (deg) NE2001 YMW16 Halpha (ff=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1) NH Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Difference between the predicted DM and true DM of all globular clusters that host pulsars, sorted by increasing |b|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Predictions are made for each globular cluster given its distance and location on the sky, using four different models for the warm ISM of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The two literature models are NE2001 (blue solid thick curve) and YMW16 (orange solid thin curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We also convert the measured Hα emission measures (green dashed curve) and HI columns (red dots and shaded region) along each sightline to DM predictions using methods described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We also include predictions for FRB 20220319D assuming a location beyond the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' and PSR J0325+6744 (total separation of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 deg), have DMs of 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 pc cm−3 and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 pc cm−3, and we consider these as conservative and less conservative estimates re- spectively of DMISM along the burst sightline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We note that the predicted NE2001 and YMW16 DMISM values along the sightlines towards these pulsars are lower than along the burst sightline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' THE HOST ENVIRONMENT OF FRB 20220319D We obtained long-slit optical spectroscopy of IRAS 02044+7048 using the Double Spectrograph (DBSP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Oke & Gunn 1982) mounted on the Palo- mar 200-inch Hale telescope on 2022 June 01 (UT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The observations were undertaken under conditions of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2′′ seeing, at an airmass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9, and used a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5′′ slit positioned to include both the FRB location and the galaxy nucleus at a position angle of 104◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Two 600 s exposures were obtained with the D55 dichroic and the 316/7500 grating on the red arm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' we only considered data from the red arm of the spectrograph given the significant extinction along this sightline (AV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='209;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Schlafly & Finkbeiner 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The two-dimensional spec- tral data were reduced according to standard procedures using a PypeIt-based pipeline (Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We then defined the trace function using observations of the standard star Feige 34, and extracted spectra in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5′′ windows along the slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Feige 34 was also used Mass of the Milky Way CGM 9 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Observed and derived parame- ters of the host galaxy of FRB 20220319D, IRAS 02044+7048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1σ errors in the last significant figures are given in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Parameter Value Redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0111(4) Luminosity distance (Mpc) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 Effective radius (kpc) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7(2) FRB projected offset (kpc) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3(3) log M∗ (M⊙) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='93(7) log Za 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2) Internal AV b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 SFR (M⊙ yr−1)c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7) Inclination (deg) 23(3) aMetallicity with respect to solar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' b V -band extinction corresponding to a uniform dust slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' c Averaged over the past 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 6 4 2 0 2 Offset along slit (kpc) 0 10 20 30 40 50 H flux (1040 erg s 1) FRB Nucleus Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Total Hα flux observed at different lo- cations within our longslit spectroscopic observations of IRAS 02044+7048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Measurements were obtained at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5′′ in- tervals along a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5′′ longslit oriented at a position angle of 104◦, which covered both the FRB position and the cen- ter of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The locations of the galaxy nucleus and FRB 20220319D are indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' for flux calibration, and all spectra were corrected for extinction according to the Fitzpatrick & Massa (2007) extinction curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We detected Hα emission at several locations along the slit, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The strongest Hα emission is observed at the galaxy core, from which we also detect lines corresponding to [NII] and [SII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These lines were together used to derive a redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0111 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0004 for IRAS 02044+7048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although we do not detect Hβ or [OIII], the ratio log [NII]/Hα = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='02 indicates a < 10% chance of the nuclear ionization being purely due to star formation activity (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' There is evidence of Hα emission from spiral-arm features on either side of the nucleus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' these features are too extended to represent HII regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The FRB appears associated with the eastern arm, although not at the center of its Hα radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' No additional evidence for or against an arm association can be gleaned from the residuals of a fit of the IRAS 02044+7048 image in Figure 3 to a two-dimensional Sersic profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In order to determine global properties of IRAS 02044+7048, we derived and modeled its spec- tral energy distribution (SED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We executed aperture photometry on archival images from the Panoramic Survey Telescope and Rapid Response System (Pan- STARRS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2016), Two Micron All Sky Survey (2MASS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2006), and ALLWISE (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021) surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We identified an elliptical aperture that captured the i-band extent of the galaxy in Pan-STARRS data, and masked likely foreground objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We then used this aperture, convolved with the PSF of each survey, to measure extinction-corrected AB magnitudes in all filters used by Pan-STARRS, 2MASS, and WISE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We modeled this SED using the Prospector stellar population synthesis modeling code (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We ran Prospector using recommended tech- niques and priors and a non-parametric star-formation history (SFH) model (Leja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2019), and sampled from the posterior using emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Non-parametric models for the SFH result in less bias in both stellar-mass (M∗) and star-formation rate (SFR) estimates, because specific SFH models are not excluded a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We included a model for dust re- radiation in the likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The spectral energy distribution of IRAS 02044+7048 is shown in Figure 6, together with the results from the Prospector analysis, and derived maximum aposteriori probability parame- ters are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The SFR of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 M⊙ yr−1 was derived by integrating the SFH over the past 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' IRAS 02044+7048, and the location of FRB 20220319D within it, are largely consistent with previous results on FRB host galaxies and environ- ments (Bhandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Heintz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Boch- enek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Mannings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Bhandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Most FRB host galaxies have detectable ongoing star formation, and some exhibit signatures of addi- tional nuclear ionization sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The stellar mass of IRAS 02044+7048 is entirely consistent with the distri- 10 Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Left: Prospector fit to the spectral energy distribution (SED) of the host galaxy of FRB 20220319D, IRAS 02044+7048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The SED measurements from archival PanSTARRS, 2MASS, and WISE data are shown in red, together with representative 10% errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Filter transmission curves are shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The maximum aposteriori probability (MAP) model photometry is shown in green, together with the MAP spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 100 spectra generated using random draws from the posterior distributions of the model parameters are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Right: Measured SFH of IRAS 02044+7048 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The results from 100 random draws from the posterior distributions are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' bution of masses found for both repeating and so far non-repeating FRB hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The potential association of FRB 20220319D with a spiral-arm feature is consistent with the results of Mannings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2021) based on Hubble Space Telescope imaging of eight FRB hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In the absence of very long baseline interferometry, our data do not have sufficient angular resolution to deter- mine whether or not the FRB originates from within an association of young stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Tendulkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The location of FRB 20220319D at an offset of ∼ 85% of the effective radius from the galaxy nucleus is consistent with the range of previously observed FRB offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' However, the SFR of IRAS 02044+7048 is rather high for its stellar mass, with respect to the observed sam- ple of hosts of so far non-repeating FRBs (Bhandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Only one so far non-repeating FRB (20190102C, at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2912) has a comparably high specific SFR (sSFR), of log sSFR = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although IRAS 02044+7048 is consistent with a location on the star-forming main sequence of galaxies (Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2014), non-repeating FRB hosts as a population are con- sistent with originating from below this sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' It is possible that FRBs from more actively star-forming hosts are selected against in existing radio surveys due to propagation effects in the host ISM (Seebeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021), in which case it is not surprising that IRAS 02044+7048 has a face-on orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' THE MASS OF THE MILKY WAY CGM The constraints on the extragalactic DM contribution along the sightline towards FRB 20220319D are severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Adopting the two estimates discussed above for DMISM from pulsars nearby to the FRB on the sky, we find possible upper limits on the extragalactic DM contribu- tion (DMCGM+DMIGM+DMhost) of either 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 pc cm−3 or 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 pc cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Even with our conservative estimates for DMISM, FRB 20220319D has the lowest extragalactic DM yet measured for an FRB localized to a host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We thus have the opportunity to stringently bound the heretofore poorly constrained value of DMCGM, and thus directly bound the mass of the Galactic CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The IGM likely contributes 7 pc cm−3 towards IRAS 02044+7048, assuming an IGM baryon fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The detection of extended Hα emission from the galaxy itself, and in particular at the location of FRB 20220319D, indicates that the WIM component of the ISM is present, although we do not know where within the ISM column the FRB source is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 We simply assume a nominal value of DMhost = 10 pc cm−3 for our analysis below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Thus, we find that for the FRB 20220319D sightline, two possible upper limits on DMCGM are 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 pc cm−3 and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 pc cm−3, depending on the assumption for DMISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These upper limits are comparable to previous results from the closest known FRB source (FRB 20200120E), in a globular cluster as- sociated with M81, which provide possible upper limits on DMCGM of 32 pc cm−3 and 42 pc cm−3] depending on the choice of NE2001 or YMW16 for DMISM (Kirsten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The constraints from FRB 20200120E are af- fected by a reliance on the NE2001/YMW16 models for 2 With a measurement of scattering in the host ISM, it is possible to estimate DMhost (Cordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' However, upper limits on the temporal broadening of FRB 20220319D of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 ms are not usefully constraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Model spectrum (MAP) Model photometry (MAP) Observed photometry Flux Density [maggies] 10 10 Wavelength [A]5 4 - SFR [Mo/yr] 3 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 18 0 2 4 10 12 Lookback Time [Gyr]Mass of the Milky Way CGM 11 DMISM, and by stronger assumptions for the halo DM of M81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A recent synthesis of models for the Galactic CGM DM by Keating & Pen (2020) highlighted the wide range of extant predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In this work, we use a representa- tive set of models to demonstrate the implications of the constraint on DMCGM from FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The mod- els are illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We also consider measure- ments of the DM contributed by sightlines through the halo from pulsars in the LMC (Ridley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2013) and in the distant (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Baumgardt & Vasiliev 2021) globular cluster M53 (Kulkarni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although the LMC is at b ≈ −33◦ and M53 is at b ≈ +80◦, we conservatively subtract the lowest DMISM to be found in either the NE2001 or YMW16 models of 18 pc cm−3 to estimate the halo DM contributions along their sightlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In the top two panels of Figure 7, we show the resulting upper limits on DMCGM towards M53 and the LMC, with ranges corresponding to the range of associated pulsar DMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The bottom-left panel shows the upper limit on DMCGM from FRB 20220319D, placed at the virial radius of the Milky Way halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In all cases we show model predictions for DMCGM assuming a spher- ically symmetric galactocentric baryon halo, but with the DM evaluated along sightlines from the position of the Earth (GRAVITY Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We evaluate the modified NFW (mNFW) and Maller & Bul- lock (2004) profiles following Prochaska & Zheng (2019), with a total Milky Way mass of Mtot = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 × 1012M⊙, and a halo baryon fraction of fCGM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The Pen (1999) model is evaluated for the same Mtot and with the core radius set to the halo virial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The semi- empirical Miller & Bregman (2013) and Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2017) models and the semi-analytic Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2020) model are all evaluated as given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Finally, in the bottom-right panel of Figure 7, we also show the model predictions for the halo baryon density along the LMC sightline, together with constraints from ram-pressure stripping analyses of dwarf galaxies (Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Putman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021) and modeling of OVII and OVIII emission measurements (Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The data are in favor of models that predict lower to- tal CGM DMs and lower inner densities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Pen 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Miller & Bregman 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The DMs of M53 pulsars and FRB 20220319D deliver consistent constraints on the range of models, albeit on very dif- ferent radial distance scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The constraints from LMC pulsars are less impactful, but nonetheless also exclude the three models (Maller & Bullock 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Prochaska & Zheng 2019) considered here that predict larger values of DMCGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We note that our treat- ment of the CGM contribution to the DMs of LMC pul- sars is far more conservative than that of Anderson & Bregman (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' this is motivated by the uncertainties discussed above in estimating DMISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The DM con- straints are consistent with the most robust density es- timates that we consider (Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The density estimates are however subject to the uncertainties discussed in §1, and we proceed by considering only the constraints on DMCGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We now derive constraints on the mass of the Galac- tic CGM, MCGM, by finding β models that are consis- tent with the FRB 20220319D and M53 DMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' When converting the electron column density to a mass col- umn density, we assume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='18 proton masses per elec- tron following Yamasaki & Totani (2020), which corre- sponds to roughly solar metallicity gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The β profile is widely adopted in the field to convert (column-)density estimates to halo masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Miller & Bregman 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020), as it is empirically motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The electron number density at a radius r is given by n(r) = n0 � 1 + � r rc �2�−3β/2 , (6) where n0 is a central density, and rc is a core radius that we fix to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='47 kpc following Miller & Bregman (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The exact value of the core radius is unimportant for our conclusions regarding MCGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Typical values of β found using various CGM tracers are in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 ≲ β ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Miller & Bregman 2013, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We consider three con- straints on DMCGM: the lower and upper constraints from FRB 20220319D, and the lower constraint from the M53 pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For each constraint, we derive correspond- ing values of MCGM from an integrated β profile for dif- ferent values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A useful joint constraint on MCGM and β is not possible with the data in hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The results are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For typical values of β of between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5, the DM of FRB 20220319D implies an upper limit on log MCGM of between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The M53 pul- sars are even more constraining, limiting log MCGM to below a value of between 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We consider the FRB 20220319D constraints to be more conserva- tive because the M53 estimate of DMCGM is very sen- sitive to the assumed DMISM along its sightline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The mass constraints all favor values of fCGM < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 (re- call fCGM = MCGMΩM MtotΩb ) for our fiducial value of Mtot = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5×1012M⊙, with M53 potentially accommodating val- ues of fCGM ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 for reasonable values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 12 Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Distance (kpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='DM (pc cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='M53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='mNFW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='MB04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='MB13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Pen99 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Faerman1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Faerman2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Distance (kpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='DM (pc cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='LMC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='mNFW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='MB04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='MB13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Pen99 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Distance (kpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='DM (pc cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='FRB 20220319 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='mNFW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='MB04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='MB13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Pen99 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Faerman1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Faerman2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Distance (kpc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Baryon number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='density (cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Density (LMC direction) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='mNFW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='MB04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='MB13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Pen99 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Faerman1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Faerman2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Illustration of the constraining power of different measurements of the CGM DM and density on a range of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In all panels, curves show predictions of the default modified NFW (mNFW) model of Prochaska & Zheng (2019) (blue), the model of Maller & Bullock (2004) (orange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' MB04), the model of Miller & Bregman (2013) (green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' MB13), a Pen (1999) model (Pen99) assuming a core radius set to the virial radius (red), and the models of Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2017) (purple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Faerman1) and Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' (2020) (brown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Faerman2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Top panels: upper limit on the CGM DM towards M53 (left) and the LMC (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In both cases, the minimum DM (18 pc cm−3) across the sky in either the NE2001 or YMW16 models has been subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The ranges indicate the range of DMs of pulsars associated with each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Bottom left: upper limit on the CGM DM towards FRB 20220319D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The range indicates different assumptions for the ISM DM that correspond to the two closest pulsars on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Bottom right: constraints on the CGM density compared with the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The dark disks indicate measurements using ram-pressure stripping modeling of the LMC (Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2015) and modeling of diffuse X-ray line emission (Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The lower limits displayed as triangles indicate estimates using ram-pressure stripping modeling of a large selection of Milky Way satellites (Putman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' DISCUSSION Our results demonstrate that the total Galactic baryon mass is likely significantly lower than the cos- mological average for a halo as massive as the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The mass constraints we derive are consistent with some previous observational results from analyses of OVII and OVIII emission and absorption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Li & Bregman 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Kaaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020) and ram-pressure stripping of the LMC (Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2015), but inconsis- tent with other results that posit higher CGM masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Yamasaki & Totani 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Analyses of the dynamics of Milky Way satellites im- ply a total virial mass of Mtot = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3) × 1012M⊙ (Watkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2010), a recent Gaia-based analysis of the dynamics of Milky Way globular clusters yields Mtot = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3) × 1012M⊙ (Posti & Helmi 2019), and an analysis of the dynamics of the Magellanic Stream yields Mtot = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3) × 1012M⊙ (Craig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' It is possible that a full accounting for the orbit of the LMC would reduce the estimates of Mtot by ∼ 15% (Correa Magnus & Vasiliev 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' However, in all cases fCGM ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 is implied by FRB 20220319D, and fCGM ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 is implied by the M53 DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Our constraints are affected by different systematic effects to previous results, and do not suffer from uncertainties in model- ing the chemical and thermal states of the CGM, nor from uncertainties in modeling the interaction of satel- lite galaxies with the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These values are consistent with several simulations of the CGM contents of galaxies like the Milky Way that account for the effects of kinetic and thermal feedback on reducing the halo baryon con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The observational constraints on MCGM are affected by uncertainties in identifying the CGM DM contri- bution along the sightlines of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For the low- latitude FRB 20220319D sightline, we adopted conser- vative estimates of DMISM, and there are uncertainties at the level of a few pc cm−3 in the DMIGM and DMhost Mass of the Milky Way CGM 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='6 log MCGM (M ) fCGM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 fCGM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='2 fCGM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1 FRB lower FRB upper M53 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Upper limits on the CGM mass for different assumptions for the radial density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We adopt a β model (see text for details), and show limits corresponding to the upper and lower ends of the range of constraints from FRB 20220319D (dashed and solid lines respectively), and from the most constraining pulsar in M53 (red solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Typical values of β found in the literature are ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For these values, the inferred fCGM values are typically below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5, as indicated by blue horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We therefore consider the constraints on MCGM based on FRB 20220319D to be robust to uncertainties in DMCGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The impact of uncertainty in DMISM for the M53 sightline is greater given the low values of DMCGM under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although we attempt to be con- servative, even a few-unit increase in DMCGM would sig- nificantly raise the derived upper limits on MCGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Our analysis is also sensitive to how we model the distribu- tion of baryons in the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' First, we do not account for sightline to sightline variance, which simulations sug- gest may be at the level of ≳ 10% (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Second, it may be that a β profile is not the correct model to adopt, and our con- straints are sensitive to the specific value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Guidance on these points from simulations for future analyses of DMCGM will be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Finally, although we assume a specific helium abundance in translating the DM con- straints to a total CGM mass, our results are sensitive to variations in this assumption only at the < 10% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The discovery of FRB 20220319D has major implica- tions for the search for FRBs in the local universe, and for future FRB studies of the CGM of the Milky Way and other galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' First, we have shown that for low Galactic latitudes (|b| ≲ 15◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' see Figure 4) there can be significant uncertainties in the widely used NE2001 and YMW16 models for DMISM, which may lead to nearby extragalactic FRBs being missed in surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' In all cases we recommend that FRB surveys save candi- dates at DMs below the model values of DMISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The possibility of missing FRBs with low extragalactic DMs also needs to be considered when using FRB samples to directly infer a characteristic DMCGM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Platts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These studies are also likely to be impacted by the uncertainties in deriving DMCGM that in most cases will be on the order of the values of DMCGM allowed by our analysis (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We recommend that in- ferences of DMCGM with FRBs focus on high-latitude sightlines towards FRBs interferometrically localized to nearby galaxies, wherein uncertainties in DMIGM and DMhost can be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Nonetheless, if the charac- teristic DMCGM is indeed on the order of 10 pc cm−3, a very large number of local FRBs will be required to suppress variance between halo sightlines, and uncer- tainties in other DM contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These samples may be furnished by future coherent all-sky radio monitors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Finally, consistent with previ- ous studies (Keating & Pen 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Kirsten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022), our constraints on fCGM firmly exclude the fiducial pa- rameterizations of the widely used modified-NFW and Maller & Bullock (2004) models for the CGM DM as described by Prochaska & Zheng (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' This impacts studies that have assumed the correspondingly large val- ues of DMCGM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Ravi 2019), as well as forecasts for the contributions of the CGM of intervening galaxies to FRB DMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The host of FRB 20220319D, IRAS 02044+7048, ap- pears to have an unusually large specific SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' This is despite FRB 20220319D being the closest so far non- repeating FRB yet discovered, and the strong evolution towards higher SFRs of the star-forming main sequence of galaxies with redshift (Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Our anal- ysis of the SFH of the IRAS 02044+7048 host galaxy suggests that most stars were formed in a burst in the last ≲ 2 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Thus it is likely that the progenitor of FRB 20220319D does not have a delay time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', age from its formation epoch) in excess of this timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although several progenitor scenarios remain consistent with this constraint, the additional possible association of the FRB source with a spiral arm may imply a pro- genitor age that is a fraction of a ∼ 100 Myr dynamical timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We note that it is not straightforward to com- pare our inferred SFR, which is averaged over the past 100 Myr of the SFH, to the Hα-based SFRs common in the FRB literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Bhandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022), which are sensitive to the last ∼ 10 Myr (Kennicutt 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' A direct comparison of non-parametric SFH estimates may lead to a more complete view of the formation channels of FRB progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS We present the DSA-110 discovery and interfero- metric localization of the nearest so far non-repeating 14 Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' FRB (20220319).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The burst was observed with a high S/N of 79, a DM of 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='95 pc cm−3, and has a lin- ear polarization fraction of 16 ± 3% and an RM of 50 ± 15 rad m−2 (Figure 1, and Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The FRB originated from the position (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' J2000, decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' J2000) = (02:08:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7(1), +71:02:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='9(6)), where uncertainties in the last significant figures are given in parenthe- ses (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We associate FRB 20220319D with a spiral-arm feature of a face-on star-forming galaxy at a distance of 50 Mpc (IRAS 02044+7048;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Figures 3 and 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' IRAS 02044+7048 is moderately mas- sive (log M∗ = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='07), with an approximately solar metallicity and an SFR averaged over the past 100 Myr of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 M⊙ yr−1 (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The low DM of FRB 20220319D motivates a sensitive new constraint on the baryon mass of the Milky Way CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Our con- clusions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The spatial coincidence of FRB 20220319D and IRAS 02044+7048 (false alarm probability of < 10−4), and a potential RM detection for FRB 20220319D that is in excess of the Galac- tic foreground, both imply a secure association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' However, the DM of FRB 20220319D is somewhat lower than the predicted Galactic ISM DM along its sightline from leading models for the WIM dis- tribution (NE2001 and YMW16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Cordes & Lazio 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We show through an anal- ysis of the DMs towards pulsar-hosting globular clusters with Gaia parallax distances (Figure 4) that there are significant uncertainties in NE2001 predictions for DMISM for |b| ≲ 15◦, and that YMW16 is particularly uncertain at low latitudes in the second quadrant, where FRB 20220319D is located (see also Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We find that estimates for DMISM based on Hα EMs perform compara- bly well to the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We conclude that models for DMISM towards FRB 20220319D can accom- modate an extragalactic origin, and urge FRB sur- veys to consider candidate bursts at DMs below standard model predictions for DMISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The SFH of IRAS 02044+7048 is consistent with most stars being formed in a burst within the past two Gigayears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' This, together with the pos- sible location of FRB 20220319D within a spi- ral arm of IRAS 02044+7048, suggests a mod- erately low delay time for the burst progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Despite its nearby distance, the specific SFR of IRAS 02044+7048 derived from SED modeling is larger than all but one so far non-repeating FRB, although the comparison with literature samples of host galaxies would be aided by the widespread use of non-parametric SFH analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We find conservative upper limits on the Galactic CGM contribution to the DM of FRB 20220319D of either 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 pc cm−3 or 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3 pc cm−3: the two val- ues are based on DMISM-estimates from the DMs of two pulsars nearby on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These limits ex- clude some literature models (Maller & Bullock 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Faerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Prochaska & Zheng 2019) for the baryon distribution in the CGM (Fig- ure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We find that consistent results are ob- tained from estimates of the CGM DM contribu- tions towards pulsars in the LMC and in the dis- tant (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 kpc) high-latitude globular cluster M53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The M53 pulsars in particular provide interest- ing constraints in the inner regions of the Milky Way halo that complement the FRB constraint, although the M53 DMCGM estimate is sensitive to the assumed ISM contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We derive an upper limit on the mass of baryons in the Galactic CGM of log MCGM ≲ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='8 − 11 from FRB 20220319D, and log MCGM ≲ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='7 from M53 (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The range corresponds to the range of assumed indices (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='4–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5) of the β pro- file for the baryon halo density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' For a fiducial total mass (baryons and dark matter) of the Milky Way of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='5 × 1012M⊙, and an as- sumed baryon-disk mass of 6×1010M⊙, the Milky Way contains ≪ 60% of the cosmological-average baryon mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' This is consistent with a baryon census, predicted by galaxy-formation simulations, wherein these “missing” baryons are expelled from halos like the Milky Way into the IGM through ki- netic and thermal feedback from AGN, supernovae and massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Although our analysis relies on just a few sightlines, the conservative upper bounds on MCGM assuming spherical symmetry are robust to variations in the density distribution of halo baryons, and to assumptions on the chem- ical and thermal state of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Studies of unseen baryons with FRB propagation sig- natures, like DMs, promise to transform our understand- ing of the distribution of baryons around and in be- tween galaxies (Ravi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' As more FRBs like FRB 20220319D are localized to nearby galaxies (see also Kirsten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 2022), constraints on the CGM con- tent of the Milky Way will continue to improve, and direct measurements of MCGM may be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' These measurements are likely to be important in assembling an in-situ picture of the processes whereby a galaxy grows from and impacts its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Mass of the Milky Way CGM 15 The authors thank staff members of the Owens Valley Radio Observatory and the Caltech radio group, includ- ing Kristen Bernasconi, Stephanie Cha-Ramos, Sarah Harnach, Tom Klinefelter, Lori McGraw, Corey Posner, Andres Rizo, Michael Virgin, Scott White, and Thomas Zentmyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Their tireless efforts were instrumental to the success of the DSA-110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' The DSA-110 is supported by the National Science Foundation Mid-Scale Innovations Program in Astronomical Sciences (MSIP) under grant AST-1836018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' We acknowledge use of the VLA calibra- tor manual and the radio fundamental catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' Some of the data presented herein were obtained at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1038/s41586-021-04354-w Koo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Heiles, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', & Reach, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1992, ApJ, 390, 108.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' 1991, Nature, 349, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='1038/349047a0 Leja, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=', Carnall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} +page_content='3847/1538-4357/ab960a' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAzT4oBgHgl3EQfEvrf/content/2301.01000v1.pdf'} diff --git a/8tFLT4oBgHgl3EQftS_u/content/tmp_files/2301.12151v1.pdf.txt b/8tFLT4oBgHgl3EQftS_u/content/tmp_files/2301.12151v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0ed4348baec4e0cd2058cada2f3a52ac1bc21f8 --- /dev/null +++ b/8tFLT4oBgHgl3EQftS_u/content/tmp_files/2301.12151v1.pdf.txt @@ -0,0 +1,1280 @@ +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +Jona Klemenc * 1 Holger Trittenbach * 1 +Abstract +Regulation, legal liabilities, and societal con- +cerns challenge the adoption of AI in safety and +security-critical applications. One of the key con- +cerns is that adversaries can cause harm by manip- +ulating model predictions without being detected. +Regulation hence demands an assessment of the +risk of damage caused by adversaries. Yet, there +is no method to translate this high-level demand +into actionable metrics that quantify the risk of +damage. +In this article, we propose a method to model and +statistically estimate the probability of damage +arising from adversarial attacks. We show that +our proposed estimator is statistically consistent +and unbiased. In experiments, we demonstrate +that the estimation results of our method have +a clear and actionable interpretation and outper- +form conventional metrics. We then show how +operators can use the estimation results to reliably +select the model with the lowest risk. +1. Introduction +Adversarial perturbations are a security risk since an adver- +sary can use them to alter machine learning model predic- +tions without being noticed by a human (Yuan et al., 2019; +Ren et al., 2020). For instance, think of an upload filter that +uses a machine learning model to identify prohibited mate- +rial on social media platforms, such as copyright-protected +images or hate speech. By perturbing prohibited content, +an adversary may bypass an upload filter even though the +content appears identical to the original prohibited content +to the human observer. When machine learning models +are deployed in systems that take or enable action in phys- +ical environments, the security risks can result in safety +hazards (Deng et al., 2020). An example is autonomous +driving, where a false classification, e.g., of a stop sign, can +severely damage property and life. Given these potentially +*Equal contribution +1neurocat, Berlin, Germany. Correspon- +dence to: Jona Klemenc , Holger +Trittenbach . +severe consequences, regulators have made clear that one +must treat risk from adversaries seriously. Broad regulation, +such as the “EU AI Act” (European Commission, 2021), +as well as domain-specific norms, such as safety standards +for autonomous vehicles (e.g., ISO 21448 (SOTIF) (The +British Standards Institution, 2022), UL 4600 (Underwriters +Laboratories Inc., 2022), and ISO PAS 8800 (International +Organization for Standardization, 2023 (forthcoming)), ex- +plicitly require an assessment of the risk arising from adver- +sarial attacks. Consequently, operators of machine learning +models strive to manage the risk that stems from adversarial +perturbations (Piorkowski et al., 2022). While the motiva- +tion and intentions are clear, neither the regulation nor the +academic literature currently provides sufficient technical +guidelines on how to assess the “risk of adversarial attacks”. +A risk assessment of adversarial attacks requires a reliable +estimate that an adversary causes damage. Intuitively, this +probability of damage describes how likely an adversary can +find perturbations that go undetected and alter model predic- +tions. The probability of damage hence depends on the capa- +bilities of an adversary and the effectiveness of measures put +in place to detect adversarial perturbations. Previous work +has focused on evaluating adversaries by comparing adver- +sarial attacks (Yuan et al., 2019; Ren et al., 2020) with each +other, e.g., by a per-attack drop in accuracy (Brendel et al., +2020). These evaluations do not consider the likelihood that +measures detect adversarial perturbations. Instead, evalua- +tions assume a threshold (Croce et al., 2021; Maho et al., +2021) on the perturbation size beyond which perturbations +are detected with certainty. This is a stark simplification. +Practical experience shows that a suitable threshold is hard +to find or may not exist. Other proposed evaluation metrics +that compare perturbation sizes (Carlini et al., 2019) are +not viable alternatives as they are either prone to outliers or +suffer from statistical bias. It is an open question of how to +reliably estimate the probability of damage and how to use +the estimate to select machine learning models. +In this article, we propose a statistical approach to assessing +the risk caused by adversarial attacks. Our main contri- +bution is a model-agnostic estimator for the probability of +damage that is statistically unbiased and consistent. Our +estimator explicitly considers the probability of detecting +perturbations instead of assuming a hard threshold. As it +turns out, calculating estimates for the probability of detec- +arXiv:2301.12151v1 [cs.LG] 28 Jan 2023 + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +tion efficiently is challenging for large sample sizes (see +Section 4). Hence, we propose a strategy to make estimates +efficient (see Section 4.2), even when querying the detector, +e.g., a human in the loop, is out of reach (see Section 4.3). +The estimates resulting from our method allow comparing +different models with each other to select the model with the +lowest risk of damage caused by adversarial attacks. Our +experiments demonstrate that our estimator is more reliable +than existing metrics in adversarial robustness benchmarks. +2. Notation +Let X be a data space, and X = ⟨x1, . . . , xI⟩ ⊆ X a sample +of I observations. A machine learning model M : X → O +is a function that maps the data space to a prediction space +O. O differs depending on the model task, e.g., for image +classification it might be the space of logit scores or class +labels; for object detection it might be the space of bounding +boxes and classification scores. When there are multiple +models, we index them as M1, . . . , MJ. +We further define a ground truth as a function g: X → O +that assigns each observation a value of the prediction space +that is considered “correct” by some gold standard. We say +a prediction is “incorrect” if M(x) ̸= g(x). We say that +two model predictions “disagree” for a pair of observations +if M(x) ̸= M(x′), x, x′ ∈ X. Herein, we do not specify +further what it means to disagree since the specifics usually +depend on the model task and application. For instance, in +one case, one would say two models disagree if the argmax +of their logit outputs are different; in another case, they +disagree if the ranking of their top-k logit scores differs. +We use PS to denote probability distribution functions over +some space S with the corresponding probability density +function dPS. The hat notation indicates empirical esti- +mates, e.g., an empirical distribution ˆP. +We say an estimator of a probability function is unbiased +if the mean of the sampling distribution of the estimator is +equal to the true probability. An estimator that converges to +the estimated value with increasing sample size is consis- +tent (see Cram´er, 1999, p. 351). +3. Fundamentals and Related Work +We expect operators to assess the risk of using a machine +learning model in a security-critical application as an ex- +pected value of the total damage, i.e., the product of the +occurrence probability of damage when operating a model +P dam(M) and the expected size of the damage Cdam. +Operational Risk(M) = P dam(M) × Cdam +(1) +We assume that Cdam is constant and independent of whether +the operator relies on machine learning or any other system, +e.g., a human in the loop. We focus on the risk of a malicious +adversary seeking to intentionally manipulate predictions +in a way that can inflict damage. We do not consider other +categories of machine-learning-related security and privacy +risks, such as the risks of model stealing (Tram`er et al., +2016) and model inversion (Fredrikson et al., 2015). +Adversarial Risk. The adversarial machine learning com- +munity often uses the term “risk” in the sense of Adver- +sarial Risk (AR). AR is concerned with the empirical risk +estimation of a model under small perturbations. Formally, +the adversarial risk is the expected loss over perturbations +within a neighborhood Nϵ(x) +AR(M) = Ex∼X +� +sup +x′∈N (x) +l(M(x′), g(x)) +� +(2) +with loss function l (Uesato et al., 2018). With ϵ=0, Equa- +tion 2 reduces to the standard empirical risk. Typically, the +neighborhood N is constrained to an epsilon ball around +x, i.e., N(x) ≡ Bϵ(x) for a fixed perturbation budget ϵ. +Details of this definition differ across the literature (Pydi +& Jog, 2021). For instance, instead of the loss against a +ground truth, one may compute the loss of prediction change +l(M(x), M(x′)) if the ground truth is unknown (Diochnos +et al., 2018). +Related research focuses on average risk under ran- +dom (Levy & Katz, 2021; Rice et al., 2021) and natural +perturbations (Pedraza et al., 2021; Hendrycks & Dietterich, +2019; Schwerdtner et al., 2020). Here, one measure of inter- +est is the Error-Region Risk (ERR) (Diochnos et al., 2018), +i.e., the probability that a successful perturbation exists +ERR(M) = Px∈X [∃x′ ∈ Bϵ(x) : M(x′) ̸= g(x′)] +(3) +However, estimations of ERR rely on random perturbations +drawn from a uniform distribution (Diochnos et al., 2018; +Webb et al., 2018). Thus, it does not account for “an explicit +and effective adversary” (Webb et al., 2018). +Benchmarks. Adversarial Risk has inspired several bench- +marks that compare the effectiveness of individual adver- +sarial attacks and defenses. The comparisons are based +on either the success rate of attacks for a defined pertur- +bation budget ϵ, see RobustBench (Croce et al., 2021) +and RoBIC (Maho et al., 2021), or the average perturba- +tion size attacks require to find successful perturbations, +see RobustVision (Brendel et al., 2020). Such com- +parisons are attack-centric, i.e., effective in evaluating at- +tacks against each other. However, estimating P dam requires +model-centric evaluations that measure how likely an adver- +sary can find successful adversarial perturbations given a set +of attacks and perturbation budgets (cf. Carlini et al., 2019). +Instead of evaluating adversarial risk for a defined budget, +one can also plot accuracy or attack success rates against a + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +perturbation budget (Dong et al., 2020; Carlini et al., 2019). +Such plots are useful to investigate the effectiveness of at- +tacks for varying perturbation budgets. However, visual +inspection of plots does not scale beyond the comparison of +a few attacks. It is an open question how one can use these +curves to estimate the operational risk; we will come back +to this question in Section 5.2. +In summary, there are a variety of evaluation methods that +seek to capture some element of adversarial risk. They pro- +vide a set of measures for attack benchmarks but are not +immediately applicable to estimate operational risk. The +reason is that they (i) do not account for the probability of +detecting adversarial attacks (see Section 4) and (ii) are ei- +ther biased or inconsistent and hence not useful as statistical +estimates (see Section 5.2). +4. Estimating the Probability of Damage +The operational risk of a machine learning model depends +on the probability of damage P dam, see Equation 1. Intu- +itively, P dam is the joint probability of finding a successful +perturbation (Succ) from the space of adversarial perturba- +tions AdvM(X) and of a detector, e.g., a human in the loop, +not detecting it (¬ Det). Formally, we can express P dam as +the joint probability +P dam :=Px∼AdvM(X)(Succ(x), ¬Det(x)) += Px∼AdvM(X)(Succ(x)) +� +�� +� +Probability of Attack Success +× Px∼AdvM(X)(¬Det(x) | Succ(x)) +� +�� +� +Probability of Detection +(4) +Considering this equation, a natural solution to estimat- +ing P dam seems to be Monte Carlo Simulation: One can +draw a sample from the space of adversarial perturbations +AdvM(X) to estimate the probability of attack success +Px∼AdvM(X)(Succ(x)) and then pass the successful per- +turbations on to a detector to estimate the probability of de- +tection Px∼AdvM(X)(¬Det(x) | Succ(x)). However, one +must be careful with designing this sampling process – a +naive sampling of random perturbations is not sufficient here. +An adversary is efficient and uses all means available to find +successful perturbations. The adversary does not search +through the infinite space of random perturbations but in- +stead uses adversarial attacks, i.e., efficient optimization +methods that make heuristic assumptions on the perturba- +tion space. Thus, naive random sampling underestimates an +efficient adversary. Valid Monte Carlo Simulation requires +to simulate an adversary, i.e., to run attacks that are in line +with the resources and capabilities of the adversary. +While Monte Carlo Simulation is conceptually sound, it has +two significant limitations in our context: +Small Sample Size: If detection requires a human in the +loop, collecting data on detection will inevitably be time- +consuming: a detector must inspect each successful pertur- +bation individually. This may significantly limit the feasible +sample size and reduce the estimation quality. +Model-Specific Estimates: Probability estimates are specific +to the machine learning model and attacks used to search +for perturbations. This is because the conditional probabil- +ity of detection Px∼AdvM(X)(¬Det(x) | Succ(x)) relies +on AdvM(X) which is specific to the model and attacks. +Consequently, one must repeat the entire estimation of P dam +for any change in the choice of models or attacks. Because +detection is costly, the estimation becomes impractical for +operators who have frequent model iterations and face a +research field that frequently produces novel attacks. +Both limitations stand in the way of obtaining an estimate +of P dam in many practical settings. +In this section, we propose a sampling method that over- +comes both limitations. We first establish a rigorous formal +framework for the probability of attack success (Section 4.1). +We then turn to the estimation of the probability of detection +(Section 4.2) and introduce a method that is not limited by +Small Sample Size and Model-Specific Estimates. Lastly, +we look at the special case of providing an estimation for +P dam when there is no explicit detector – a setting that often +occurs in academic benchmarks (Section 4.3). The result is +a formal expression of the probability of damage that has +both a clear interpretation and an unbiased and consistent +estimator. +4.1. Estimating the Probability of Attack Success +So far, we have defined P dam as an estimate over the space +of adversarial perturbations AdvM(X). However, one typi- +cally only has access to a sample X ∼ X, used for training +or testing of a machine learning model. The space of adver- +sarial perturbations is defined implicitly by a pushforward +of X along a function ΠM : x �→ x′ +Px∼AdvM(X)(Succ(x), ¬Det(x)) += Px∼X (Succ(ΠM(x)), ¬Det(ΠM(x))) +(5) +We call ΠM the attack strategy. +One can use a sample X and an attack strategy ΠM to +simulate an adversary. To illustrate, think of an adversary +that wants to evade an upload filter for copyright-protected +material. The adversary would select a copyright-protected +image and manipulate it via an attack strategy ΠM such +that the classifier of the upload filter predicts it to be +non-protected content. To estimate the probability that +the adversary finds such a successful perturbation, one +can simulate the adversary by sampling images from the +data distribution of interest to the adversary, here the + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +copyright-protected images, and then manipulate the im- +ages with the attack strategy the adversary is expected to use. +An attack strategy depends on the choice of search algo- +rithms that an efficient adversary has at their disposal to +search for successful adversarial perturbations. The most +effective search algorithms known today are adversarial at- +tacks, i.e., heuristics to find small successful perturbations. +Many adversarial attacks require access to the model, e.g., +to obtain inference results or gradients with respect to the +model input. An adversary with restricted model access +can only use a subset of adversarial attacks that do not rely +on gradient calculations. We define the set of applicable +adversarial attacks as +A = {ak : (x, M) �→ x′}k∈K +where each ak ∈ {a1, a2, . . . , aK} is an attack that has +access to an input x and potentially restricted access to +a model M. In practice, an adversary also may have a +limited computational budget, which forces them to select a +computationally feasible subset A′ ⊂ A; the selection of a +good subset of adversarial attacks under budget restrictions, +however, is a question orthogonal to our current article. +An adversary executes an attack strategy as follows. First, +the adversary uses A to generate candidate perturbations +CandA(x, M) = {a(x, M) | a ∈ A} +Out of the candidates, the adversary filters the ones x′ ∈ +CandA(x, m) where Succ(x′) = True∧¬Det(x′) = True. +Filtering for Succ(x′) is straightforward if the adversary can +access the model predictions. For instance, if the adversary +is interested in an untargeted misclassification, e.g., chang- +ing the prediction from “copyright-protected” images to any +other class, the success filter is +CandSucc +A +(x, M) = {x′ ∈CandA(x, M)|M(x′)̸=M(x)} +where the sample X are images that initially are classified +as M(x) = “copyright-protected”. +However, filtering is not possible for an adversary because it +requires access to ¬Det(x′), e.g., a human in the loop. If an +adversary would have access to the detector, then there is no +need to use adversarial attacks that minimize perturbation +sizes. The adversary could instead directly optimize for +evading the detector with adaptive attacks (Tramer et al., +2020). To minimize the chance of being detected without ac- +cess to ¬Det(x′), the adversary thus makes an assumption: +small perturbations are less likely to be detected than large +ones. The adversary then uses the perturbation size as a +proxy for detectability and selects the smallest perturbation +per observation among all successful perturbations. This +Algorithm 1: Monte Carlo Simulation of P dam +Input +:X, M, A, Det, d +Output : �P dam +1 r ← 0 +2 for x ∈ X do +▷ Outer Loop +3 +dmin ← ∞ +4 +for a ∈ A do +▷ Inner Loop +5 +x′ ← a(x, M) +6 +if M(x′) ̸= M(x) ∧ d(x′, x) < dmin then +7 +x′′ ← x′, dmin ← d(x′, x) +8 +end +9 +end +10 +if x′′ ̸= x and ¬Det(x′′) then +11 +r ← r + 1 +12 +end +13 end +14 return +r +|X| +means that an operator has to assume the worst-case, i.e., +the smallest successful perturbation for each observation. +A difficulty for the adversary, and in turn also for the simu- +lation, is to select a distance metric to measure perturbation +size that correlates well with the chance of detection. A +common choice are Lp metrics, but there is an active debate +about which metrics align well with human perception, (see +for instance Zhang et al., 2018). With this in mind, we can +now introduce a formal definition of the attack strategy: +Definition 4.1 (Attack Strategy). An attack strategy is a +function +ΠM +A (x)= +� +� +� +arg min +x′∈CandSucc +A +(x,M) +d(x′, x), if CandSucc +A +(x, M)̸=∅ +x +otherwise, +of type ΠM +A : X → X that returns the smallest perturbation +obtained by applying a set of applicable adversarial attacks +A to an observation x given a distance metric d. +We say an attack strategy is successful if ΠM +A (x) ̸= x. Since +an attack strategy relies on empirical evaluations, it yields +an upper bound on the minimum perturbation required for +any attack strategy to be successful. +Definition 4.2 (Smallest Upper Bound on Perturbation Size). +The smallest upper bound on the perturbation size obtained +by a successful attack strategy is +dA(x, M) = +� +d(ΠM +A (x), x), if ΠM +A (x) ̸= x +∞ +otherwise, +where d is a distance metric on X. +We can now define an algorithm to find an estimate for P dam +by Monte Carlo Simulation. Algorithm 1 illustrates the idea: + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +iterate over observations in a sample X (Line 2) and indi- +vidual attacks of an attack strategy A (Line 4), measure the +attack success (Line 6) and return the ratio of observations +for which an attack strategy was successful, i.e., the ones +that are not detected (Line 10). The result is an unbiased +and consistent estimate of the probability of attack success +for an attack strategy. +4.2. Estimating the Probability of Detection +Algorithm 1 reveals the two limitations that we introduced +earlier. First, samples sizes |X| must be small if evaluating +Det(x′) is costly (Small Sample Size). This, in turn, means +that the sample size and the number of attacks to evaluate +the success of an attack strategy must be small. The small +sample sizes limit the quality of the estimate. Second, the +evaluation of Det occurs after generating x′. Since x′ de- +pends on M, the result of the estimation is model-dependent +(Model-Specific Estimates). One must repeat the estimation +of P dam for each model. We now show how one can over- +come both limitations. +4.2.1. OVERCOMING SMALL SAMPLE SIZE +One way to overcome Small Sample Size is to substitute +the detector with a surrogate function that is inexpensive to +evaluate. Specifically, if there is a function F that substitutes +¬Det, such that F(x) = ¬Det(ΠM +A (x)), one can use F +instead of ¬Det to evaluate the detector during a Monte +Carlo Simulation. We can express this in commutative +diagram notation as +ΠM +A (x) +¬Det(ΠM +A (x)) +¬Det +F +Finding a suitable F is difficult since the domain of F is +the high-dimensional observation space. In particular, es- +timating a function in a high-dimensional space may still +require large sample sizes. However, under the assumption +that the detection probability correlates well with the pertur- +bation size, one can first map x to a distance and then use +the distance as the domain of the detection function. With +this assumption, the commutative diagram changes to +dA(x) +x +ΠM +A (x) +¬Det(ΠM +A (x)) +F ′ +¬Det +where F ′ is a function of type F ′ : R → {0, 1}, and the +image of dA the distance space induced by a metric d. +We further say that such an F ′ has the point-wise detection +commutation property if +∀x ∈ X : Succ(ΠM +A (x)) ⇒ F ′(dA(x)) = ¬Det(ΠM +A (x)). +Such an F ′ may not exist. +Think of two observations +x, ˜x ∈ X, x ̸= ˜x with x′ = ΠM +A (x) and ˜x′ = ΠM +A (˜x), +where dA(x) = dA(˜x) but ¬Det(x′) ̸= ¬Det(˜x′), i.e., +both observations have the same upper bound on the per- +turbation size. Then, there is no F ′ that has the detection +commutation property. +Fortunately, we do not require a point-wise detection com- +mutation. Since �P dam is a probabilistic estimate, it suffices +that F ′ is equivalent to the probability of detection. +Definition 4.3 (Probabilistic Detection Commutation Prop- +erty). We say a function F ′ : R → R has the probabilistic +detection commutation property if +Px∼X ′(¬Det(ΠM +A (x))) = Ex∼X ′(F ′(dA(x))), +(6) +where X ′ = {x ∈ X : Succ(ΠM +A (x))}. +Given an F ′ that has the probabilistic detection commu- +tation property, one can obtain P dam using the cumulative +distribution function of dA, the Attack Success Distribution. +Definition 4.4 (Attack Success Distribution). +ASDA,M(τ) := Px∼AdvM(X)(dA(x, M)≤ τ), τ ≥ 0 (7) +Formally, this leads to the following theorem. +Theorem 4.5. Let F ′ : R → R be a function that fulfills +the probabilistic detection commutation property. Then +P dam = +� ∞ +0 +F ′(τ) dASDA,M(τ) dτ. +Proof. See Appendix A. +Using Theorem 4.5, the key to an estimation of P dam is an +estimation of ASDA,M. Given a sample X ∼ X, one can +approach the Attack Success Distribution with the following +empirical distribution function (cf. Dong et al., 2020). +Definition 4.6 (Attack Success Ratio). +ASRA,M(τ) = |{x ∈ X | dA(x, M) ≤ τ}| +|X| +Empirical distribution functions are unbiased and consis- +tent estimators of their distribution functions. Applying +this general property to ASRA,M allows us to formulate an +unbiased and consistent estimator for P dam. +Theorem 4.7. The estimator +�P dam = +� ∞ +0 +F ′(τ) dASRA,M(τ) dτ += +1 +|X| +� +x∈X,dA(x)̸=∞ +F ′(dA(x)) +(�) +is an unbiased, consistent estimator of P dam. +Proof. See Appendix A. + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +4.2.2. OVERCOMING MODEL-SPECIFIC ESTIMATES +What is left to discuss is how to construct an F ′ that fulfills +the probabilistic commutation property. A natural choice +is to use the probability of detection conditioned on τ. In- +tuitively, this probability is the ratio of observations with +dA(x) = τ that one expects to be detected. Formally, we +define this as a probability function. +Definition 4.8 (Detection Probability Function). +ΨA,M(τ) = Px∼X (¬Det(ΠM +A (x)) | dA(x) = τ) +With a suitable distance measurement, we can assume that +the perturbation size statistically determines the detection +probability function. In this case, the detection probability +does not depend on the choice of a model, i.e., +ΨA,M1(τ) = Px∼X (¬Det(ΠM1 +A (x)) | dA(x) = τ) += Px∼X (¬Det(ΠM2 +A (x)) | dA(x) = τ) += ΨA,M2(τ). +where M1 ̸= M2. One can show that ΨA,M is indeed a +suitable choice for F ′. +Theorem 4.9. The Detection Probability Function has the +probabilistic detection commutation property. +Proof. See Appendix B. +Combining Theorem 4.9 with Theorem 4.5 gives +Corollary 4.10. The probability of damage P dam is +P dam = +� ∞ +0 +ΨA,M(τ) dASDA,M(τ) dτ +Furthermore, the estimator +�P dam = +� ∞ +0 +ΨA,M(τ) dASRA,M(τ) dτ += +1 +|X| +� +x∈X,dA(x)̸=∞ +ΨA,M(dA(x)) +is an unbiased, consistent estimator of P dam. +A useful implication of Corollary 4.10 is that one can further +refine ΨA,M to include prior knowledge and assumptions +in the calculation of P dam. For instance, recall that an ad- +versary uses an attack strategy ΠM +A to find the smallest +perturbation for an observation. The rationale is that the ad- +versary expects that the chances of being detected increase +with the perturbation size. With this assumption, one can +simplify ΨA,M to monotonic non-decreasing functions or +even a logistic curve. One must then only query the detector +with a data sample and then fit a posterior for ˆΨA,M, e.g., +with Bayesian estimation, see Appendix C. A further take- +away from this section is that the estimate ˆΨA,M depends +on the application but not on a specific model. Think of our +upload filter example. Upload filters are used in different ap- +plications, e.g., to detect copyright infringement of portrait +photographs (Application A) and violent content in pictures +(Application B). For each application, one must estimate +a detection probability function: ˆΨphoto +A,M for Application A +and ˆΨviolent +A,M +for Application B. However, within Applica- +tion A, ˆΨphoto +A,M can be used to estimate the risk for different +copyright detection models, e.g., trained with different hy- +perparameter settings. Likewise, ˆΨviolent +A,M +can be used to +estimate the risk of different violence detection models. This +reduces the number of detection probability functions one +has to fit by querying a detector from one per model to one +per application. Thus, the independent estimation of the +detection probability function mitigates both Small Sample +Size and Model-Specific Estimates, and hence reduces the +effort for risk estimation. +4.3. Estimations without a Detector +In some cases, collecting a sample from a detector to fit +ΨA,M is infeasible. For instance, think of academic bench- +marks that compare adversarial attacks or defense methods. +There, querying a human detector is often beyond the scope +of the study. One may not even have an indication of which +magnitude of perturbation size is actually required for a +detection to be successful. In such a case, selecting and +estimating a suitable detection probability function is not +possible. +However, one can still obtain a relative comparison of the +operational risk between models. One way is to assume that +the average sensitivity of models to adversarial perturbations +is similar to the sensitivity of a potential detector. Formally, +this gives an average detection function +ˆΨavg +A,M(τ) = 1 − 1 +J +J +� +j=1 +ASRA,Mj(τ) +(8) +where M1, M2, . . . , Mj, . . . , MJ are the models to com- +pare. Intuitively, using ˆΨavg +A,M as a detection function means +that models more sensitive to adversarial examples than the +average will obtain a high operational risk. +If all ASRA,MJ estimates are based on the same sample +X, one can rearrange Equation 8 to make its computation +efficient. We first define W(τ) to count the combination +of observations and models where the adversarial example +with the smallest perturbation size is further away than τ. +W(τ) = |{(i, j) | i ∈ [I], j ∈ [J], dA(xi, Mj) > τ}| . +We then have + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +0.0 +0.1 +0.2 +0.3 +8 +255 +2 +255 +Perturbation Size (L∞) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ASR +Baseline +Engstrom-Robust +Rice-Overfit +Carmon-Semi +Calculated ΨA,M +Figure 1. Solid, colored: ASR of the different models on 200 +observations. Dashed, black: ΨA,M estimated according to Sec- +tion 4.3. Solid, black: Vertical lines marking L∞ ∈ { +2 +255, +8 +255}. +ˆΨavg +A,M(τ) = 1 − +� +j=1...J ASRA,Mj(τ) +J += 1 +J +� +� � +j=1...J +(1 − ASRA,Mj(τ)) +� +� += +1 +|X| · J +� +� � +j=1...J +(|X| − |{x ∈ X | dA(x) ≤ τ}|) +� +� += +1 +|X| · J +� +j=1...J +|{x ∈ X | dA(x) > τ}| += +1 +|X| · J W(τ) +With Corollary 4.10, we have +�P dam = +1 +|X|2 · J +� +j=1...J +W(dA(xi, Mj)) +(9) +Algorithm 2 in Appendix D summarizes the estimation of +�P dam using Equation 9. The algorithm helps identify the +model with the lowest probability of damage even if collect- +ing a sample from a detector is infeasible. +5. Experiments +This section demonstrates that �P dam provides a consistent +and unbiased evaluation to compare the robustness of ma- +chine learning models without the need to choose a threshold +on the perturbation size. Our experimental setup is represen- +tative of how academic benchmarks comparing adversarial +robustness are typically constructed. Our goal is to under- +line the usefulness of our metric in common setups using +open-source models and attack implementations. Hence, our +choice of models, attacks, and parametrization is arbitrary +and can be replaced with any other use-case.1 +1Our +implementations +and +results +are +available +at +https://github.com/duesenfranz/risk_scores_ +paper_code. +Table 1. Summary statistics of the model robustness over 200 +observations. Smaller values are better for all metrics but MPS. +The best values are highlighted in bold. +Model +�P dam +ASR +� 2 +255 +� +ASR +� 8 +255 +� +MPS +Baseline +0.76 +0.70 +1.00 +0.00018 +Engstrom-Robust +0.44 +0.16 +0.48 +0.00020 +Rice-Overfit +0.43 +0.20 +0.42 +0.00119 +Carmon-Semi +0.33 +0.13 +0.33 +0.00095 +5.1. Setup +We compare the operational risk of publicly available mod- +els on CIFAR-10 (Krizhevsky et al., 2009). +Models. The list of models includes a baseline classifier and +three other models trained to achieve high adversarial ro- +bustness. We obtained all models from public repositories.2 +Baseline (Croce et al., 2021): A baseline model trained +without a specific focus on robustness. +Carmon-Semi (Carmon et al., 2019): A model trained +for robustness with a semi-supervised learning method. +Engstrom-Robust (Engstrom et al., 2019): A model +trained for robustness by adversarial training. +Rice-Overfit (Rice et al., 2020): A model trained for +robustness by a combination of adversarial training and +a focus on minimizing overfitting. +Attacks. We define an attack strategy based on a set of +attacks A based on Foolbox (Rauber et al., 2020), an open- +source attack library. We use Projected Gradient Descent +(PGD), PGD with Adam optimizer, and DeepFool. +Parametrization. We use RobustBench (Croce et al., 2021) +to run attacks on observations in the test set. We instantiate +each attack with eight different values for epsilon, i.e., |A| = +24, set d = d∞, and consider a perturbation successful if +the model prediction does not agree with the ground truth. +Metrics. Next to our metric �P dam, we compute three alter- +native metrics. We compute ASR (τ), with two thresholds +τ ∈ { 2 +255, 8 +255}: the fraction of observations with at least +one successful adversarial example with perturbation size +L∞ ≤ τ. ASR (τ) is the maximum likelihood estimator for +the adversarial risk using the 0-1 loss (cf. Section 3). The +threshold +8 +255 is arbitrary but common (e.g., see Rice et al., +2020). Another common approach is to estimate the popu- +lation parameters of the perturbation sizes dA(x), x ∈ X, +(e.g., see Carlini et al., 2019). This approach is not reliable +since estimates are either not robust to outliers (e.g., the +average over dA(x), see Appendix E) or biased by nature +(e.g., the median). To demonstrate the issue of relying on +perturbation size as a metric, we compute the size of the +smallest perturbation that alters the model prediction, the +2All models can be downloaded using RobustBench (Croce +et al., 2021). + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +25 +50 +75 +100 +125 +150 +175 +200 +Number of Observations +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +MPS +Baseline +Engstrom-Robust +Rice-Overfit +Carmon-Semi +50 +100 +150 +200 +Number of Observations +0.00 +0.01 +0.02 +MPS +(a) MPS with increasing sample size. +50 +100 +150 +200 +Number of Observations +0.00 +0.25 +0.50 +0.75 +�P dam +(b) �P dam with increasing sample size. +Figure 2. Convergence of robustness statistics. Each line plots a statistic measured on n ∈ [20, 200] observations. The error band is the +5% and the 95% percentile, calculated by sampling 50 times with replacement. +Minimal Perturbation Size (MPS). MPS is non-robust to +outliers and biased. +5.2. Results +Figure 1 plots the attack success rate (ASR) against the per- +turbation size. Since this is an academic benchmark, we do +not have access to a human detector in the loop. Hence, we +proceed as outlined in Section 4.3 to calculate ΨA,M. Based +on Figure 1, an operator would choose Carmon-Semi, the +model with the lowest ASR for all perturbation sizes. The +second best is Rice-Overfit since the ASR (green line) +is lower than the one of Engstrom-Robust (yellow line) +on most of the perturbation spectrum. The Baseline per- +forms poorly: an adversary can find successful perturbations +for each observation even for a small perturbation budget. +Table 1 summarizes the ASR plot with the metrics �P dam, +ASR +� 2 +255 +� +, ASR +� 8 +255 +� +and MPS. The selection based on +�P dam corresponds to the visual inspection: Carmon-Semi +is the model with the lowest value ( �P dam = 0.33), followed +by Rice-Overfit ( �P dam = 0.43). A benefit of �P dam is +that it remains actionable even if there are too many models +for a visual inspection. +The ASR(τ) metrics, however, contradict each other. +ASR +� 8 +255 +� +suggests that Rice-Overfit is more robust +than Engstrom-Robust; ASR +� 2 +255 +� +suggests the oppo- +site. Further, ASR (τ) = 1.0 for all models if the perturba- +tion size is unconstrained, i.e., τ → ∞. Thus selecting any +threshold on ASR is arbitrary, and results are volatile. +MPS suggests that Rice-Overfit is the most robust +model. However, MPS is biased: with increasing sample +size, its value approaches the true size of the smallest ad- +versarial example, which is close to 0 for all models, see +Figure 2(a). Indeed, if adversaries can choose from an infi- +nite number of observations, they likely find an adversarial +example with a very small perturbation. MPS has high +variance for small sample sizes n < 100, i.e., results are +insignificant. On the other hand, �P dam is consistent and un- +biased, i.e., the metric converges to its true expected value +with increasing sample size, see Figure 2(b). +In summary, our experimental results confirm our theoretical +analyses. Neither ASR (τ) nor MPS are reliable metrics for +selecting a robust model. �P dam allows for relative model +robustness comparisons, even when no detector is available. +6. Conclusions +Estimating the damage caused by adversarial attacks is diffi- +cult. Standard Monte Carlo Simulation is inefficient because +it is limited to small sample sizes and model-specific esti- +mates. A consequence is that resulting metrics do not give +reliable estimates of model robustness. This prevents opera- +tors of machine learning models from translating high-level +regulations on AI safety and security into actionable techni- +cal requirements. +In this article, we put forward an original approach to quan- +tifying the risk of adversarial attacks that overcomes current +limitations. To this end, we first decompose the damage +caused by adversarial attacks into the probability that an +attacker is successful and the probability that an attack goes +undetected. We then propose an unbiased and consistent +estimator for both quantities. For cases where one does not +have access to a detector, we provide an alternative method +that allows comparing the risk between models. The results +are interpretable statistical estimates that provide an empiri- +cal basis for operators to select the model with the least risk +of damage from adversarial attacks. +Acknowledgments +This work was partially funded by the German BMWI +project KI-LOK. + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +References +Brendel, W., Rauber, J., Kurakin, A., Papernot, N., Veliqi, +B., Mohanty, S. P., Laurent, F., Salath´e, M., Bethge, M., +Yu, Y., Zhang, H., Xu, S., Zhang, H., Xie, P., Xing, E. P., +Brunner, T., Diehl, F., Rony, J., Hafemann, L. G., Cheng, +S., Dong, Y., Ning, X., Li, W., and Wang, Y. Adversarial +vision challenge. In The NeurIPS ’18 Competition, pp. +129–153. Springer, 2020. +Carlini, N., Athalye, A., Papernot, N., Brendel, W., Rauber, +J., Tsipras, D., Goodfellow, I., Madry, A., and Kurakin, +A. On evaluating adversarial robustness. arXiv, 2019. +Carmon, Y., Raghunathan, A., Schmidt, L., Duchi, J. C., +and Liang, P. S. 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In Conference on Computer Vision +and Pattern Recognition (CVPR), 2018. + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +Appendix +A. Proof of Theorem 4.5 and Theorem 4.7 +Proof. Starting with Equation 5, we have +P dam = Px∼D(Succ(ΠM +A (x)), ¬Det(ΠM +A (x))) += Px∼D(¬Det(ΠM +A (x)) | Succ(ΠM +A (x))) +× Px∼D(Succ(ΠM +A (x))) += Ex∼D(F ′(dA(x)) | Succ(ΠM +A (x))) +× Px∼D(Succ(ΠM +A (x))) += Ex∼D( +� +F ′(dA(x)), if dA(x) < ∞ +0 +otherwise +) +(*) += +� ∞ +0 +F ′(τ) dASDA,M(τ) dτ, +where the last line is because ASD is the distribution func- +tion of dA and because of the Law of the Unconscious +Statistician (LOTUS). What remains to be proven is the +unbiasedness and consistency of �P dam and the equality in +Line �. The latter is a direct result of the equality +ASRA,M = +1 +|X| +� +x∈X,dA(x)̸=∞ +χ[dA(x),∞), +where χ denotes the characteristic function. For unbiased- +ness and consistency, note that �P dam in Line � is the sample +mean of the expected value in Line *, and therefore an +unbiased and consistent estimator of P dam. +B. Proof of Theorem 4.9 +Proof. We have to prove that +Px∼D′(¬Det(ΠM +A (x))) = Ex∼D′(ΨA,M(dA(x))). +With the Law of Total Probability, we have +Px∼D′(¬Det(ΠM +A (x))) += E˜x∼D′(Px∼D′(¬Det(ΠM +A (x)) | dA(x) = dA(˜x))) += E˜x∼D′(ΨA,M(dA(˜x))) += Ex∼D′(ΨA,M(dA(x))) +C. Sketch of Fitting a Detection Probability +Function ΨA,M with Logistic Regression +Figure 3 sketches how a detection probability function can +be estimated with logistic regression using only 30 samples. +D. Algorithm for �P dam without a Detector +Algorithm 2 calculates P dam if no detector is available, see +Equation 9. The algorithm consists of two parts: +0 +2 +4 +6 +8 +10 +d(x′, x) +0.0 +0.5 +1.0 +¬Det(x′) +Figure 3. Sketch of fitting a detection probability function ΨA,M +with logistic regression. +Blue: +Scatterplot for d(x′, x) = +d(a(x, M), x), a ∈ A and ¬Det(x′). +Red: Estimate ˆΨA,M +by logistic regression. +25 +50 +75 +100 +125 +150 +175 +200 +Number of Observations +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +MPS +Baseline +Engstrom-Robust +Rice-Overfit +Carmon-Semi +50 +100 +150 +200 +Number of Observations +0.00 +0.05 +0.10 +mean(dA(x)) +Figure 4. Convergence of the average perturbation size. Each line +plots the average perturbation size, measured on n ∈ [20, 200] +observations. The error band is the 5% and the 95% percentile, +calculated by sampling n observations 50 times with replacement. +Part A calculates the smallest perturbation sizes dA(x) of +the adversarial examples for all models and observa- +tions x and collects them in two arrays: +(a) +A sorted array W which contains the smallest suc- +cessful perturbations of all models. +(b) +A double array ASR, which contains the smallest +successful perturbations organized into subarrays +per model. +Part B computes �P dam of each model Mi by comparing +ASR[i] with W: It sums the indices in W of the small- +est successful perturbations for the model Mi. It then +divides the result by the square of the number of obser- +vations and by the number of models. +Finally, the algorithm returns �P dam. + +Selecting Models based on the Risk of Damage Caused by Adversarial Attacks +Algorithm 2: Estimation of P dam without Detector +Input +:x1, . . . , xI, M1, . . . , MJ, A, d +Output : �P dam for M1, M2, . . . , MJ +1 W ← [] +2 ASR ← [] +3 �P dam ← [] +4 for j = 1 . . . J do +▷ Part A +5 +dmin ← ∞ +6 +ASR[j] ← [] +7 +for i = 1 . . . I do +8 +for a ∈ A do +9 +x′ ← a(xi, Mj) +10 +if M(x′) ̸= Mj(xi) ∧ d(x′, xi) < dmin +then +11 +x′′ ← x′ +12 +dmin ← d(x′, x) +13 +end +14 +end +15 +if x′′ ̸= xi then +16 +push(W, d(x′′, xi)) +17 +push(ASR[j], d(x′′, xi)) +18 +else +19 +push(W, ∞) +20 +end +21 +end +22 end +23 sort descending(W) +24 for j = 1 . . . J do +▷ Part B +25 +�P dam[j] ← 0 +26 +for τ ∈ ASR[j] do +27 +�P dam[j] ← �P dam[j] + index(W, τ) +28 +end +29 +�P dam[j] ← +� +P dam[j] +I2·J +30 end +31 return �P dam +E. Average Perturbation Size with Increasing +Sample Size +Figure 4 plots the average perturbation size calculated on a +subset of the 200 observations. The non-robustness to out- +liers causes the wide error bands, which leads to an overlap +of the percentiles of Carmon-Semi and Rice-Overfit +for estimates based on fewer than 50 observations. In con- +trast, �P dam yields statistically significant results even for +estimates on fewer than 50 observations, see Figure 2. + diff --git a/8tFLT4oBgHgl3EQftS_u/content/tmp_files/load_file.txt b/8tFLT4oBgHgl3EQftS_u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1e83384ea9c01926ce606061e19b71b226cff29 --- /dev/null +++ b/8tFLT4oBgHgl3EQftS_u/content/tmp_files/load_file.txt @@ -0,0 +1,819 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf,len=818 +page_content='Selecting Models based on the Risk of Damage Caused by Adversarial Attacks Jona Klemenc * 1 Holger Trittenbach * 1 Abstract Regulation, legal liabilities, and societal con- cerns challenge the adoption of AI in safety and security-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' One of the key con- cerns is that adversaries can cause harm by manip- ulating model predictions without being detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' Regulation hence demands an assessment of the risk of damage caused by adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' Yet, there is no method to translate this high-level demand into actionable metrics that quantify the risk of damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' In this article, we propose a method to model and statistically estimate the probability of damage arising from adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' We show that our proposed estimator is statistically consistent and unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' In experiments, we demonstrate that the estimation results of our method have a clear and actionable interpretation and outper- form conventional metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' We then show how operators can use the estimation results to reliably select the model with the lowest risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' Introduction Adversarial perturbations are a security risk since an adver- sary can use them to alter machine learning model predic- tions without being noticed by a human (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' For instance, think of an upload filter that uses a machine learning model to identify prohibited mate- rial on social media platforms, such as copyright-protected images or hate speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' By perturbing prohibited content, an adversary may bypass an upload filter even though the content appears identical to the original prohibited content to the human observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' When machine learning models are deployed in systems that take or enable action in phys- ical environments, the security risks can result in safety hazards (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' An example is autonomous driving, where a false classification, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=', of a stop sign, can severely damage property and life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' Given these potentially Equal contribution 1neurocat, Berlin, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFLT4oBgHgl3EQftS_u/content/2301.12151v1.pdf'} +page_content=' Correspon- dence to: Jona Klemenc d and the corresponding Shannon entropies HS(≡d) +and HS(≡d′) of the model class distributions given by ≡d and ≡d′, we have +HS(≡d) < HS(≡d′) when d′ is at most n/2, and +HS(≡d) = H(≡d′) when d is at least n/2. +A similar result also follows for expected Boltzmann entropies HB(≡d) and +HB(≡d′), but with the orders reversed, that is HB(≡d) > HB(≡d′) for d′ at +most n/2. +To get a better sense of the relative sizes of the classes when n and d are +altered, we prove an asymptotic characterization of the class distributions +as n → ∞ and d is a function of n. Let us say that ≡d(n) has a dominating +class if with limit probability one, a random model of size n belongs to a +maximum size class in ≡d(n). Similarly, all classes in ≡d(n) are vanishing if +with limit probability zero, a random model of size n belongs to a maximum +size class. Then the following results hold as n → ∞. +• If d(n) ≤ n/2|τ| − f(n) where f(n) = ω(√n), then ≡d(n) has a domi- +nating class. +• If d(n) ≥ n/2|τ| − f(n) where f(n) = o(√n), then ≡d(n) has no domi- +nating class. +• If d(n) ≥ n/2|τ| + f(n) where f(n) = ω(√n), then every class in ≡d(n) +is vanishing. +One corollary of these results is that for d(n) ≤ n/t−f(n), if f(n) = ω(√n), +then with limit probability one, two random models of size n cannot be +separated in GMLUd(n). +Finally, we also give a non-asymptotic variant +of the characterization of the class distributions for ≡d including explicit +bounds on d for separating the cases where ≡d has a majority class or not. +By a majority class, we mean a class containing more than half of all models +in M. +Concerning related work, as already mentioned, it is well known that +entropy and Kolmogorov complexity are related. Indeed, for computable +distributions, Shannon entropy links to Kolmogorov complexity to within +a constant. This result is discussed, e.g., in [11], [4], [10]. However, it is +shown in [15] that the general link fails for R´enyi and Tsallis entropies. See +3 + +for example [4], [10], [15] for R´enyi and Tsallis entropies. The first connec- +tion between logical formula length and entropy has—to our knowledge— +been obtained in [8], [7], where expected Boltzmann entropy is shown to be +asymptotically equivalent to description complexity. +Concerning further related work, we will next discuss the proof tech- +niques used in the current paper. For proving bounds on formula sizes, we +use formula size games for the logics GMLUd. +Indeed, variants of stan- +dard Ehrenfeucht-Fra¨ıss´e games and (graded) bisimulation games would not +suffice, as we need to deal with formula length, and thereby with all logi- +cal operators, including connectives. The formula size games for the logics +GMLUd will be developed below based on a similar game used in [8], [7]. +That game builds on the game for standard modal logic ML used and devel- +oped in [5] for proving a nonelementary succinctness gap between first-order +logic and ML. The first formula size game, developed by Razborov in [13], +dealt with propositional logic. A later variant of the game was defined by +Adler and Immerman for CTL in [1]. Designing the games for GMLUd is +relatively straightforward and based directly on similar earlier systems, but +using them requires some nontrivial combinatorial arguments. +In addition to games, we also make use of a range of techniques for esti- +mating model class sizes and description complexity. These include Stirling’s +approximations and Chernoff bounds. In particular, to obtain our results, +we prove new estimates on r-associated Stirling numbers, which may be of +independent interest. +As a brief summary of our paper, the main objective is to elucidate the +general picture of how description length relates to model class size. This +also builds links between logic and notions of entropy. The logic GMLU is +suitable for the current study, and it even allows simple access to a chain +of increasingly expressive logics GMLUd via increasing d. The concluding +section discusses possibilities for generalizing to further logics. While the +current paper focuses on theory, the notion of description complexity is also +relevant in a range of applications. For example, in some currently active +research on explainability in AI, minimal length specifications can be used +as explanations of longer formulas. For work on this topic see, e.g., [2], [6]. +The plan of the paper is as follows. After the preliminaries in Section +2, we prove crucial lower bounds for description complexity in Section 3 +using games. In Section 4 we prove a monotone connection between model +class size and description complexity, and in Section 5 we investigate phase +transitions of class size distributions by varying n and d. Section 6 concludes +the paper. +4 + +2 +Preliminaries +We first define the logics studied in this work. +Let τ be a finite set of +proposition symbols. We consider τ to be fixed throughout the entire paper. +The syntax of graded universal modal logic GMLU[τ] is generated as +follows (the syntactic choices will be explained later on): +ϕ :=♦≥kψ | ■ 0}. We first define a model M0 as follows. For each i ∈ I, i ̸= 1, the +type πi is realized precisely ni times. The type π1 is realized n − � +i̸=1 ni +times. Additionally, the point 1 is of type π1. Intuitively the model M0 is a +model of the class Mn, where all points not fixed by the tuple n are of type +π1. We set An := {(M0, 1)}. +9 + +For the set Bn we define models Mi→j for some pairs (i, j) ∈ supp(n) × +supp(n) as follows. For i ̸= 1 and any j ∈ supp(n), the model Mi→j is +obtained from the model M0 by changing one point of type πi to type πj. +If i = 1 and nj = d, then the model Mi→j is obtained from M0 by changing +|π1|M0 − d + 1 points w ̸= 1 of type π1 to type πj. If i = 1 and nj < d, +no model is defined for the pair (i, j). We set Bn := {(Mi→j, 1) | i, j ∈ +supp(n)}. +In terms of their tuples n′, the models Mi→j have n′ +i = ni − 1 and +n′ +j = nj + 1, except if nj = d, in which case n′ +j = nj = d. For all other +indices ℓ, n′ +ℓ = nℓ. All models (M0, 1) and (Mi→j, 1) are propositionally +equivalent as they realize the type π1 in the point 1. +Let P = (r, A, B) be a position of the game GAMEd(r, An, Bn). +We +define a directed graph G(A, B) := (V, E), where V = supp(n) and (i, j) ∈ E +if there are propositionally equivalent (M0, w) ∈ A and (Mi→j, v) ∈ B, or +vice versa with respect to A and B. We call a set C ⊆ supp(n) a cover of +G(A, B) if for every (i, j) ∈ E, we have that either i ∈ C or both j ∈ C and +nj < d. The cost r(C) of a cover C is +r(C) := +� +i∈C +ni. +Intuitively, the definition of a cover means that including an index i ∈ I +generally covers all edges to and from i. The exception is that when ni = d, +incoming edges are not covered. +In the proof of the following lemma we use the notation Md(A) = {M | +(M, w) ∈ A, w ∈ W} for the set of models that have at least one pointed +version in the set A of pointed models. We also denote by tp(A) the set of +propositional types realized in the set A of points in a model M that will +below be clear from the set of points. +Lemma 3.3. Let P = (r, A, B) be a position of the game GAMEd(r, An, Bn) +and let +R(P) := min{r(C) | C is a cover of G(A, B)}. +If r < R(P), then D has a winning strategy from position P. +Proof. We show for all possible moves of S that either the condition r < +R(P) is maintained or D has a winning strategy for some other reason. +Since the definition of the graph G(A, B) is symmetrical with respect to A +and B, we only need to handle one of each pair of dual moves. +p-move: Since 0 < r < R(P), we have propositionally equivalent models on +both sides of the game and thus D wins if S makes any p-move. +∨-move: Let A1, A2 ⊆ A and r1, r2 ≥ 1 be the choices of S and let P1 = +(r1, A1, B) and P2 = (r2, A2, B). +For each (i, j) ∈ E there is a pair of +10 + +propositionally equivalent models (M, w) ∈ A and (M′, v) ∈ B as witnesses. +Since A1∪A2 = A, each model (M, w) ∈ A is in A1 or A2. The set B remains +unchanged so for each (i, j) ∈ E we have (i, j) ∈ E1 or (i, j) ∈ E2. Now if +r1 ≥ R(P1) and r2 ≥ R(P2), then there is a minimal cover C1 of position P1 +with r(C1) ≤ r1 and the same for P2. Now the set C := C1 ∪ C2 is a cover +of position P with r(C) ≤ r1 + r2 < r, which is a contradiction. Therefore +we have r1 < R(P1) or r2 < R(P2) and D can maintain the condition by +choosing such a position. +♦≥d-move: Let k ≤ d be the number chosen by S. The following position +is P ′ = (r − k, A′, B′). We first note that if M0 ∈ Md(A) ∩ Md(B), then S +must choose k points from the version in A and n − k + 1 points from the +version in B. Thus the next position P ′ will have propositionally equivalent +versions (M0, w) ∈ A′ and (M0, v) ∈ B′. +By Lemma 3.2 this gives D a +winning strategy so we assume M0 is only present on one side of the game. +Case A: M0 ∈ Md(A). For each (M0, w) ∈ A, S chooses a set PM0,w of k +points. Let PM0 := � +w∈W PM0,w. We consider the following cases: +1) We have � +πi∈tp(PM0) ni > k. Let (i, j) ∈ E. Since the model Mi→j +only differs from M0 by n′ +i = ni − 1 and n′ +j ≥ nj, the model Mi→j has +at least k points with types from tp(PM0). Thus, when S chooses the set +NMi→j of n − k + 1 points, it contains at least one point with a type from +tp(PM0). Thus the pair of propositionally equivalent models is maintained +and (i, j) ∈ E′. +2) We have � +πi∈tp(PM0) ni = k. Let (i, j) ∈ E. Assume πi /∈ tp(PM0) or +πj ∈ tp(PM0). Now the model Mi→j has at least k points with types from +tp(PM0) and (i, j) ∈ E′ as in case 1. +Assume then that πi ∈ tp(PM0) and πj /∈ tp(PM0). All edges of this kind +being eliminated is an acceptable worst case, so we assume that (i, j) /∈ E′. +Summing up Case A, the worst case is that S chooses a set tp(PM0) +of types with � +πi∈tp(PM0) ni = k ≤ d and eliminates all edges with πi ∈ +tp(PM0) and πj /∈ tp(PM0). +Case B: M0 ∈ Md(B). For each (M0, w) ∈ B, S chooses a set NM0,w of +n − k + 1 points. Let NM0 := � +w∈W NM0,w and let Π be the complement of +tp(NM0). We note that ni < d for each πi ∈ Π since M0 only has at most +k − 1 points with types from Π. +Let (i, j) ∈ E and assume πi ∈ Π or πj /∈ Π. +Now πi /∈ tp(NM0) +or πj ∈ tp(NM0) so the model Mi→j has at least n − k + 1 points with +types from tp(NM0). Thus any set PMi→j,v of k points contains at least one +point with a type from tp(NM0). Thus the pair of propositionally equivalent +models is maintained and (i, j) ∈ E′. +Now assume πi /∈ Π and πj ∈ Π. If Mi→j has at least n − k + 1 points +with types from tp(NM0), then (i, j) ∈ E′ as above. We thus assume that +Mi→j has less than n − k + 1 points with types from tp(NM0), meaning it +11 + +has at least k points with types from Π. Since nj < d and Mi→j differs from +M0 only by n′ +i = ni − 1 and n′ +j = nj + 1, the only remaining option is that +Mi→j has exactly k points with types from Π. Thus by the same reasoning +M0 has exactly k − 1 points with types from Π and since k − 1 < d, we have +� +i∈Π ni = k − 1. We again accept S eliminating these edges as a worst case +and assume (i, j) /∈ E′. +Summing up Case B, the worst case is that S chooses a set Π of types +with � +πi∈Π ni = k −1 < d and eliminates all edges with πi /∈ Π and πj ∈ Π. +We now consider the condition r − k < R(P ′) in the following position. +From the above arguments we see that the only way for S to eliminate +edges moving from G(A, B) to G(A′, B′) is to choose in each model (M0, w) +or (Mi→j, v) in A exactly all points that satisfy some set Π of types. If +M0 ∈ Md(A), we have � +πi∈Π ni = k ≤ d and S can eliminate all edges from +types in Π to other types. If M0 ∈ Md(B), we have � +πi∈Π ni = k − 1 and +ni < d for all πi ∈ Π. In this case S can eliminate all edges from other types +to types in Π. In both cases, the set C = {i ∈ supp(n) | πi ∈ Π} covers all +eliminated edges. The cost of C is r(C) = � +i∈C ni ≤ k. Let C′ be a cover +of P ′ with minimal cost so r(C′) = R(P ′). Now C ∪ C′ is a cover of P so +r < R(P) ≤ R(P ′) + r(C) ≤ R(P ′) + k. Thus r − k < R(P ′). +♦=d-move: Let k < d be the number chosen by S. The following position +is P ′ = (r − k − 1, A′, B′). As for the ♦≥d-move, we may assume that M0 is +only present on one side of the game. +Case A: M0 ∈ Md(A). For each (M0, w) ∈ A, S chooses a k point set +PM0,w and NM0,w = W \ PM0,w. +We denote PM0 := � +w∈W PM0,w and +NM0 := � +w∈W NM0,w. We consider the following cases: +1) We have � +πi∈tp(PM0) ni > k. Thus there are propositionally equivalent +w ∈ PM0 and v ∈ NM0. This means that in the following position P ′ there +are propositionally equivalent versions of the model M0 on both sides of the +game. D has a winning strategy from P ′ by Lemma 3.2. +2) We have � +πi∈tp(PM0) ni = k. Note that since k < d, also ni < d for all +πi ∈ tp(PM0). Let (i, j) ∈ E. +Assume πi, πj /∈ tp(PM0). Now the model Mi→j has exactly k points +with types from tp(PM0). Thus if S chooses a k + 1 point set PMi→j, then +one of those points v′ has a type πℓ /∈ tp(PM0). By the definition of the +models, nℓ ̸= 0 so there is a point w′ in the model M0 of type πℓ. Now +there are propositionally equivalent (Mi→j, v′) ∈ A′ and (M0, w′) ∈ B′ so +(i, j) ∈ E′. +Similarly, if S chooses for Mi→j a set NMi→j of n − k + 1 points, then +this set contains at least one point v′ of a type πℓ ∈ tp(PM0). Now there +is a point w′ ∈ PM0 of type πℓ. Thus there are propositionally equivalent +(M0, w′) ∈ A′ and (Mi→j, v′) ∈ B′ so (i, j) ∈ E′. +If πi ∈ tp(PM0) or πj ∈ tp(PM0), we assume (i, j) /∈ E′. As for the +12 + +♦≥d-move, this is an acceptable worst case for our lower bound. +Summing up Case A, the worst case is that S chooses a set tp(PM0) of +types with � +πi∈tp(PM0) ni = k < d and eliminates all edges (i, j) ∈ E with +πi ∈ tp(PM0) or πj ∈ tp(PM0). +Case B: M0 ∈ Md(B). For each (M0, w) ∈ B, S chooses either a set PM0,w +of k + 1 points or a set NM0,w of n − k + 1 points. There are three cases: +1) Assume first that there are (M0, w), (M0, w′) ∈ B with sets PM0,w and +NM0,w′ chosen. Since k + 1 + n − k + 1 > n, there is v ∈ PM0,w ∩ NM0,w′. In +the following position P ′ the model (M0, v) is on both sides of the game so +by Lemma 3.2 D has a winning strategy from P ′. +2) Assume then that S chooses a set PM0,w of k+1 points for each (M0, w) ∈ +B and let PM0 = � +w∈W PM0,w. Let (i, j) ∈ E and let (Mi→j, v) ∈ A and +(M0, w) ∈ B be the corresponding models. +Assume πi /∈ tp(PM0) or πj ∈ tp(PM0). Now the model Mi→j has at +least k+1 points with types from tp(PM0). Thus the set NMi→j,v of size n−k +has at least one point with a type from tp(PM0). Thus the propositionally +equivalent pair of models is maintained and (i, j) ∈ E′. +Assume πi ∈ tp(PM0) and πj /∈ tp(PM0). If Mi→j has at least k+1 points +with types from tp(PM0), the above argument again works and (i, j) ∈ E′. +Since k < d and Mi→j only differs from M0 by n′ +i = ni − 1 and n′ +j ≤ nj + 1, +the only remaining option is that Mi→j has exactly k points with types from +tp(PM0). We obtain � +i∈tp(PM0) n′ +i = k. We again assume as a worst case +that (i, j) /∈ E′. +3) Finally assume that S chooses a set NM0,w of n − k + 1 points for each +(M0, w) ∈ B and let NM0 = � +w∈W NM0,w. Let Π be the complement of +tp(NM0). +Assume πi ∈ Π or πj /∈ Π. Now πi /∈ tp(NM0) or πj ∈ tp(NM0) so the +model Mi→j has at least n−k+1 points with types from tp(NM0). Thus the +set PMi→j,v of size k has at least one point with a type from tp(NM0). Thus +the propositionally equivalent pair of models is maintained and (i, j) ∈ E′. +Assume πi /∈ Π and πj ∈ Π. If Mi→j has at least n − k + 1 points with +types from tp(NM0), the above argument again works and (i, j) ∈ E′. +We assume Mi→j has less than n−k+1 points with types from tp(NM0), +meaning it has at least k points with types from Π. Since k < d and Mi→j +only differs from M0 by n′ +i = ni − 1 and n′ +j ≤ nj + 1, the only remaining +option is that Mi→j has exactly k points with types from Π. We obtain +� +i∈Π n′ +i = k. We again assume as a worst case that (i, j) /∈ E′. Edges +eliminated this way are ones from other types to types in Π. In particular, +we note that for any edge (i, j) eliminated this way, nj < d since M0 has at +most n − (n − k + 1) = k − 1 points of type πj. +Summing up Case B, the worst case is that S chooses a set Π of types +and eliminates either all edges (i, j) ∈ E with πi ∈ Π or πj /∈ Π or all edges +13 + +(i, j) ∈ E with πi /∈ Π or πj ∈ Π. In both cases � +i∈Π n′ +i = k for the model +Mi→j and for the latter case nj < d. +We now consider the condition r−k−1 < R(P ′) in the following position. +From the above arguments, we see that the only way for S to eliminate +edges moving from G(A, B) to G(A′, B′) is to choose in each model (M0, w) +or (Mi→j, v) in A exactly all points that satisfy some set Π of types. If +M0 ∈ Md(A), we have � +πi∈Π ni = k < d and S can eliminate all edges to +and from the set Π of types. If M0 ∈ Md(B), we have � +πi∈Π n′ +i = k < d and +S can eliminate either all edges to or all edges from the set Π of types. In +particular, if nj = d for πj ∈ Π, then only outgoing edges can be eliminated. +In all cases, the set C = {i ∈ supp(n) | πi ∈ Π} covers all eliminated edges. +The cost of C is r(C) = � +i∈C ni ≤ k + 1 since � +i∈C ni ≤ � +i∈C n′ +i + 1. +Let C′ be a cover of P ′ with minimal cost so r(C′) = R(P ′). Now C ∪ C′ +is a cover of P so r < R(P) ≤ R(P ′) + r(C) ≤ R(P ′) + k + 1. +Thus +r − k − 1 < R(P ′). +A lower bound for the description complexity of an arbitrary class Mn +can now be obtained by simply calculating the minimum cost of a cover of +G(An, Bn). +Lemma 3.4. Let Mn be a class with ni = d for at least two different i ∈ I. +Then C(Mn) ≥ � +i∈I +ni. +Proof. We assume that n1 = d. We consider the graph G(An, Bn) = {(i, j) ∈ +supp(n) | i ̸= 1 or nj = d}. This graph has edges from all types πi with +ni ̸= 0 to each other, with the exception that there are no edges from π1 to +πj with nj < d. +We first note that C = supp(n) is a cover of G(An, Bn) with cost � +i∈I ni. +For C′ ̸= C if i /∈ C′ with i ̸= 1, then by the definition of a cover, the edge +(i, 1) ∈ E is not covered, since i /∈ C and n1 = d. If 1 /∈ C′, then let j ∈ I +be the other index with nj = d besides 1. Now the edge (1, d) ∈ E is not +covered since 1 /∈ C and nj = d. We see that C is a minimal cost cover and +by Lemma 3.3 and Theorem 3.1 the claim holds. +Lemma 3.5. Let Mn be a class with ni = d for at most one i ∈ I. Let πj +be one of the types with the most realizing points in Mn. Then C(Mn) ≥ +� +i∈I\{j} ni = n − |πj|. +Proof. We assume n1 = max{ni | i ∈ I}. Thus π1 is one of the types with +the most realizing points and if n1 < d, then ni < d for all i ∈ I. +We consider the graph G(An, Bn). From the definition we get G(An, Bn) = +{(i, j) ∈ supp(n) | i ̸= 1}. This graph has edges from all types πi with ni ̸= 0 +to each other, with the exception that there are no edges originating from +1. +14 + +We first note that C = supp(n)\{1} is a cover of G(An, Bn) with r(C) = +� +i∈I\{1} ni = n − |π1|. Clearly supp(n) is a cover with higher cost. +Let C′ ⊂ supp(n). If there are i, j ∈ supp(n) with i, j /∈ C′, then the +edge (i, j) or (j, i) is in E and is not covered. If there is only one i ∈ supp(n) +with i /∈ C′ and i ̸= 1, then r(C′) = � +i∈I\{j} ni ≥ r(C) since π1 is one of +the types with the largest ni. We see that C is a minimal cost cover and by +Lemma 3.3 and Theorem 3.1 the claim holds. +We sum up the above lemmas into the Theorem below: +Theorem 3.6. Let n be an (n, d)-admissible tuple and let Mn be the cor- +responding equivalence class of ≡d. If ni = d for at least two i ∈ I, then +C(Mn) ≥ � +i∈I ni. Otherwise C(Mn) ≥ � +i∈I\{j} ni = n − |πj|. +4 +A monotone connection +Consider the following strict partial orders on the set of all (n, d)-admissible +tuples. +1. n kkdd+1 +d +kd! +⇔ n > (ni + 1)kkdd+1 +d +kd! ++ +k +� +ℓ=1 +nℓ +Using nℓ < d, which holds for every 1 ≤ ℓ ≤ k, and 1 ≤ kd ≤ t gives us the +desired result. +Lemma 4.5. Fix d ∈ Z+ and let n be an (n, d)-admissible tuple. Suppose +that n is sufficiently large with respect to d and |τ|. Then for every i ∈ I +such that ni < d we have that +|Mn| < |Mn′|, +where n′ +j = d, when j = i, and n′ +j = nj otherwise. +Proof. For notational simplicity we assume that nℓ < d iff ℓ ≤ k. By Lemma +4.3 the inequality that we need to establish is +� +n +n1, . . . , ni, . . . , nk, m +� +· kd! · +�m +kd +� +≥d +< +� +n +n1, . . . , ni−1, ni+1, . . . , nk, m + ni +� +· (kd + 1)! · +�m + ni +kd + 1 +� +≥d +, +where m = n − �k +ℓ=1 nℓ and kd = t − k. Note that since n is large enough +w.r.t. d, kd ≥ 1. Simplifying this gives us the equivalent inequality +�m +kd +� +≥d +< +m!ni! +(m + ni)! · (kd + 1) · +�m + ni +kd + 1 +� +≥d +It follows from Lemma 4.1 that +�m + ni +kd + 1 +� +≥d +≥ (kd + 1)m+ni +(kd + 1)(kd+1)d +and +�m +kd +� +≥d +≤ km +d +kd! +18 + +Hence we only have to show that +km +d +kd! < +m!ni! +(m + ni)! · (kd + 1) · (kd + 1)m+ni +(kd + 1)(kd+1)d +or equivalently that +(kd + 1)(kd+1)d +(kd + 1)ni+1kd!ni! < m!(kd + 1)m +km +d (m + ni)! += +(kd + 1)m +km +d +�ni−1 +j=0 (m + ni − j) += +� kd + 1 +kd +� �� � +>1 +�m� ni−1 +� +j=0 +(m + ni − j) +Recall 0 ≤ ni < d and 1 ≤ kd ≤ t. Hence in the above inequality the left +hand side is a constant, and thus it suffices to show that the right hand side +formula tends to infinity as n grows (recall m = n−�k +ℓ=1 nℓ, i.e., m is just n +minus a constant). However, this is clear, since the right hand side is of the +form f(n)/g(n), where f grows exponentially w.r.t. n while g grows only +polynomially w.r.t. n. +4.3 +Connecting size and description complexity +Let Pn,d denote the set of all (n, d)-admissible tuples. We define a natural +partial order ⪯ on Pn,d as follows: n ⪯ n′ if and only if ni ≤ n′ +i, for every +i ∈ I. By writing n ≺ n′ we mean that n ⪯ n′ and n ̸= n′. +Lemma 4.6. Suppose that n is sufficiently large with respect to d and |τ|. +Let n and n′ be (n, d)-admissible tuples such that n ≺ n′. Then +|Mn| < |Mn′| +Proof. It suffices to show that if n′ is an immediate successor of n, then +|Mn| < |Mn′|. Now, there must exist exactly one i ∈ I such that n′ +i = ni + 1 +and n′ +j = nj for every j ̸= i. If n′ +i < d, then the claim follows from Lemma +4.4, while if n′ +i = d, then the claim follows from Lemma 4.5. +Recall the constant cτ = 2|τ|(2|τ| + 1) − 1 from the previous section. We +define yet another partial order ⪯τ on Pn,d as follows: n ⪯τ n′ if and only +if the following two conditions hold: +1. For every i ∈ I we have that ni ≤ n′ +i. +2. Either n = n′ or � +i∈I(n′ +i − ni) > cτ. +19 + +Roughly speaking n ⪯τ n′ means that if the tuples are distinct, then the +distance between them w.r.t. to the order ⪯ is more than cτ. Again, by +writing n ≺τ n′ we mean that n ⪯τ n′ and n ̸= n′. For example (1, d) ≺τ +(2 + cτ, d), but (1, d) ̸⪯τ (1 + cτ, d). +Lemma 4.7. Suppose that n is sufficiently large with respect to d and |τ|. +Let n and n′ be (n, d)-admissible tuples such that n ≺τ n′. Then we have +that +C(Mn) < C(Mn′) +Proof. Suppose first that d occurs at least twice in n′. Theorem 3.6 entails +that C(Mn′) ≥ � +i∈I n′ +i. We now have two cases based on whether or not d +occurs at least twice in n. Suppose first that d occurs exactly once in n say, +nj = d. Then Mn can be defined by a formula of size +cτ + +� +i∈I−{j} +ni < +� +i∈I−{j} +n′ +i ≤ C(Mn′) +and hence C(Mn) < C(Mn′). On the other hand, if d occurs at least twice +in n, then Mn can be defined by a formula of size +cτ + +� +i∈I +ni < +� +i∈I +n′ +i ≤ C(Mn′) +and hence C(Mn) < C(Mn′). +Suppose then that d occurs exactly once in n′, say n′ +j = d. Theorem 3.6 +entails that C(Mn′) ≥ � +i∈I−{j} n′ +i. Since n ≺τ n′, we have that ni < d, for +every i ̸= j. Since d must occur at least once in n — as n is (n, d)-admissible +— we have that nj = d. Now Mn can be defined by a formula of size +cτ + +� +i∈I−{j} +ni < +� +i∈I−{j} +n′ +i ≤ C(Mn′) +and hence C(Mn) < C(Mn′). +Recall the partial orderings ≤s and ≤c on Pn,d, which were introduced at +the beginning of this section. The following theorem formalizes a connection +between sizes of model classes and their description complexities. +Theorem 4.8. If n is sufficiently large with respect to d, then +⪯τ ⊆ ≤s ∩ ≤c . +In particular, if n and n′ are two distinct ⪯τ-comparable tuples, then +|Mn| < |Mn′| ⇔ C(Mn) < C(Mn′). +20 + +Proof. Suppose that n ≺τ n′. Lemmas 4.6 and 4.7 guarantee that if n is +large enough w.r.t. d, then |Mn| < |Mn′| and C(Mn) < C(Mn′). Hence +n ≤s n′ and n ≤c n′, which proves the first claim. The second claim follows +directly from the first. +The intuitive content of the above theorem is that the partial order ⪯τ +approximates both ≤s and ≤c. Hence ⪯τ can be viewed as a highly non- +trivial monotone connection between ≤s and ≤c. +We note that the above theorem also works for Boltzmann entropy. This +is because the order ≤s is the same as the order based on Boltzmann entropy, +since logarithm is an increasing function. We formulate the latter of the two +claims as a corollary. +Corollary 4.9. Assume n is sufficiently large with respect to d. If n and n′ +are two distinct ⪯τ-comparable tuples, then +HB(Mn) < HB(Mn′) ⇔ C(Mn) < C(Mn′). +5 +The phase transitions of class size distributions +In this section we move our attention from single classes to the entire pro- +bability distribution given by ≡d. By allowing d to depend on n, we obtain +qualitative results which link the growth rate of d with the emergence of a +dominating class, i.e., a class which contains almost all the models. For fixed +n, we also obtain quantitative results on how the relationship between d and +n determines whether there exists a class in ≡d which contains majority of +all the models of size n. We will also point out consequences of these results +on the “average-case” expressive power of GMLUd. +Throughout this section we continue to use our previous convention that +t = 2|τ| denotes the number of types π of the alphabet τ. We start with the +following observation, which gives a sense of what happens in the distribu- +tion, when the counting depth is increased. +Proposition 5.1. Suppose that d < d′ ≤ n/2. Then +HS(≡d) < HS(≡d′) and HB(≡d) > HB(≡d′). +Furthermore, for every n/2 ≤ d ≤ d′ we have that +HS(≡d) = HS(≡d′) and HB(≡d) = HB(≡d′). +Proof. By Proposition 2.1 HS(≡d) + HB(≡d) = |τ|n. Hence it suffices to +establish the claims in the case of Boltzmann entropy. We first note that if +n/2 ≤ d ≤ d′, then HB(≡d) = HB(≡d′), since the logic GMLUd can already +specify each structure up to isomorphism. +21 + +Suppose then that d < d′ ≤ n/2. Consider the class Mn, where n = +(0, . . . , 0, d, d). As the depth is increased to d′ > d, this class is divided to +at least two smaller classes with tuples (0, . . . , 0, d, d′) and (0, . . . , 0, d′, d). +Meanwhile, clearly no class increases in size so we see that HB(≡d) > +HB(≡d′). +We take this opportunity to point out an easy corollary of the previous +result. In [7] it was established that HB(≡n) ∼ |τ|n, by which we mean +that limn→∞ HB(≡n)/|τ|n = 1. When combined with Proposition 5.1, this +result yields quite directly the following. +Corollary 5.2. For any counting depth d(n), we have +HB(≡d(n)) ∼ |τ|n +Proof. W.l.o.g. we assume that d(n) ≤ n, for every n. Since log(|M|) ≤ |τ|n, +for any class M, we have that HB(≡d(n)) ≤ |τ|n. Since HB(≡n) ∼ |τ|n, for +every ε > 0 we have that if n is large enough, then HB(≡n) ≥ (1 − ε)|τ|n. +Since HB(≡d(n)) ≥ HB(≡n), for every ε > 0 we have that HB(≡d(n)) ≥ +(1 − ε)|τ|n, provided that n is sufficiently large. Combining these bounds +yields the desired result. +Thus, from an asymptotic point of view, the dependence of the counting +depth d on n has no effect on the Boltzmann entropy of ≡d. By virtue of +Proposition 2.1, the same is true for the Shannon entropy of ≡d. +We proceed now to further analyze the effect of counting depth on the +distribution. To formulate some of our results, we will use standard asymp- +totic notation, which we recall here. Let f, g : N → R>0. We use f = o(g) +to denote that limn→∞ f(n)/g(n) = 0. Furthermore, we use f = ω(g) to +denote that limn→∞ f(n)/g(n) = ∞. +We first show that if the counting depth grows slowly enough with respect +to n, then the distribution contains a class which contains almost all of the +models of size n. More formally, fix a function d : Z+ → N and thereby +a sequence ≡d(n) of equivalence relations. A class sequence M(n) with +respect to the sequence ≡d(n) is a function that outputs a single class for each +individual equivalence relation in the sequence. Each class sequence M(n) +is naturally associated with the corresponding probability sequence +pn := p≡d(n)(M(n)) = |M(n)| +2n|τ| . +We say that ≡d(n) has a dominating class if there exists a class sequence +M(n) such that pn → 1 as n → ∞. Then the class sequence M(n) is said to +dominate ≡d(n). Intuitively, a class (sequence) is dominating if a random +model of size n belongs to it with limit probability one. +Our proof uses the well-known Chernoff bounds. The following lemma +will require the lower-tail estimate, while the upper-tail estimate will be +used later in this section. +22 + +Proposition 5.3 (Chernoff bounds [12]). Let X := �n +i=1 Xi be a sum of +independent 0-1-valued random variables, where Xi = 1 with probability p +and Xi = 0 with probability 1 − p. Then for every δ ≥ 0 we have that +(Lower tail) Pr[X ≤ (1 − δ)np] ≤ e−δ2 np +2 +(Upper tail) Pr[X ≥ (1 + δ)np] ≤ e−δ2 np +2+δ +Lemma 5.4. For any f(n) = ω(√n), if the counting depth is d(n) ≤ n/t − +f(n), then the class sequence Mn, where n = (d(n), . . . , d(n)), dominates +≡d(n). +Proof. We will show that for each f(n), such that f(n) = ω(√n) and f(n) ≤ +n/t, we have that with limit probability one a random model realizes each +type more than n/t − f(n) times. +Given a type π, we let Xπ denote a +random variable which counts the number of times π is realized. Now Xπ := +� +1≤i≤n Xπ,i, where Xπ,i is an indicator random variable for the event that +the element i realizes the type π. Since the success probability of Xπ,i is +t−1, we have that E(Xπ) = n/t. +Now, Chernoff bound give us that for every n and for every 0 ≤ δ we +have +Pr +� +Xπ ≤ (1 − δ)n +t +� +≤ e−δ2 n +2t . +Note that δ can indeed depend on n. Setting δ(n) := +� +2tg(n)/n, for any +g(n) = ω(1), we obtain that +e−δ2 n +2t = e−g(n) → 0. +Furthermore +δ(n)n +t = +√ +2t · 1 +t +� �� � +=:C +· +� +g(n) · √n = C +� +ng(n). +Hence for any function g(n) = ω(1) we have that with high probability the +type π is realized more than n/t − C +� +ng(n) many times. +Since π was +arbitrary, it follows from the union bound that with high probability every +type π is realized more than n/t − C +� +ng(n) times. Now, if f(n) = ω(√n), +then by setting g(n) = (f(n)/C√n)2 we obtain that every type is realized +more than n/t − f(n) times. +We showed that when the counting depth is low enough, the distribution +has a dominating class. Next we will show that for a high enough counting +depth, there is no dominating class. For this, we will utilize the following +inequality version of Stirling’s approximation, due to Robbins [14]. +23 + +Proposition 5.5 (Stirling’s approximation [14]). For all n ∈ N, n > 0, we +have +√ +2πn +�n +e +�n +e +1 +12n+1 < n! < +√ +2πn +�n +e +�n +e +1 +12n +Lemma 5.6. For any f(n) = o(√n), if the counting depth is d(n) ≥ n/t − +f(n), then ≡d(n) has no dominating class as n → ∞. +Proof. Let M(n) be an arbitrary sequence of classes. We will show that +there is some n0 such that for every n ≥ n0 the probability of a model +belonging to class M(n) is at most half. +Let Mn be a class of the sequence M(n) with i, j ≤ t such that ni ̸= nj. +Let n′ be obtained from n by switching the numbers ni and nj in the tuple. +We clearly have |Mn′| = |Mn| by symmetry so the probability of a model +belonging to Mn is at most half. +It remains to show that classes of the sequence M(n) with tuples re- +peating only one number have probability at most half, when n is large +enough. Let d ≥ n/t − f(n), where f(n) = o(√n). Assume d < n/t and let +M = Mn, where n = (d, . . . , d). Note that this is the only (n, d)-admissible +tuple that repeats only one number. We show that this class M has proba- +bility less than half if n is large enough. In fact, we establish the stronger +claim that the sequence of such classes with tuples (d(n), . . . , d(n)) has limit +probability 0. +The size of the above class M is given by the sum +|M| = +� +n1+···+nt=n +ni≥d +� +n +n1, . . . , nt +� +. +First we note that a multinomial coefficient is largest, when the numbers +n1, . . . , nt are equal. Using this and Stirling’s approximation, we get +� +n +n1, . . . , nt +� +≤ +� +n +n +t , . . . , n +t +� +≤ +√ +2πn(n +e )ne +1 +12n +(�2π n +t ( n +et) +n +t e +1 +12 n +t +1)t += +√ +2πn(n +e )ne +1 +12n +�2π n +t +t(n +e )n 1 +tn e +t +12 n +t +1 += +� +tt +(2π)t−1 · +1 +√ +nt−1 · eg(n) · tn +≤ +� +tt +(2π)t−1 · +1 +√ +nt−1 · 2n|τ| +24 + +The exponent of e above is +g(n) = +1 +12n − +t +12n +t + 1 = +1 +12n − +t2 +12n + t += 12n + t − 12t2n +12n(12n + t) += (1 − t2)12n + t +12n(12n + t) . +Clearly g(n) < 0 for all positive n so eg(n) < 1 and the above estimate holds. +By the stars and bars method, the original sum has +�tf(n) + t − 1 +t − 1 +� +≤ (2t)t−1 · f(n)t−1 +terms so the final estimate is +|M| ≤ +� +tt +(2π)t−1 · (2t)t−1 · f(n)t−1 +√ +nt−1 · 2n|τ| +Recall that there are 2n|τ| models of size n in total and f(n) = o(√n). +We obtain the following limit: +lim +n→∞ +� +tt +(2π)t−1 · (2t)t−1 · f(n)t−1 +√ +nt−1 · 2n|τ| +2n|τ| += 0. +We see that the sequence of classes Mn with n = (d, . . . , d) has limit pro- +bability 0. Thus for a single such class, the probability is certainly at most +half if n is large enough. +Now assume d ≥ n/t, and consider the sequence of classes Mn, where n = +(n/t, . . . , n/t). This is again the only (n, d)-admissible tuple that repeats +only one number. The size of these classes is given by +� +n +n +t ,..., n +t +� +and it is easy +to see from the above that this sequence also has limit probability 0. Thus +a single such class has probability less than half for large enough n. +Intuitively, a class (sequence) is vanishing, if a random model of size n +belongs to it with limit probability zero. To define this notion formally, fix +a sequence ≡d(n). We say that all classes in ≡d(n) are vanishing as n → ∞ +if for all class sequences M(n) for ≡d(n), we have pn → 0 as n → ∞. We +now show that if the counting depth is high enough, then all classes in the +distribution sequence are vanishing. +Lemma 5.7. For any f(n) such that f(n) = ω(√n), if the counting depth +is d(n) ≥ n/t + f(n), then all classes in ≡d(n) are vanishing as n → ∞. +Proof. Fix some function f(n) such that f(n) = ω(√n). +We first show +that with limit probability zero a random model realizes each type less +than n/t + f(n) times. +Without loss of generality we can assume that +25 + +f(n) ≤ n−n/t. Given a type π, we let Xπ denote the same random variable +as in the proof of Lemma 5.4. Chernoff bound give us again that for every +n and for every 0 < δ ≤ 1 we have +Pr +� +Xπ ≥ (1 + δ)n +t +� +≤ e−δ2 +n +t(2+δ) ≤ e−δ2 n +3t +Note that δ can depend on n. Setting δ(n) := +� +3tg(n)/n, for any g(n) such +that g(n) = ω(1) and g(n) ≤ n/(3t) (to guarantee that δ(n) ≤ 1), we get +that +e−δ2 n +3t = e−g(n) → 0. +Furthermore +δ(n)n +t = +√ +3t · 1 +t +� �� � +=:C +· +� +g(n) · √n = C +� +ng(n). +Hence for any function g(n) such that g(n) = ω(1) and g(n) ≤ n/(3t) we +have that with high probability the type π is realized less than n +t −C +� +ng(n) +many times. Since π was arbitrary, it follows from the union bound that +with high probability every type π is realized less than n +t − C +� +ng(n) many +times. By setting g(n) = (f(n)/C√n)2 we obtain that every type is realized +less than n/t − f(n) times. +Now consider an arbitrary class sequence M(n). We want to show that +for every ε > 0 we have that pn < ε, provided that n is sufficiently large. +Consider a class M(n) and let n be the corresponding (n, d(n))-admissible +tuple, i.e., M(n) is the class Mn. If there is i ∈ I such that ni = d(n) = +n/t + f(n), then it follows from the previous result that p≡d(n)(Mn) < ε, as +long as n is sufficiently large. Suppose then that ni < n/t + f(n), for every +i ∈ I. In this case the class Mn is an isomorphism class, which means that +the probability that a random model belongs to Mn is simply +� +n +n1, . . . , nt +� +/2n|τ|, +which, as calculated in the proof of Lemma 5.6, is at most a constant times +1/ +√ +nt−1. This latter quantity is certainly less than ε, provided that n is +sufficiently large. +Hence p≡d(n)(Mn) < ε, provided that n is sufficiently +large. +We gather the above results in the following theorem: +Theorem 5.8. The following statements hold for counting depth d(n) as +n → ∞. +• If d(n) ≤ n/t−f(n) where f(n) = ω(√n), then ≡d(n) has a dominating +class. +26 + +• If d(n) ≥ n/t − f(n) where f(n) = o(√n), then ≡d(n) has no domina- +ting class. +• If d(n) ≥ n/t + f(n) where f(n) = ω(√n), then every class in ≡d(n) +is vanishing. +We point out a corollary of the above result. When the counting depth +is too low, almost all models are in the same dominating class in terms +of GMLUd definability. Conversely, if the counting depth is high enough, +GMLUd can separate the models into classes that vanish. These observations +directly give us the following result: +Corollary 5.9. Let f(n) = ω(√n). +If d(n) ≤ n/t − f(n), then with +limit probability one, two random models of size n cannot be separated in +GMLUd(n). If d(n) ≥ n/t + f(n), then with limit probability one, two ran- +dom models of size n can be separated in GMLUd(n). +We say that a class is a majority class, if it contains more than half of +all the models. The following theorem is a quantitative version of Theorem +5.8. +Theorem 5.10. Let n ∈ Z+. +• If d ≤ n/t − c1 +√n, where +c1 := 1 +t +� +2t ln(2t), +then the distribution ≡d for models of size n has a majority class. +• If d ≥ n/t − c2 +√n, where +c2 := +� +π +2t3(4t)1/(t−1) < c1 +then the distribution ≡d for models of size n does not have a majority +class. +Proof. Suppose first that d ≤ n/t − c1 +√n. Set δ = (tc1)/√n, in which case +δ · n +t = c1 +√n and δ2 · n/(2t) = (tc1)2/(2t). Applying Chernoff bound and +the union bound we obtain, in a similar manner as in the proof of Lemma +5.4, that with probability strictly greater than (1 − te−(tc1)2/(2t)) every type +is realized at least n/2 − c1 +√n-times. A quick calculation shows that this +latter probability is equal to 1/2 (hence the choice of c1). +Consider then the case d ≥ n/t − c2 +√n. +Let M = Mn, where n = +(d, . . . , d). Using the estimate from the proof of Lemma 5.6 with f(n) = +c2 +√n, we obtain +|M| +2n|τ| ≤ +� +tt +(2π)t−1 · (2t)t−1 · (c2 +√n)t−1 +√ +nt−1 += 1/2 +27 + +Thus M is not a majority class. +The same reasoning as in the proof of +Lemma 5.6 shows there is no other majority class. +We of course obtain a corresponding quantitative version of Corollary +5.9. +Corollary 5.11. Let n ∈ Z+. If d ≤ n/t − c1 +√n, then the probability that +two random models of size n can be separated in GMLUd is less than 1/4. +If d ≥ n/t − c2 +√n, then the probability that two random models of size n +can be separated in GMLUd is at least 1/2. +6 +Conclusion +We have established an interesting monotone connection between model +class sizes and description complexities, also obtaining related results for +entropy. Furthermore, we have characterized the phase transitions of model +class size when the domain size n and expressive power (in the form of count- +ing depth d) is altered. These results elucidate the interplay of class size and +formula length. While focusing on GMLUd, the results have been intended +to give a general overview of related phenomena. Thereby, an obvious future +direction involves investigating how these results lift into the framework of +first-order logic. There, natural parametrizations—analogous to varying the +counting depth d—can possibly be obtained by using quantifier depth and +the number of variables. +Moving beyond first-order logic, it would be interesting to investigate +the expressively Turing-complete logic CL, or computation logic, introduced +in [9]. Studying description complexities within that framework would lead +to an even closer link to Kolmogorov complexity. +Acknowledgments. Antti Kuusisto and Miikka Vilander were supported +by the Academy of Finland project Explaining AI via Logic (XAILOG), +grant number 345612 (Kuusisto). Antti Kuusisto was also supported by the +Academy of Finland project Theory of computational logics, grant numbers +324435, 328987, 352419, 352420, 352419, 353027. +References +[1] Micah Adler and Neil Immerman. An n! lower bound on formula size. +ACM Trans. Comput. Log., 4(3):296–314, 2003. +[2] Pablo Barcel´o, Mika¨el Monet, Jorge P´erez, and Bernardo Subercaseaux. +Model interpretability through the lens of computational complexity. In +Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina +Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information +28 + +Processing Systems 33: Annual Conference on Neural Information Pro- +cessing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, +2020. +[3] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. 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The American Math- +ematical Monthly, 62(1):26–29, 1955. +[15] Andreia Teixeira, +Armando Matos, +Andre Souto, +and Luis Fil- +ipe Coelho Antunes. +Entropy measures vs. Kolmogorov complexity. +Entropy, 13(3):595–611, 2011. +30 + diff --git a/99FST4oBgHgl3EQfbzgH/content/tmp_files/load_file.txt b/99FST4oBgHgl3EQfbzgH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf6fa5e97b86f78a66d37ee971ac50a27a5bf84d --- /dev/null +++ b/99FST4oBgHgl3EQfbzgH/content/tmp_files/load_file.txt @@ -0,0 +1,971 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf,len=970 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content='13800v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content='LO] 31 Jan 2023 A monotone connection between model class size and description length Reijo Jaakkola Tampere University Finland Antti Kuusisto Tampere University University of Helsinki Finland Miikka Vilander Tampere University Finland Abstract This paper links sizes of model classes to the minimum lengths of their defining formulas, that is, to their description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Limiting to models with a fixed domain of size n, we study description com- plexities with respect to the extension of propositional logic with the ability to count assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' This logic, called GMLU, can alternati- vely be conceived as graded modal logic over Kripke models with the universal accessibility relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' While GMLU is expressively complete for defining multisets of assignments, we also investigate its fragments GMLU(d) that can count only up to the integer threshold d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' We fo- cus in particular on description complexities of equivalence classes of GMLU(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' We show that, in restriction to a poset of type realiza- tions, the order of the equivalence classes based on size is identical to the order based on description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' This also demonstrates a monotone connection between Boltzmann entropies of model classes and description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Furthermore, we characterize how the relation between domain size n and counting threshold d determines whether or not there exists a dominating class, which essentially means a model class with limit probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' To obtain our results, we prove new estimates on r-associated Stirling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' As another cru- cial tool, we show that model classes split into two distinct cases in relation to their description complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' 1 Introduction This paper investigates how sizes of model classes are linked to the minimum lengths of formulas needed to define the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' In the scenarios we consider, we first fix a class M of models that share a domain of the same finite size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The model classes M ⊆ M are then studied with respect to the extension of propositional logic with the ability to count propositional assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' We call this logic GMLU, as it can alternatively be defined as graded modal 1 logic over Kripke models with the universal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' In order to obtain more fine grained results, we parameterize GMLU and study also its fragments GMLUd that can count only up to the threshold d ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' This also enables us to demonstrate how the relationship between minimum formula lengths and model class sizes develops when we gradually increase the expressive power of the logic used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' For a model class M ⊆ M, the description complexity of M with respect to GMLUd is simply the minimum length of a formula of GMLUd needed to define M, if such a formula exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' In this paper we are particularly interested in the description complexi- ties Cd(M) of the logical equivalence classes M determined by GMLUd over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Let us write M ≡d M′ if the models M, M′ ∈ M satisfy the same set of formulas of GMLUd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Note that Cd(M) of an equivalence class M of ≡d can also be regarded as the description complexity of each model M ∈ M, as the expressive power of GMLUd suffices precisely to describe M up to the equi- valence ≡d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' From this perspective, description complexity is analogous to Kolmogorov complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' There exist well known links between Kolmogorov complexity and Shannon entropy, see for example [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The recent work in [8],[7] demonstrates a way to conceive related results also in the scenario where relational structures are classified via logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' In particular, it is shown that the expected Boltzmann entropy of the equivalence classes of GMLU is asymptotically equivalent to the expected description complexity (with respect to GMLU) times the size of the vocabulary considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' It is also shown that for d = 1, the greatest equivalence class of GMLUd has maxi- mum description complexity among the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' This paper builds on those results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Firstly, as a crucial tool for our proofs, we establish a classification of description complexities into two distinct classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' This division is based on the numbers ni of elements realizing different propositional types i ∈ I in models of the described model class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' here I is just an index set for the types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The division is then determined by whether or not ni = d for at least two different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Using this, we establish a strong connection between model class sizes and description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' For each model M ∈ M, let nM denote the tuple (ni)i∈I that gives the numbers ni of points realizing propositional types in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Furthermore, instead of recording numbers ni greater than the counting threshold d, simply put d in nM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' We define a poset (M, ⪯τ) over the models, where τ is the vocabulary and the order ⪯τ is based on comparing the tuples nM coordinatewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The order ⪯τ is directly inherited also by the classes of ≡d such that M ⪯τ M′ if and only if for some (or equivalently, all) models M and M′ in the respective classes, we have M ⪯τ M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' We will prove that for all classes M and M′ of ≡d such that M and M′ are ⪯τ-comparable, we have |M| < |M′| ⇔ Cd(M) < Cd(M′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' 2 In other words, over ⪯τ, the ordering of model classes according to size is identical to the ordering based on description complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' This is an intimate link between syntax and semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' As a corollary, we obtain a correspon- ding relationship between Boltzmann entropies and description complexities of model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' We then investigate how the classes of ≡d behave when we alter the domain size n and counting threshold d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Note that increasing d corresponds to moving to more and more expressive logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' First we observe that for thresholds d and d′ > d and the corresponding Shannon entropies HS(≡d) and HS(≡d′) of the model class distributions given by ≡d and ≡d′, we have HS(≡d) < HS(≡d′) when d′ is at most n/2, and HS(≡d) = H(≡d′) when d is at least n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' A similar result also follows for expected Boltzmann entropies HB(≡d) and HB(≡d′), but with the orders reversed, that is HB(≡d) > HB(≡d′) for d′ at most n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' To get a better sense of the relative sizes of the classes when n and d are altered, we prove an asymptotic characterization of the class distributions as n → ∞ and d is a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Let us say that ≡d(n) has a dominating class if with limit probability one, a random model of size n belongs to a maximum size class in ≡d(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Similarly, all classes in ≡d(n) are vanishing if with limit probability zero, a random model of size n belongs to a maximum size class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Then the following results hold as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' If d(n) ≤ n/2|τ| − f(n) where f(n) = ω(√n), then ≡d(n) has a domi- nating class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' If d(n) ≥ n/2|τ| − f(n) where f(n) = o(√n), then ≡d(n) has no domi- nating class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' If d(n) ≥ n/2|τ| + f(n) where f(n) = ω(√n), then every class in ≡d(n) is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' One corollary of these results is that for d(n) ≤ n/t−f(n), if f(n) = ω(√n), then with limit probability one, two random models of size n cannot be separated in GMLUd(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Finally, we also give a non-asymptotic variant of the characterization of the class distributions for ≡d including explicit bounds on d for separating the cases where ≡d has a majority class or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' By a majority class, we mean a class containing more than half of all models in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Concerning related work, as already mentioned, it is well known that entropy and Kolmogorov complexity are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Indeed, for computable distributions, Shannon entropy links to Kolmogorov complexity to within a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' This result is discussed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=', in [11], [4], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' However, it is shown in [15] that the general link fails for R´enyi and Tsallis entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' See 3 for example [4], [10], [15] for R´enyi and Tsallis entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The first connec- tion between logical formula length and entropy has—to our knowledge— been obtained in [8], [7], where expected Boltzmann entropy is shown to be asymptotically equivalent to description complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Concerning further related work, we will next discuss the proof tech- niques used in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' For proving bounds on formula sizes, we use formula size games for the logics GMLUd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Indeed, variants of stan- dard Ehrenfeucht-Fra¨ıss´e games and (graded) bisimulation games would not suffice, as we need to deal with formula length, and thereby with all logi- cal operators, including connectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The formula size games for the logics GMLUd will be developed below based on a similar game used in [8], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' That game builds on the game for standard modal logic ML used and devel- oped in [5] for proving a nonelementary succinctness gap between first-order logic and ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The first formula size game, developed by Razborov in [13], dealt with propositional logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' A later variant of the game was defined by Adler and Immerman for CTL in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Designing the games for GMLUd is relatively straightforward and based directly on similar earlier systems, but using them requires some nontrivial combinatorial arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' In addition to games, we also make use of a range of techniques for esti- mating model class sizes and description complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' These include Stirling’s approximations and Chernoff bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' In particular, to obtain our results, we prove new estimates on r-associated Stirling numbers, which may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' As a brief summary of our paper, the main objective is to elucidate the general picture of how description length relates to model class size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' This also builds links between logic and notions of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The logic GMLU is suitable for the current study, and it even allows simple access to a chain of increasingly expressive logics GMLUd via increasing d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The concluding section discusses possibilities for generalizing to further logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' While the current paper focuses on theory, the notion of description complexity is also relevant in a range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' For example, in some currently active research on explainability in AI, minimal length specifications can be used as explanations of longer formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' For work on this topic see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=', [2], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The plan of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' After the preliminaries in Section 2, we prove crucial lower bounds for description complexity in Section 3 using games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' In Section 4 we prove a monotone connection between model class size and description complexity, and in Section 5 we investigate phase transitions of class size distributions by varying n and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Section 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' 4 2 Preliminaries We first define the logics studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' Let τ be a finite set of proposition symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' We consider τ to be fixed throughout the entire paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'} +page_content=' The syntax of graded universal modal logic GMLU[τ] is generated as follows (the syntactic choices will be explained later on): ϕ :=♦≥kψ | ■ 0, +(1) +where σ > 0 is the shape parameter and λ > 0 is the scale parameter, it says T ∼ WE(λ, σ). +Two particular models under this parametrization are obtained for σ = 1 and σ = 1/2, which +represents the exponential and the Rayleigh models with means λ and λ +� +π/2, respectively. In +this work, we focused on those models. However, if σ is unknown, we assume that it can be + +4 +replaced by consistent estimate. The ρ% survival time is given by +tρ = λ[− log(ρ)]σ +(2) +The regression structure can be incorporated in (1) by making +log(λi) = x⊤ +i β +(3) +where β is a p-vector of unknown parameters and xi is a vector of predictors related to the ith +observation. +In lifetime data, there is the censoring restriction, i.e, if T1, . . . , Tn are a random sample +from (1), instead of Ti, we observe, under right censoring, ti = min(Ti, Li), where Li is the +censoring time, independent of Ti, i = 1, . . . , n. Here, we consider an hybrid censoring scheme, +where the study is finalized when a pre-fixed number r ≤ n out of n observations have failed, as +well as when a prefixed time, say L1 = · · · = Ln = L, has been reached. The type I censoring +is a particular case for r = n and the type II censoring appears when L1, . . . , Ln = +∞. +Additionally, we add the non-informative censoring assumption, i.e., the random variables Li +does not depend on λ. Usually, the regression modeling considers the distribution of Yi = log(Ti) +instead of Ti, which is an accelerated lifetime model form, see Kalbfleisch and Prentice.9 The +distribution of Yi is of the extreme value form with pdf given by +f(yi; xi) = 1 +σ exp +�yi − µi +σ +− exp +�yi − µi +σ +�� +, +−∞ < yi < ∞, +(4) +where µi = log λi. From this moment, we assume that σ is known, Then, the log-likelihood +function derived from (4) is given by +ℓ(β) = +n +� +i=1 +� +δi +� +−n log σ + yi − µi +σ +� +− exp +�yi − µi +σ +�� +. +The total score function and the total Fisher information matrix for β are, respectively, +U β = σ−1X⊤W 1/2v and K = Kββ = σ−2X⊤W X, where X = (x1, . . . , xn)⊤, the model ma- +trix, assuming rank(X) = p, W = diag(w1, . . . , wn), wi = E +� +exp +� yi−µi +σ +�� +and v = (v1, . . . , vn)⊤, +vi = +� +−δi + exp +� yi−µi +σ +�� +w−1/2 +i +. It can observed that the value of wi depends on the mechanism + +5 +of censoring. That means wi = q × +� +1 − exp +� +−L1/σ +i +exp(−µi/σ) +�� ++ (1 − q) × (r/n), where +q = P +� +W(r) ≤ log Li +� +and W(r) denotes the rth order statistic from W1, . . . , Wn. Note that +q = 1 and q = 0 for types I and II censoring, respectively, as showed in Magalhães et al.5 +The maximum likelihood estimator (MLE) of β, �β, is the solution of U β = 0. The �β can +not be expressed in closed-form. It is typically obtained by numerically maximizing the log- +likelihood function using a Newton or quasi-Newton nonlinear optimization algorithm. Under +mild regularity conditions and in large samples, +�β ∼ Np +� +β, K−1 +ββ +� +, +(5) +approximately. The classic Wald test10 statistic is +WMLE = +� +C�β − Cβ(0)�⊤ � +C � +KC⊤�−1 � +C�β − Cβ(0)� +, +(6) +where C is a matrix of contrasts m × p. Under the null hypothesis H : Cβ = Cβ(0), WMLE +has a χ2 +p distribution up to an error of order n−1. The null hypothesis is rejected for a given +nominal level, α say, if the test statistic exceeds the upper 100(1 − α)% quantile of the χ2 +p +distribution. +3 +MCP-Mod General approach +We briefly summarize the two-stage procedure discussed by Pinheiro et al.3 We consider +that time-to-event responses tij for doses xi given to jth subject for i = 0, . . . , p and j = 1, . . . , ni +can be described by the Weibull distribution with scale parameters λi and shape parameter σ +defined in (1) and (3). Then, we consider the parameter µ as our response for the dose-response +model such that it could be defined as the median survival time (2) for ρ = 0.5 or, alternatively, +log(λ). +Initially, a set of candidate dose-response models µi = f(xi, θ) is considered such that each +model can be rewritten as function of standardized model as below + +6 +fm(x, θ) = θ0 + θ1f 0 +m(x, θ0) +(7) +for m = 1, . . . , M. +In this work, we consider (M = 5) standardized models: (i) linear: +f 0(x, θ0) = x; (ii) emax : f 0(x, θ0) = x/(x + ED50) where ED50 can be interpreted as the dose +that produces the desired response on 50% of subjects; (iii) exponential: f 0(x, θ0) = exp{x/δ}− +1, where δ is the exponential rate; (iv) logistic: f 0(x, θ0) = 1/(1 + exp{(ED50 − x)/δ}) (v) +beta: f 0(x, θ0) = β(δ1, δ2)(x/scal)δ1(1 − x/scal)δ2. +3.1 +MCP-step +In this step, a set of contrasts corresponding to the candidate models will be tested. For +each candidate model, an optimal contrast copt that maximizes the probability of rejecting the +hypothesis of non-signal dose-response is derived assuming that the candidate model is correct +and guess estimates for θ0: +copt ∝ S−1 +� +µ0 +m − µ0 +m +⊤S−11 +1S−11⊤ +� +, +(8) +where µ0 +m = (f 0 +m(x1, θ0), . . . , f 0 +m(xp, θ0)). Then, the test of hypotheses for proof-of-concept can +be translated to H0 : copt +m µ = 0 vs. H1 : copt +m µ > 0 for candidate model m based on the Wald +test statistic +W (m) = (copt +m ˆµ)⊤ � +Copt ˆSCopt⊤�−1 +m,m copt +m ˆµ +(9) +where Copt is the matrix m × p of optimal contrast with [A]m,m denoting the mth diagonal +element of matrix A for m = 1, . . . , M. Critical values for tests are derived based on the joint +distribution for W = (W (1), . . . , W (M)) allowing one to calculate multiplicity adjusted p-values +controlling the FWER at a prespecificed nominal type I error α. + +7 +3.2 +Mod-step +In this step, the estimation of non-linear dose responses models is performed in two stages. In +the first stage, the parameters µ = [µ1, . . . , µp] are estimated using standard software packages +with analysis of variance (ANOVA) parametrization for the design matrix resulting into a +separate parameter µi for each dose level xi for i = 1, . . . , D. Let ˆµ denote the estimated +dose-response parameter vector under ANOVA parametrization, such that we assume that ˆµ +follows approximately Np(µ, S), where S is the covariance matrix of ˆµ that is estimated by ˆS. +In particular for µ = log(λ), we have µ = β and S = K1 +ββ. +In the second stage, the non-linear dose-response model f(x, θ) is fitted by minimizing the +generalized linear squares (GLS) criterion: +Ψ(θ) = (ˆµ − f(x, θ))′ ˆS +−1(ˆµ − f(x, θ)) +(10) +with respect to θ. +Then, the minimum effective dose can be estimated as � +MED = {x|f(x, ˆθ) > f(0, ˆθ) + ∆}, +where ∆ is a clinical meaningful threshold and ˆθ minimizes (10). +4 +Improved inference +4.1 +Bias correction +Inferences based on maximum likelihood method depend strongly on asymptotic properties. +Among these properties, the MLE is approximately non-biased, in other words, E( �β − β) = +O(n−1), which is essential to define the mean of the normal distribution of �β. +Therefore, +likelihood inferences based on asymptotic approximation may not be reliable when sample sizes +are small or moderate, and two approaches are available to correct the MLE. +The corrective approach. +The bias of the MLE can be written as E( �β − β) = B(β) + +O(n−2), where B(β) is a term of order O(n−1), a function of the derivatives of the log-likelihood +function. Cox and Snell7 proposed a bias-corrected maximum likelihood estimator (BCE), that +can be expressed as �β = �β − B( �β), where B(β) is the term of order n−1, evaluated in �β and +E(�λ − λ) = O(n−2), i.e., less biased then MLE of β. The Cox and Snell’s method is known as + +8 +a corrective approach because the MLE is calculated and then, the bias correction is applied. +For the censored Weibull regression model, the expression of B( �β) has the form +B( �β) = − 1 +2σ3P Zd (W + 2σW ′) 1, +(11) +where P = K−1X⊤, Z = XK−1X⊤, Zd is a diagonal matrix with diagonal given by the +diagonal of Z, W ′ = diag(w′ +1, . . . , w′ +n), w′ +i = −σ−1L1/σ +i +exp{−L1/σ +i +exp(−µi/σ) − µi/σ} and 1 +is a n-dimensional vector of ones. +The preventive approach. +As alternative to the corrective approach, Firth11 proposed the +following modification in the score vector: +U ⋆ +β = Uβ − KββB(β), +(12) +where B(β) is given by (11). The estimator ˇβ, solution of U ⋆ +β = 0, has a bias of order O(n−2). +This is a preventive approach because the procedure already computes a less biased estimator +than the regular MLE. +4.2 +Covariance correction +From the general result of Magalhães et al.,8 we derived the specific matrix expression for +the MLE and BCE second-order covariance matrices for the censored Weibull regression model +and it is given by +Covτ +2(β⋆) = K−1 + K−1 � +∆ + ∆⊤� +K−1 + O(n−3), +(13) +where ∆ = −0.5∆(1) + 0.25∆(2) + 0.5τ2∆(3) with +∆(1) = 1 +σ4X⊤W ⋆ZdX, +∆(2) = − 1 +σ6X⊤ � +W Z(2)W − 2σW Z(2)W ′ − 6σ2W ′Z(2)W ′� +X, +∆(3) = 1 +σ5X⊤W ′W ⋆⋆X, +W ⋆ = diag(w⋆ +1, . . . , w⋆ +n), w⋆ +i = wi(wi − 2) − 2σw′ +i + στ1(w′ +i + 2σw′′ +i ), Z(2) = Z ⊙ Z, with ⊙ + +9 +representing a direct product of matrices (Hadamard product), W ⋆⋆ is a diagonal matrix, with +Z(W +2σW ′)Zd1 as its diagonal, W ′′ = diag(w′′ +1, . . . , w′′ +n), w′′ +i = −σ−1w′ +i +� +L1/σ +i +exp(−µi/σ) − 1 +� +, +τ = (τ1, τ2) = (1, 1) indicating the second-order covariance matrix of the MLE β⋆ = �β denoted +by Cov2( �β) and τ = (0, −1) indicating the second-order covariance matrix of the BCE β⋆ = �β +denoted by Cov2( �β). +4.3 +Wald-type test +Let Cov−1 +2 ( �β) and Cov−1 +2 ( �β) the inverse of Cov2( �β) and Cov2( �β), respectively and con- +sidering also the partitions and the notation for the Fisher information matrix discussed in the +introductory section, we can propose two modification to the Wald test in (6): +WMLE2 = +� +C �β − Cβ(0)�⊤ � +C �Cov2( �β)C⊤�−1 � +C �β − Cβ(0)� +, +(14) +WBCE = +� +C �β − Cβ(0)�⊤ � +C � +KC⊤�−1 � +C �β − Cβ(0)� +, +(15) +WBCE2 = +� +C �β − Cβ(0)�⊤ � +C �Cov2( �β)C⊤�−1 � +C �β − Cβ(0)� +, +(16) +WFirth = +� +C ˇβ − Cβ(0)�⊤ � +C ˇ +KC⊤�−1 � +C ˇβ − Cβ(0)� +, +(17) +where �Cov2( �β) is the matrix Cov2(β) evaluated at �β, � +K is the Fisher information evaluated +at �β, �Cov2( �β) is the matrix Cov2( �β) evaluated at �β, ˇ +K is the Fisher information evaluated +at ˇβ,. Under H, WMLE, WBCE, WBCE2, WFirth follow a χ2 +p distribution. +4.4 +Improved Estimator strategies +In the supplemental material and the next section, we studied the statistical properties of +improved estimators for β and its covariance matrix with the following strategies: the classi- +cal ML estimator with the Fisher information as covariance matrix (MLE); the classical ML +estimator with the corrected covariance matrix defined in (13) (MLE2); the bias corrected esti- +mator (BCE) given in (11) with the Fisher information as covariance matrix; the bias corrected +estimator with the corrected covariance matrix (BCE2); and the Firth estimator defined in (12) +with its Fisher information as covariance matrix. + +10 +5 +Simulation study +In our collaborative work, investigators wanted to establish a dose-response relationship +between a new inhibitor agent for pancreatic cancer in combination with a given dose of gem- +citabine in mouse models. Based on preliminary data, a survival median time of 4 months was +estimated in control-treated KPC mice model such that previous studies showed no survival +benefit with only gemcitabine.12 Assuming a Weibull distribution with σ = 0.5, we calculated +the placebo effect equal to 1.57 and maximum effect of 2.96 considering a hazard ratio of 4. +Investigators were interested in the minimum effective dose yielding a minimum hazard ratio +of 2 yielding ∆ = 0.693. +Model +Constraints +Guess estimates/True parameters +True MED +θ0 +θ1 +θ2 +Linear +- +E0 = 1.569 +δ = 0.0139 +- +- +Emax +50% at x4 +E0 = 1.569 +EMax = 2.079 +ED50 = 50.000 +25.00 +Exponential +10% at x1 +E0 = 1.569 +E1 = 0.017 +δ = 22.756 +84.51 +Logistic +10% at x3 +80% at x4 +E0 = 1.569 +EMax = 1.391 +ED50 = 40.329 +δ = 6.976 +40.37 +Beta +30% at x2 +E0 = 1.569 +EMax = 1.386 +δ1 = 0.749 +δ2 = 1.049 +10.61 +Table 1: Scenarios (Emax, Exponential, Logistic and Beta) and candidate models (Linear, +Emax, Exponential, Logistic and Beta) defined based on, respectively, true parameters and +guess estimates. True parameters/guess estimates were calculated based on placebo effect of +1.57, maximum effect of 2.96 and constraints with scale parameter of 120 for Beta model. True +MED was calculated based on ∆ = 0.693. +Five scenarios (constant, emax, exponential, logistic and beta model) presented in Table 1 +were studied. True parameters were defined based on the aforementioned placebo and maxi- +mum effects, and the percent of maximum effect that is achieved at givens dose as discussed in +Bornkamp et al.13 For each scenario, the doses 0, 5, 25, 50 and 100 (mg/kg) were considered +such that true MED was calculated as continuous dose given in Table 1. Five candidate mod- +els (linear, emax, exponential, logistic and beta model) were considered with guess estimates +defined as the true parameters to calculate the optimal contrasts in (8). For each model, the +null hypothesis of non-signal of the new inhibitor agent was tested at 5% significance level, and +we used Akaike Information Criteria (AIC) to choose the model to estimate MED when more +than one model rejected the non-signal hypothesis. + +11 +For both steps of the MCP-Mod framework, the five strategies (MLE, MLE2, BCE, BCE2 +and Firth) were evaluated based on a Monte Carlo simulation study with 100,000 replicates for +sample sizes of 5, 10, 15, 20 and 25 mice per dose and censoring rate of 10%, 25% and 50%. The +following operating characteristics were studied: (i) convergence rate of GLS algorithm in the +MCP-Mod General framework; (ii) Probability of incorrectly detecting a dose-response signal, +i.e., type I error; (iii) Probability of correctly detecting a dose-response signal, i.e., power; (iv) +Probability of selecting the true model given that there is a signal; (v) Bias of MED estimate +and (vi) RMSE of MED estimate. +5.1 +Results +In Figure 1, convergence rates when calculating estimators and applying them as input for +the MCP-Mod framework is presented for different censoring rates and true models. When +censoring is 10%, there is no difference among estimators; when censoring is 25%, similar con- +clusion can be drawn but the estimators Firth, MLE2 and BCE2 have lower convergence rates +than MLE and BCE for the true model Exponential; when censoring is 50%, the lower conver- +gence rates of the estimators Firth, MLE2 and BCE2 are also observed in other scenarios with +dose-relationship signal. These lower convergence rates are attained because small sample sizes +in combination with high censoring rates result into doses with no events (in our case, deaths), +therefore, very large estimates for the regression coefficients are obtained and, consequently, +singular covariance matrix estimates. +For the MCP-step, type I error probability is displayed for different censoring rates and +true models in Figure 2. When censoring is 10%, corrected estimators (MLE2, BCE, BCE2, +Firth) show observed type I error probability closer to the nominal value (0.05) than MLE for +all sample sizes. In particular, MLE shows type I error probability more than twice than the +nominal value for sample size of 5 and almost twice the nominal value for 10 mice per dose; +when censoring is 25%, corrected estimators are slightly conservative for small sample sizes +of 5 and 10, while type I error probability of MLE is still inflated for all sample sizes; when +censoring is 50%, corrected estimators are overly conservative for all sample sizes and MLE +produces the nominal type I probability. As sample size increases, it is expected that type I +error probability converges to the nominal value (0.05) for all estimators. + +12 +In Figure 3A, the probability of correctly detecting dose-response signal in the MCP-step +is showed as function of censoring rates and true models. It is expected that MLE show higher +probability than corrected estimators given that the power function is inflated as seen in Figure +2. A fair comparison would require us to re-adjust critical values to reject the null hypothesis +such that the observed type I probability was set at 0.05 for all estimators. +Nonetheless, +the probability of correctly detecting dose-response signal is similar among estimators when +censoring rate is 10% because of the large effect sizes. When censoring rate is 25%, all estimators +are still comparable except when the true model is Exponential with BCE showing a similar or +superior performance than MLE; BCE2 having a higher probability than MLE for sample sizes +from 10 to 25; Firth estimator displaying comparable performance with MLE only when sample +size is at least 15 mice per dose; and MLE2 having the poorest performance for all sample sizes. +For 50% of censoring, there is no clear pattern, but MLE presents the highest probability when +the sample size is 5 mice per dose in three out of four scenarios, while corrected estimators +have poorer performance that becomes comparable with MLE as sample size increases, except +MLE2. +In Figure 3B, the probability of correctly selecting dose-response model using AIC is cal- +culated given that we selected at least one model in the MCP-step, i.e., it is a conditional +probability. Conclusions are somewhat similar to 3A: when censoring is 10%, MLE shows a +slight higher probability for Exponential, Logistic and Beta as true models such that this pat- +tern becomes more prominent for censoring rates of 25% and 50%. For Emax as true model, +BCE and BCE2 show consistently higher performance when censoring rates are 25% and 50% +and no differences can be seen for censoring rate of 10%. +For Mod-step, we calculated the relative bias and RMSE of � +MED estimator in Figure 4A +and B, respectively, for different censoring rates and true models. When censoring is 10%, MLE +and corrected estimators have negligible differences for bias and RMSE; when censoring is 25%, +discrepancies can be see for bias and RMSE when true models are Emax and Exponential such +that MLE consistently shows a better performance than corrected estimators; when censoring +is 50%, conclusions are hard to be drawn for sample sizes of 5 and 10 such that estimators +perform similar - but MLE2 - for sample sizes greater than 15 mice per dose. + +13 +6 +Concluding Remarks +We have derived improved inferences based on the Wald statistic for WRM particularizing +general results from Cox and Snell7 and Magalhães et al.,8 which complements previous results +for improved inference based on likelihood ratio, Rao score and gradient statistics discussed in +Magalhães and Gallardo.6 Few authors have presented improved inference for survival models +with small sample sizes under the classical approach: Cordeiro and Colosimo14,15 and Medeiros16 +derived those statistics, respectively, for censored exponential regression models (ERM), a par- +ticular case of censored WRM. Also for ERM, Lemonte17 presented the second-order covariance +matrix of the MLE. +We have also proposed to use bias-corrected (BCE, BCE2 and Firth) and second-order +covariance matrices (MLE2, BCE2) as input for the general MCP-Mod framework introduced +by Pinheiros et al.,3 which addresses the issue of relying on the asymptotic properties of MLE, +which might not be valid for small sample sizes. +To the best of our knowledge, this work +is the first attempt to apply refined estimators for small sample sizes in the general MCP- +Mod framework. Two simulations studies were performed to study the properties of refined +estimators in an usual context of regression models and relevant operating characteristics in +the MCP-Mod framework. +In the simulation study presented in the Supplementary material, we showed numerical +evidences that BCE and Firth estimator have lower bias and RMSE than MLE; second-order +covariance matrices evaluated at MLE and BCE are closer to their respective empirical covari- +ance matrices in comparison to the first-order covariance matrices; and Wald statistics derived +from the combination between bias-corrected estimators and second-order covariance matrices +yielded type I error probability closer to the nominal value than the standard Wald statistic +with no loss of power. +In the simulation study for the MCP-Mod framework, we have found that refined estimators +and second-order covariance matrices approximate type I error probability of its nominal value +in the MCP-step, while there are negligible differences between MLE and refined estimators in +the probability of correctly detecting the dose-response signal, probability correctly selecting the +dose-response model, bias and RMSE when censoring rates are up to 25%. For censoring rate +of 50%, we found convergence issues for the corrected estimators and second-order covariance + +14 +matrices and poorer performance in the assessed operating characteristics. +In conclusion, we recommend the use of refined estimators for small sample sizes when +the expected censoring rate is at most 25% in the MCP-Mod framework. In the context of +basic science with limited sample sizes and large effect sizes, we do not expect large censoring +rates in mouse experiments. In human trials, smaller effect sizes are pursued such that larger +sample sizes are required. In this case, refined estimators are not needed because all estimators +present comparable performance for large sample sizes. Nonetheless, we showed that type I +error probability is still inflated even for sample sizes of 25 subjects per dose, therefore, the use +of refined estimators would avoid to dedicate further efforts on non-promising drugs. +We hope that refined estimators allow statisticians to implement the MCP-Mod framework +with small sample sizes accelerating the pre-clinical and clinical drug development process. +Similar ideas can be applied to other distributions such as Binomial, Negative Binomial and +Poisson, and they are currently under investigation. + +15 +10% +25% +50% +Constant +Emax +Exponential +Logistic +Beta +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample size per dose +MCP Convergence Rate +Method +MLE +MLE2 +BCE +BCE2 +Firth +Figure 1: Convergence rate when calculating MLE, MLE2, BCE, BCE2 and Firth estimators +and applying them as input for the MCP-Mod framework as function of censoring rate and +true model. +10% +25% +50% +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +Sample size per dose +Type I error probability +Method +MLE +MLE2 +BCE +BCE2 +Firth +Figure 2: Type I error probability when in the MCP-step using MLE, MLE2, BCE, BCE2 and +Firth estimators as function of censoring rate. + +16 +10% +25% +50% +Emax +Exponential +Logistic +Beta +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +Sample size per dose +Probability of Correctly Detecting Dose Response Signal +A +10% +25% +50% +Emax +Exponential +Logistic +Beta +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +Sample size per dose +Probability of Correctly Selecting Dose Response Model +B +Method +MLE +MLE2 +BCE +BCE2 +Firth +Figure 3: A. Probability of correctly detecting dose-response signal when in the MCP-step +using MLE, MLE2, BCE, BCE2 and Firth estimators as function of censoring rate and true +model. B. Probability of correctly selecting dose-response model using AIC in the Mod-step +using MLE, MLE2, BCE, BCE2 and Firth estimators as function of censoring rate and true +model. + +17 +10% +25% +50% +Emax +Exponential +Logistic +Beta +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +−50% +−25% +0% +25% +−75% +−50% +−25% +0% +−80% +−60% +−40% +−20% +0% +−50% +−25% +0% +Sample size per dose +Relative Bias (%) +A +10% +25% +50% +Emax +Exponential +Logistic +Beta +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +10 +15 +20 +25 +30 +0 +50 +100 +150 +200 +10 +20 +30 +5 +10 +15 +Sample size per dose +RMSE +B +Method +MLE +MLE2 +BCE +BCE2 +Firth +Figure 4: A. Bias of � +MED derived from the MCP-Mod framework when using MLE, MLE2, +BCE, BCE2 and Firth estimators as function of censoring rate and true model. B. Root-Mean- +Square Error of � +MED derived from the MCP-Mod framework when using MLE, MLE2, BCE, +BCE2 and Firth estimators as function of censoring rate and true model. + +18 +References +1 Jardim Denis L, Groves Eric S, Breitfeld Philip P, Kurzrock Razelle. Factors associated +with failure of oncology drugs in late-stage clinical development: a systematic review Cancer +treatment reviews. 2017;52:12–21. +2 Bretz Frank, Pinheiro José C, Branson Michael. Combining multiple comparisons and mod- +eling techniques in dose-response studies Biometrics. 2005;61:738–748. +3 Pinheiro José, Bornkamp Björn, Glimm Ekkehard, Bretz Frank. Model-based dose finding un- +der model uncertainty using general parametric models Statistics in medicine. 2014;33:1646– +1661. +4 Carroll Kevin J. On the use and utility of the Weibull model in the analysis of survival data +Controlled clinical trials. 2003;24:682–701. +5 Magalhães Tiago M., Gallardo Diego I., Gómez H. W.. Skewness of maximum likelihood +estimators in the Weibull censored data Symmetry. 2019;11:1351. +6 Magalhães Tiago M., Gallardo Diego I.. Bartlett and Bartlett-type corrections for censored +data from a Weibull distribution SORT - Statistics and Operations Research Transactions. +2020;44:127–140. +7 Cox David R., Snell E. J.. A general definition of residuals Journal of the Royal Statistical +Society. Series B (Methodological). 1968;30:248–275. +8 Magalhães Tiago M., Botter Denise A., Sandoval Mônica C.. A general expression for second- +order covariance matrices - an application to dispersion models Brazilian Journal of Proba- +bility and Statistics. 2020. To appear. +9 Kalbfleisch John D., Prentice Ross L.. The statistical analysis of failure time data. +New +Jersey: John Wiley & Sons2 ed. 2002. +10 Wald Abraham. Test of statistical hypotheses concerning several parameter when the number +of observations is large Transactions of the American Mathematical Society. 1943;54:426–482. +11 Firth David. Bias reduction of maximum likelihood estimates Biometrika. 1993;80:27–38. + +19 +12 Olive Kenneth P, Jacobetz Michael A, Davidson Christian J, et al. Inhibition of Hedgehog +signaling enhances delivery of chemotherapy in a mouse model of pancreatic cancer Science. +2009;324:1457–1461. +13 Bornkamp Björn, Pinheiro José, Bretz Frank, others . MCPMod: An R package for the design +and analysis of dose-finding studies Journal of Statistical Software. 2009;29:1–23. +14 Cordeiro Gaus M., Colosimo Enrico A.. Improved likelihood ratio tests for exponential cen- +sored data Journal of Statistical Computation and Simulation. 1997;56:303–315. +15 Cordeiro Gaus M., Colosimo Enrico A.. Corrected score tests for exponential censored data +Statistics & Probability Letters. 1999;44:365–373. +16 Medeiros Francisco M. C., Lemonte Artur J.. Likelihood-based inference in censored exponen- +tial regression models Communications in Statistics - Theory and Methods. 2021;50:3214– +3233. +17 Lemonte Artur J.. Covariance matrix of maximum likelihood estimators in censored exponen- +tial regression models Communications in Statistics - Theory and Methods. 2022;51:1765– +1777. + +Improved inference for MCP-Mod approach for +time-to-event endpoints with small sample sizes - +Supplementary material +Márcio A. Diniz∗ +Diego I. Gallardo† +Tiago M. Magalhães‡ +July 2, 2022 +We performed a simulation study with 10,000 Monte Carlo replicates where censored Weibull +data was generated with the regression structure established in (3). We considered scenarios +with p = 3, 5, 7 covariates following the standard normal distribution associated with the first p +components of the regression coefficient vector β = (−2, 1.5, −1, 2.5, −1.3, 1.8, −0.5). Censoring +was assumed as 10%, 25% and 50%. Bias and RMSE for MLE, Firth and BCE estimators are +presented in Supplemental Figures S1-S6. +Furthermore, we calculated matrix distances between the sampling covariance matrix - +Cov(β∗) - and the Fisher Information - K(β∗) - and the corrected covariance matrix Cov2(β∗) +from 13 evaluated at MLE (�β) and BCE (�β) in Table S1. The following distances between +matrices A and B were considered: +• d1 = max diag|A − B|; +• d2 = +� +tr((A − B)⊤(A − B)); +• d3 = � +i +� +j |aij − bij|; +Finally, we evaluated type I error testing the composite null hypothesis H : β1 = β(0) +1 +against a composite alternative hypothesis A : H is false, where β(0) +1 +is a specified vector, +β1 is a q-dimensional vector and β2 contains the remaining p − q parameters. This partition +∗Biostatistics and Bioinformatics Research Center, Samuel Oschin Cancer Center, Cedars Sinai Medical +Center, Los Angeles, California, USA, e-mail to marcio.diniz@cshs.org. +†Department of Mathematics, Engineering School, University of Atacama, Copiapó, Chile, e-mail to +diego.gallardo@uda.cl. +‡Department of Statistics, Institute of Exact Sciences, Federal University of Juiz de Fora, Juiz de Fora, +Brazil, e-mail to tiago.magalhaes@ufjf.br. +arXiv:2301.00325v1 [stat.ME] 1 Jan 2023 + +2 +β = (β⊤ +1 , β⊤ +2 )⊤ induces the corresponding partitions +K = +� +� +� +K11 +K12 +K21 +K22 +� +� +� = σ−2 +� +� +� +X⊤ +1 W X1 +X⊤ +1 W X2 +X⊤ +2 W X1 +X⊤ +2 W X2 +� +� +� and K−1 = +� +� +� +K11 +K12 +K21 +K22 +� +� +� , +where X = [X1 X2], X1, X2 being n × q and n × (p − q), respectively. We consider the +following Wald test[1] statistics: +WMLE = +� +�β1 − β(0) +1 +�⊤ � +� +K11�−1 � +�β1 − β(0) +1 +� +, +(1) +WMLE2 = +� +�β1 − β(0) +1 +�⊤ � +�Cov11 +2 ( �β) +�−1 � +�β1 − β(0) +1 +� +, +(2) +WBCE = +� +�β1 − β(0) +1 +�⊤ � +� +K11�−1 � +�β1 − β(0) +1 +� +, +(3) +WBCE2 = +� +�β1 − β(0) +1 +�⊤ � +�Cov11 +2 ( �β) +�−1 � +�β1 − β(0) +1 +� +, +(4) +WFirth = +� +ˇβ1 − β(0) +1 +�⊤ � ˇ +K11�−1 � +ˇβ1 − β(0) +1 +� +. +(5) +In the censored Weibull regression model the statistic (1)-(5) can be rewritten as +W = +� +β1 − β(0) +1 +�⊤ � +R⊤W R +� � +β1 − β(0) +1 +� +, +(6) +with R = X1 − X2C, C = +� +X⊤ +2 W X2 +�−1 X⊤ +2 W X1 represents a (p − q) × q matrix whose +columns are the vectors of regression coefficients obtained in the weighted normal linear regres- +sion of the columns of X1 on the model matrix X2 with W as a weight matrix. +Under the null hypothesis H, W has a χ2 +q distribution up to an error of order n−1. The null +hypothesis is rejected for a given nominal level, α say, if the test statistic exceeds the upper +100(1 − α)% quantile of the χ2 +q distribution. +Similarly, we evaluated power with the composite alternative hypothesis A : β⊤ = (ψ1q, 0p−q)⊤ +for ψ = 0.05, 0.10, 0.25, 0.50, 1.00, 2.00. Results are presented in Table S3. +1 +Results +1.1 +Assessing the bias for different estimation procedures + +3 +p = 3 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.06 +0.08 +0.10 +0.12 +0.14 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.06 +0.08 +0.10 +0.12 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.06 +0.08 +0.10 +0.12 +Sample size +Bias β2 +Method +MLE +BCE +Firth +p = 5 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.06 +0.09 +0.12 +0.15 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.06 +0.09 +0.12 +0.15 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Sample size +Bias β3 +Method +MLE +BCE +Firth +p = 7 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.08 +0.10 +0.12 +0.14 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.08 +0.10 +0.12 +0.14 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.08 +0.10 +0.12 +Sample size +Bias β6 +Method +MLE +BCE +Firth +Figure S1: Empirical bias for different estimators. Case σ = 0.5. + +4 +p = 3 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.075 +0.100 +0.125 +0.150 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.075 +0.100 +0.125 +0.150 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.075 +0.100 +0.125 +0.150 +Sample size +RMSE β2 +Method +MLE +BCE +Firth +p = 5 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.08 +0.12 +0.16 +0.20 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.08 +0.12 +0.16 +0.20 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Sample size +RMSE β3 +Method +MLE +BCE +Firth +p = 7 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.100 +0.125 +0.150 +0.175 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.100 +0.125 +0.150 +0.175 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.100 +0.125 +0.150 +Sample size +RMSE β6 +Method +MLE +BCE +Firth +Figure S2: Empirical RMSE for different estimators. Case σ = 0.5. + +5 +p = 3 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.12 +0.16 +0.20 +0.24 +0.28 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.12 +0.16 +0.20 +0.24 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.12 +0.16 +0.20 +0.24 +Sample size +Bias β2 +Method +MLE +BCE +Firth +p = 5 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +Sample size +Bias β3 +Method +MLE +BCE +Firth +p = 7 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.2 +0.3 +0.4 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.2 +0.3 +0.4 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.2 +0.3 +Sample size +Bias β6 +Method +MLE +BCE +Firth +Figure S3: Empirical bias for different estimators. Case σ = 1. + +6 +p = 3 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +Sample size +RMSE β2 +Method +MLE +BCE +Firth +p = 5 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +0.35 +Sample size +RMSE β3 +Method +MLE +BCE +Firth +p = 7 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.2 +0.3 +0.4 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.2 +0.3 +0.4 +0.5 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Sample size +RMSE β6 +Method +MLE +BCE +Firth +Figure S4: Empirical RMSE for different estimators. Case σ = 1. + +7 +p = 3 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.5 +0.6 +0.7 +0.8 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.5 +0.6 +0.7 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.5 +0.6 +0.7 +Sample size +Bias β2 +Method +MLE +BCE +Firth +p = 5 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.6 +0.8 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.6 +0.8 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.5 +0.6 +0.7 +0.8 +Sample size +Bias β3 +Method +MLE +BCE +Firth +p = 7 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.6 +0.8 +1.0 +Sample size +Bias β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.6 +0.8 +1.0 +Sample size +Bias β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Sample size +Bias β6 +Method +MLE +BCE +Firth +Figure S5: Empirical bias for different estimators. Case σ = 3. + +8 +p = 3 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.5 +0.6 +0.7 +0.8 +0.9 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.4 +0.5 +0.6 +0.7 +0.8 +Sample size +RMSE β2 +Method +MLE +BCE +Firth +p = 5 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.6 +0.8 +1.0 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.6 +0.8 +1.0 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.6 +0.8 +1.0 +Sample size +RMSE β3 +Method +MLE +BCE +Firth +p = 7 +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.6 +0.8 +1.0 +1.2 +Sample size +RMSE β0 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.6 +0.8 +1.0 +1.2 +1.4 +Sample size +RMSE β1 +Method +MLE +BCE +Firth +10% +25% +50% +20 +30 +40 20 +30 +40 20 +30 +40 +0.6 +0.8 +1.0 +Sample size +RMSE β6 +Method +MLE +BCE +Firth +Figure S6: Empirical RMSE for different estimators. Case σ = 3. + +9 +1.2 +Assessing the variance estimation +Table S1: Distance between different estimated covariance matrices. +n = 20 +n = 30 +n = 40 +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +c +σ +p d +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +10% 0.5 +3 d1 0.0634 0.0262 0.0593 0.0175 0.0431 0.0275 0.0403 0.0243 0.0320 0.0151 0.0297 0.0116 +d2 0.0070 0.0010 0.0059 0.0005 0.0026 0.0012 0.0022 0.0009 0.0017 0.0003 0.0015 0.0003 +d3 0.0129 0.0025 0.0115 0.0014 0.0053 0.0029 0.0044 0.0022 0.0033 0.0009 0.0031 0.0006 +5 d1 0.1046 0.0657 0.0988 0.0640 0.0616 0.0338 0.0588 0.0323 0.0485 0.0278 0.0468 0.0280 +d2 0.0232 0.0095 0.0207 0.0088 0.0085 0.0031 0.0078 0.0030 0.0051 0.0023 0.0048 0.0023 +d3 0.0642 0.0343 0.0590 0.0328 0.0257 0.0132 0.0241 0.0127 0.0168 0.0100 0.0161 0.0097 +7 d1 0.2691 0.2520 0.2694 0.2545 0.1790 0.1684 0.1783 0.1689 0.1470 0.1397 0.1463 0.1398 +d2 0.1799 0.1562 0.1786 0.1581 0.0867 0.0777 0.0863 0.0783 0.0583 0.0537 0.0579 0.0538 +d3 0.6681 0.6066 0.6620 0.6084 0.3999 0.3755 0.3982 0.3763 0.3020 0.2888 0.3004 0.2884 +1.0 +3 d1 0.1319 0.0680 0.1138 0.0419 0.0781 0.0555 0.0677 0.0488 0.0694 0.0342 0.0625 0.0265 +d2 0.0280 0.0070 0.0208 0.0028 0.0099 0.0059 0.0073 0.0037 0.0071 0.0016 0.0060 0.0012 +d3 0.0517 0.0194 0.0396 0.0061 0.0213 0.0160 0.0151 0.0094 0.0131 0.0038 0.0120 0.0025 +5 d1 0.1639 0.0630 0.1442 0.0494 0.1065 0.0512 0.0952 0.0441 0.0781 0.0452 0.0710 0.0402 +d2 0.0572 0.0103 0.0434 0.0064 0.0237 0.0059 0.0189 0.0044 0.0120 0.0031 0.0099 0.0024 +d3 0.1472 0.0461 0.1138 0.0258 0.0656 0.0233 0.0542 0.0173 0.0323 0.0114 0.0280 0.0091 +7 d1 0.2433 0.1433 0.2192 0.1368 0.1609 0.1026 0.1514 0.1032 0.1268 0.0893 0.1201 0.0882 +d2 0.1487 0.0496 0.1214 0.0459 0.0651 0.0274 0.0571 0.0267 0.0399 0.0205 0.0359 0.0200 +d3 0.4508 0.2061 0.3819 0.1819 0.2397 0.1385 0.2151 0.1325 0.1642 0.1096 0.1521 0.1068 +3.0 +3 d1 0.4335 0.2503 0.3222 0.2257 0.2526 0.1844 0.1633 0.1480 0.2072 0.0949 0.1570 0.0826 +d2 0.2772 0.0751 0.1308 0.0639 0.0996 0.0495 0.0446 0.0364 0.0695 0.0159 0.0388 0.0079 +d3 0.4908 0.1454 0.2683 0.1309 0.1865 0.0984 0.1088 0.0866 0.1349 0.0411 0.0744 0.0167 +5 d1 0.5378 0.1597 0.4069 0.1557 0.3359 0.1497 0.2678 0.1372 0.2472 0.1339 0.1933 0.1250 +d2 0.6043 0.1040 0.3160 0.0591 0.2407 0.0490 0.1397 0.0287 0.1232 0.0372 0.0726 0.0235 +d3 1.6158 0.4748 0.8530 0.2428 0.6192 0.2058 0.3797 0.0970 0.3509 0.1551 0.2081 0.0857 +7 d1 0.6234 0.2563 0.4373 0.2420 0.3905 0.1814 0.2912 0.1542 0.2880 0.1344 0.2240 0.1162 +d2 0.9863 0.2202 0.4766 0.1692 0.3766 0.0933 0.2042 0.0617 0.1996 0.0546 0.1183 0.0401 + +10 +n = 20 +n = 30 +n = 40 +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +c +σ +p d +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +d3 3.3965 1.1968 1.9409 0.9714 1.2864 0.5180 0.7472 0.3326 0.7210 0.3036 0.4882 0.2401 +25% 0.5 +3 d1 0.0793 0.0369 0.0715 0.0276 0.0525 0.0256 0.0492 0.0226 0.0402 0.0098 0.0380 0.0134 +d2 0.0099 0.0022 0.0081 0.0014 0.0043 0.0011 0.0037 0.0008 0.0027 0.0003 0.0024 0.0004 +d3 0.0188 0.0054 0.0163 0.0037 0.0084 0.0027 0.0073 0.0020 0.0056 0.0009 0.0053 0.0010 +5 d1 0.1023 0.0246 0.0916 0.0227 0.0686 0.0288 0.0634 0.0277 0.0466 0.0159 0.0437 0.0141 +d2 0.0225 0.0023 0.0182 0.0021 0.0099 0.0022 0.0085 0.0019 0.0048 0.0008 0.0042 0.0007 +d3 0.0570 0.0108 0.0483 0.0093 0.0259 0.0097 0.0219 0.0084 0.0138 0.0029 0.0127 0.0028 +7 d1 0.2113 0.1726 0.2025 0.1731 0.1410 0.1185 0.1371 0.1180 0.1114 0.0967 0.1093 0.0965 +d2 0.1134 0.0757 0.1049 0.0757 0.0483 0.0345 0.0456 0.0341 0.0287 0.0217 0.0274 0.0214 +d3 0.4040 0.3062 0.3807 0.3053 0.1978 0.1613 0.1886 0.1585 0.1283 0.1086 0.1237 0.1067 +1.0 +3 d1 0.1674 0.0780 0.1431 0.0586 0.1119 0.0528 0.0982 0.0428 0.0757 0.0426 0.0680 0.0366 +d2 0.0434 0.0077 0.0312 0.0051 0.0197 0.0041 0.0151 0.0034 0.0089 0.0030 0.0069 0.0023 +d3 0.0842 0.0176 0.0673 0.0135 0.0378 0.0108 0.0312 0.0092 0.0189 0.0076 0.0160 0.0059 +5 d1 0.2079 0.0774 0.1749 0.0671 0.1493 0.0739 0.1369 0.0766 0.0958 0.0508 0.0851 0.0456 +d2 0.0885 0.0114 0.0631 0.0082 0.0457 0.0107 0.0366 0.0112 0.0183 0.0038 0.0145 0.0033 +d3 0.2280 0.0446 0.1818 0.0321 0.1186 0.0418 0.0972 0.0396 0.0517 0.0143 0.0451 0.0130 +7 d1 0.2791 0.1399 0.2393 0.1451 0.1852 0.1011 0.1676 0.1047 0.1350 0.0755 0.1247 0.0788 +d2 0.1993 0.0447 0.1473 0.0494 0.0853 0.0259 0.0689 0.0273 0.0446 0.0148 0.0370 0.0152 +d3 0.6023 0.2093 0.4748 0.2045 0.2781 0.1079 0.2396 0.1138 0.1589 0.0749 0.1406 0.0743 +3.0 +3 d1 0.4998 0.2301 0.3203 0.1692 0.3732 0.1706 0.2704 0.1684 0.2375 0.1330 0.1685 0.1164 +d2 0.4127 0.0684 0.1597 0.0329 0.2179 0.0321 0.1133 0.0328 0.0918 0.0263 0.0434 0.0219 +d3 0.7448 0.1563 0.3189 0.0655 0.3926 0.0630 0.2255 0.0641 0.1965 0.0726 0.1000 0.0556 +5 d1 0.6530 0.1756 0.4516 0.1994 0.4249 0.1260 0.3083 0.1536 0.2991 0.1105 0.2281 0.0834 +d2 0.9047 0.0906 0.4160 0.0909 0.3938 0.0506 0.2079 0.0514 0.1868 0.0277 0.0967 0.0190 +d3 2.3326 0.3686 1.3273 0.3487 1.0084 0.2307 0.5947 0.2009 0.4849 0.1211 0.2822 0.0791 +7 d1 0.8010 0.3253 0.5387 0.2408 0.5254 0.2072 0.3900 0.2308 0.3497 0.0975 0.2608 0.1008 +d2 1.5787 0.2777 0.6635 0.1951 0.6367 0.1091 0.3239 0.1048 0.3118 0.0512 0.1607 0.0418 +d3 5.2917 1.5483 2.7006 1.1317 2.1360 0.6376 1.2488 0.5553 1.0500 0.2858 0.6471 0.2486 +50% 0.5 +3 d1 0.1158 0.0483 0.1039 0.0485 0.0776 0.0370 0.0702 0.0343 0.0540 0.0204 0.0496 0.0183 + +11 +n = 20 +n = 30 +n = 40 +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +Cov(�β) and +c +σ +p d +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +�K +Cov2(�β) +d2 0.0228 0.0036 0.0182 0.0033 0.0096 0.0018 0.0081 0.0015 0.0047 0.0006 0.0039 0.0005 +d3 0.0422 0.0083 0.0348 0.0078 0.0174 0.0035 0.0148 0.0030 0.0091 0.0018 0.0075 0.0015 +5 d1 0.1607 0.0742 0.1458 0.0761 0.0990 0.0480 0.0913 0.0451 0.0686 0.0262 0.0637 0.0244 +d2 0.0562 0.0110 0.0462 0.0118 0.0211 0.0046 0.0178 0.0040 0.0099 0.0012 0.0085 0.0011 +d3 0.1446 0.0376 0.1217 0.0388 0.0548 0.0170 0.0475 0.0152 0.0257 0.0051 0.0218 0.0046 +7 d1 0.2964 0.2236 0.2852 0.2309 0.1937 0.1616 0.1862 0.1586 0.1552 0.1358 0.1508 0.1334 +d2 0.2162 0.1162 0.2001 0.1267 0.0896 0.0606 0.0825 0.0584 0.0571 0.0430 0.0536 0.0415 +d3 0.7277 0.4459 0.6687 0.4468 0.3187 0.2389 0.2954 0.2289 0.2305 0.1906 0.2185 0.1842 +1.0 +3 d1 0.2294 0.0855 0.1953 0.0953 0.1541 0.0762 0.1304 0.0728 0.1015 0.0218 0.0860 0.0189 +d2 0.0880 0.0133 0.0607 0.0150 0.0372 0.0085 0.0269 0.0076 0.0174 0.0020 0.0124 0.0018 +d3 0.1646 0.0348 0.1166 0.0382 0.0721 0.0187 0.0541 0.0169 0.0338 0.0044 0.0246 0.0040 +5 d1 0.3220 0.1557 0.2783 0.1709 0.1877 0.0797 0.1645 0.0756 0.1337 0.0480 0.1157 0.0444 +d2 0.2137 0.0411 0.1573 0.0531 0.0739 0.0113 0.0549 0.0107 0.0375 0.0047 0.0283 0.0042 +d3 0.5431 0.1338 0.4213 0.1656 0.1826 0.0381 0.1429 0.0364 0.0977 0.0205 0.0748 0.0181 +7 d1 0.4181 0.1732 0.3656 0.2374 0.2503 0.1409 0.2212 0.1384 0.1831 0.1091 0.1651 0.1053 +d2 0.4305 0.0704 0.3286 0.1362 0.1573 0.0470 0.1218 0.0469 0.0859 0.0297 0.0685 0.0276 +d3 1.4277 0.4023 1.0993 0.5507 0.4759 0.1787 0.3801 0.1720 0.2791 0.1271 0.2325 0.1182 +3.0 +3 d1 0.6912 0.3287 0.4826 0.3279 0.4278 0.1813 0.2997 0.1798 0.3416 0.1817 0.2595 0.1766 +d2 0.7694 0.1459 0.3561 0.1466 0.2957 0.0417 0.1362 0.0399 0.1864 0.0504 0.1004 0.0463 +d3 1.3934 0.3504 0.6465 0.3234 0.5378 0.1021 0.2642 0.0922 0.3608 0.1235 0.2068 0.1108 +5 d1 0.9271 0.4391 0.6588 0.4635 0.5571 0.2677 0.4015 0.2622 0.3910 0.1621 0.2830 0.1558 +d2 1.8814 0.3978 0.9394 0.4618 0.6351 0.1231 0.3188 0.1175 0.3187 0.0517 0.1633 0.0478 +d3 4.6010 1.4400 2.3371 1.4081 1.6206 0.4609 0.8959 0.4286 0.8370 0.2154 0.4862 0.1886 +7 d1 1.1551 0.2748 0.7808 0.5265 0.6514 0.2569 0.4563 0.2763 0.4654 0.2025 0.3365 0.2006 +d2 3.4455 0.5081 1.5612 0.7768 1.0745 0.2111 0.5275 0.2168 0.5367 0.1118 0.2744 0.1042 +d3 12.2895 3.0940 6.5278 4.0423 3.7371 1.1778 2.1761 1.1681 1.8224 0.6151 1.0733 0.5555 +1.3 +Assessing the type I error + +12 +Table S2: Simulated rejection rates for H0 : βq = 0q, with different combinations of σ, censoring, +n, p and q. +σ +censoring +n +p +q +MLE +MLE2 +BCE +BCE2 +Firth +0.5 +10% +20 +3 +1 +0.0723 +0.0463 +0.0723 +0.0590 +0.0722 +2 +0.0874 +0.0474 +0.0877 +0.0627 +0.0869 +5 +1 +0.0794 +0.0415 +0.0796 +0.0580 +0.0779 +3 +0.1131 +0.0518 +0.1132 +0.0729 +0.1121 +7 +2 +0.1223 +0.0531 +0.1226 +0.0791 +0.1190 +4 +0.1495 +0.0580 +0.1492 +0.0853 +0.1460 +30 +3 +1 +0.0636 +0.0422 +0.0638 +0.0522 +0.0639 +2 +0.0770 +0.0473 +0.0769 +0.0605 +0.0767 +5 +1 +0.0806 +0.0462 +0.0806 +0.0626 +0.0793 +3 +0.0917 +0.0461 +0.0919 +0.0636 +0.0919 +7 +2 +0.0952 +0.0442 +0.0956 +0.0656 +0.0939 +4 +0.1179 +0.0503 +0.1182 +0.0733 +0.1153 +40 +3 +1 +0.0620 +0.0460 +0.0621 +0.0539 +0.0621 +2 +0.0684 +0.0471 +0.0684 +0.0570 +0.0687 +5 +1 +0.0703 +0.0430 +0.0708 +0.0561 +0.0704 +3 +0.0836 +0.0465 +0.0836 +0.0625 +0.0830 +7 +2 +0.0888 +0.0439 +0.0888 +0.0638 +0.0875 +4 +0.0993 +0.0471 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+0.9876 +0.9759 +0.9876 +0.9839 +0.9876 + +29 +References +[1] Wald Abraham. Test of statistical hypotheses concerning several parameter when the +number of observations is large Transactions of the American Mathematical Society. +1943;54:426–482. + diff --git a/D9AyT4oBgHgl3EQfevgJ/content/tmp_files/load_file.txt b/D9AyT4oBgHgl3EQfevgJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4b8ce9a6f1aa1b40ff3537b2bb4ec032d284cf2 --- /dev/null +++ b/D9AyT4oBgHgl3EQfevgJ/content/tmp_files/load_file.txt @@ -0,0 +1,4502 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf,len=4501 +page_content='Improved inference for MCP-Mod approach for time-to-event endpoints with small sample sizes Márcio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Diniz∗ Diego I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Gallardo† Tiago M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Magalhães‡ Created: July 7, 2020, updated: July 2, 2022 Abstract The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) frame- work has been recently approved by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Food and Administration and European Medicines Agency as fit-per-purpose for phase II studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' We performed two simulation studies to eval- uate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Keywords: MCP-Mod approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' small sample size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Weibull model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' bias correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' co- variance refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' MSC 2010: Primary 62Fxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' secondary 62F12 ∗Biostatistics and Bioinformatics Research Center, Samuel Oschin Comprehensive Cancer Center, Cedars Sinai Medical Center, Los Angeles, California, USA, e-mail to marcio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='diniz@cshs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' †Department of Mathematics, Engineering School, University of Atacama, Copiapó, Chile, e-mail to diego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='gallardo@uda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' ‡Department of Statistics, Institute of Exact Sciences, Federal University of Juiz de Fora, Juiz de Fora, Brazil, e-mail to tiago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='magalhaes@ufjf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00325v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='ME] 1 Jan 2023 2 1 Introduction Adequate designs for early-phase trials is essential to a successful clinical drug development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Traditionally, investigators evaluate safety in phase I trials, proof-of-concept (PoC) in phase IIa trials, and efficacy in phase IIb trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' When drugs are promising in these early stages, phase III trials are designed accruing a large number of patients to provide a definitive evidence of efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Nonetheless, Jardim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='1 showed that 20% of 80 cancer drug programs submitted between 2009 to 2015 to the Food and Drug Administration (FDA) did not have any data for phase II trials such that 31% obtained FDA approval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 46% had positive PoC with of 76% FDA- approval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' and 34% had negative PoC with only 15% of FDA-approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Therefore, attaining proof-of-concept is predictor to a successful phase III trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Bretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 proposed a framework named Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) unifying phase IIa and IIb trials into a seamless design while taking advantage of both traditional approaches for normally distributed endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Later, Pinheiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 extended this methodology to general parametric models using generalized least squares estimation allowing statisticians to consider more complex designs and other types of endpoints such as binary and time to event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Regulatory agencies have stated their approval of the MCP- Mod framework as an adequate and efficient methodology for design and analysis of phase II dose-finding studies that will guide dose selection for phase III trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The MCP-Mod framework is implemented in two steps: (i) MCP-step consists of a trend test to assess the presence of a dose response signal among a set of pre-specified candidate models while preserving FWER;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (ii) Mod-step corresponds to estimate dose-response curves in order to identify the optimal dose that achieves a desired level of response in comparison to placebo among the models that were selected in the MCP-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Therefore, the properties of the trend test defined as a Wald statistic used in the MCP-step and the maximum likelihood (ML) estimators with their covariance matrix used in the Mod-step are essential to the success- ful implementation of MCP-Mod approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' However, both steps rely on the the asymptotic properties of the ML estimators, which are only valid for large sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In cancer mouse studies, time-to-death is an endpoint that could be used to guide the identi- fication of the minimum effective dose (MED) with large expected effect sizes and small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Nonetheless, the application of the MCP-Mod framework is limited due the underly- 3 ing asymptotic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In this context, parametric survival models such as the censored Weibull regression model (WRM) was showed to be useful given that provides clinical mean- ingful interpretation based on event time ratios or hazard ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 Moreover, the asymptotic properties of the ML estimators can be studied and refined for small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Recently, Magalhães et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 obtained the skewness coefficient of the distribution of the maximum likeli- hood estimators for WRM, and Magalhães et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 derived improved test statistics for LR, score and gradient tests but not for the Wald statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Our main goal is to derive improved ML estimators for the regression parameters and its second-order covariance matrix for WRM, then use them as input to the generalized least squares procedure proposed by Pinheiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 yielding a type I error probability closer to the nominal value when testing proof-of-activity and more accurate MED estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Moreover, we particularize the results from Cox and Snell7 and Magalhães et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 that are very general to WRM and they can be in much broader context than the MCP-Mod framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The remaining paper is organized as follows: in Section 2, we revisit the Weibull distribution and its properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' in Section 3, we review the main concepts of the MCP-Mod framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' in Section 4, we introduce the improved estimators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' in Section 5, we conducted a simulation study to evalute the use of improved estimators in the MCP-Mod framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' finally, in Section 6, we presented some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 2 Weibull distribution The Weibull distribution is commonly used to analysis of time-to-event or lifetime data and a continuous random variable T is called Weibull, if its probability density function (pdf) is f(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' λ, σ) = 1 σλ1/σ t1/σ−1 exp � − (t/λ)1/σ� , t > 0, (1) where σ > 0 is the shape parameter and λ > 0 is the scale parameter, it says T ∼ WE(λ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Two particular models under this parametrization are obtained for σ = 1 and σ = 1/2, which represents the exponential and the Rayleigh models with means λ and λ � π/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In this work, we focused on those models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' However, if σ is unknown, we assume that it can be 4 replaced by consistent estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The ρ% survival time is given by tρ = λ[− log(ρ)]σ (2) The regression structure can be incorporated in (1) by making log(λi) = x⊤ i β (3) where β is a p-vector of unknown parameters and xi is a vector of predictors related to the ith observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In lifetime data, there is the censoring restriction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='e, if T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , Tn are a random sample from (1), instead of Ti, we observe, under right censoring, ti = min(Ti, Li), where Li is the censoring time, independent of Ti, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Here, we consider an hybrid censoring scheme, where the study is finalized when a pre-fixed number r ≤ n out of n observations have failed, as well as when a prefixed time, say L1 = · · · = Ln = L, has been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The type I censoring is a particular case for r = n and the type II censoring appears when L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , Ln = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Additionally, we add the non-informative censoring assumption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=', the random variables Li does not depend on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Usually, the regression modeling considers the distribution of Yi = log(Ti) instead of Ti, which is an accelerated lifetime model form, see Kalbfleisch and Prentice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='9 The distribution of Yi is of the extreme value form with pdf given by f(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' xi) = 1 σ exp �yi − µi σ − exp �yi − µi σ �� , −∞ < yi < ∞, (4) where µi = log λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' From this moment, we assume that σ is known, Then, the log-likelihood function derived from (4) is given by ℓ(β) = n � i=1 � δi � −n log σ + yi − µi σ � − exp �yi − µi σ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The total score function and the total Fisher information matrix for β are, respectively, U β = σ−1X⊤W 1/2v and K = Kββ = σ−2X⊤W X, where X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , xn)⊤, the model ma- trix, assuming rank(X) = p, W = diag(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , wn), wi = E � exp � yi−µi σ �� and v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , vn)⊤, vi = � −δi + exp � yi−µi σ �� w−1/2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' It can observed that the value of wi depends on the mechanism 5 of censoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' That means wi = q × � 1 − exp � −L1/σ i exp(−µi/σ) �� + (1 − q) × (r/n), where q = P � W(r) ≤ log Li � and W(r) denotes the rth order statistic from W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Note that q = 1 and q = 0 for types I and II censoring, respectively, as showed in Magalhães et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 The maximum likelihood estimator (MLE) of β, �β, is the solution of U β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The �β can not be expressed in closed-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' It is typically obtained by numerically maximizing the log- likelihood function using a Newton or quasi-Newton nonlinear optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Under mild regularity conditions and in large samples, �β ∼ Np � β, K−1 ββ � , (5) approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The classic Wald test10 statistic is WMLE = � C�β − Cβ(0)�⊤ � C � KC⊤�−1 � C�β − Cβ(0)� , (6) where C is a matrix of contrasts m × p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Under the null hypothesis H : Cβ = Cβ(0), WMLE has a χ2 p distribution up to an error of order n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The null hypothesis is rejected for a given nominal level, α say, if the test statistic exceeds the upper 100(1 − α)% quantile of the χ2 p distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 3 MCP-Mod General approach We briefly summarize the two-stage procedure discussed by Pinheiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 We consider that time-to-event responses tij for doses xi given to jth subject for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , p and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , ni can be described by the Weibull distribution with scale parameters λi and shape parameter σ defined in (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Then, we consider the parameter µ as our response for the dose-response model such that it could be defined as the median survival time (2) for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 or, alternatively, log(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Initially, a set of candidate dose-response models µi = f(xi, θ) is considered such that each model can be rewritten as function of standardized model as below 6 fm(x, θ) = θ0 + θ1f 0 m(x, θ0) (7) for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In this work, we consider (M = 5) standardized models: (i) linear: f 0(x, θ0) = x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (ii) emax : f 0(x, θ0) = x/(x + ED50) where ED50 can be interpreted as the dose that produces the desired response on 50% of subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (iii) exponential: f 0(x, θ0) = exp{x/δ}− 1, where δ is the exponential rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (iv) logistic: f 0(x, θ0) = 1/(1 + exp{(ED50 − x)/δ}) (v) beta: f 0(x, θ0) = β(δ1, δ2)(x/scal)δ1(1 − x/scal)δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='1 MCP-step In this step, a set of contrasts corresponding to the candidate models will be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' For each candidate model, an optimal contrast copt that maximizes the probability of rejecting the hypothesis of non-signal dose-response is derived assuming that the candidate model is correct and guess estimates for θ0: copt ∝ S−1 � µ0 m − µ0 m ⊤S−11 1S−11⊤ � , (8) where µ0 m = (f 0 m(x1, θ0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , f 0 m(xp, θ0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Then, the test of hypotheses for proof-of-concept can be translated to H0 : copt m µ = 0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' H1 : copt m µ > 0 for candidate model m based on the Wald test statistic W (m) = (copt m ˆµ)⊤ � Copt ˆSCopt⊤�−1 m,m copt m ˆµ (9) where Copt is the matrix m × p of optimal contrast with [A]m,m denoting the mth diagonal element of matrix A for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Critical values for tests are derived based on the joint distribution for W = (W (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , W (M)) allowing one to calculate multiplicity adjusted p-values controlling the FWER at a prespecificed nominal type I error α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 Mod-step In this step, the estimation of non-linear dose responses models is performed in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In the first stage, the parameters µ = [µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , µp] are estimated using standard software packages with analysis of variance (ANOVA) parametrization for the design matrix resulting into a separate parameter µi for each dose level xi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Let ˆµ denote the estimated dose-response parameter vector under ANOVA parametrization, such that we assume that ˆµ follows approximately Np(µ, S), where S is the covariance matrix of ˆµ that is estimated by ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In particular for µ = log(λ), we have µ = β and S = K1 ββ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In the second stage, the non-linear dose-response model f(x, θ) is fitted by minimizing the generalized linear squares (GLS) criterion: Ψ(θ) = (ˆµ − f(x, θ))′ ˆS −1(ˆµ − f(x, θ)) (10) with respect to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Then, the minimum effective dose can be estimated as � MED = {x|f(x, ˆθ) > f(0, ˆθ) + ∆}, where ∆ is a clinical meaningful threshold and ˆθ minimizes (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 4 Improved inference 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='1 Bias correction Inferences based on maximum likelihood method depend strongly on asymptotic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Among these properties, the MLE is approximately non-biased, in other words, E( �β − β) = O(n−1), which is essential to define the mean of the normal distribution of �β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Therefore, likelihood inferences based on asymptotic approximation may not be reliable when sample sizes are small or moderate, and two approaches are available to correct the MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The corrective approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The bias of the MLE can be written as E( �β − β) = B(β) + O(n−2), where B(β) is a term of order O(n−1), a function of the derivatives of the log-likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Cox and Snell7 proposed a bias-corrected maximum likelihood estimator (BCE), that can be expressed as �β = �β − B( �β), where B(β) is the term of order n−1, evaluated in �β and E(�λ − λ) = O(n−2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=', less biased then MLE of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The Cox and Snell’s method is known as 8 a corrective approach because the MLE is calculated and then, the bias correction is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' For the censored Weibull regression model, the expression of B( �β) has the form B( �β) = − 1 2σ3P Zd (W + 2σW ′) 1, (11) where P = K−1X⊤, Z = XK−1X⊤, Zd is a diagonal matrix with diagonal given by the diagonal of Z, W ′ = diag(w′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , w′ n), w′ i = −σ−1L1/σ i exp{−L1/σ i exp(−µi/σ) − µi/σ} and 1 is a n-dimensional vector of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The preventive approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' As alternative to the corrective approach, Firth11 proposed the following modification in the score vector: U ⋆ β = Uβ − KββB(β), (12) where B(β) is given by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The estimator ˇβ, solution of U ⋆ β = 0, has a bias of order O(n−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' This is a preventive approach because the procedure already computes a less biased estimator than the regular MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 Covariance correction From the general result of Magalhães et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=',8 we derived the specific matrix expression for the MLE and BCE second-order covariance matrices for the censored Weibull regression model and it is given by Covτ 2(β⋆) = K−1 + K−1 � ∆ + ∆⊤� K−1 + O(n−3), (13) where ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5∆(1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25∆(2) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5τ2∆(3) with ∆(1) = 1 σ4X⊤W ⋆ZdX, ∆(2) = − 1 σ6X⊤ � W Z(2)W − 2σW Z(2)W ′ − 6σ2W ′Z(2)W ′� X, ∆(3) = 1 σ5X⊤W ′W ⋆⋆X, W ⋆ = diag(w⋆ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , w⋆ n), w⋆ i = wi(wi − 2) − 2σw′ i + στ1(w′ i + 2σw′′ i ), Z(2) = Z ⊙ Z, with ⊙ 9 representing a direct product of matrices (Hadamard product), W ⋆⋆ is a diagonal matrix, with Z(W +2σW ′)Zd1 as its diagonal, W ′′ = diag(w′′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' , w′′ n), w′′ i = −σ−1w′ i � L1/σ i exp(−µi/σ) − 1 � , τ = (τ1, τ2) = (1, 1) indicating the second-order covariance matrix of the MLE β⋆ = �β denoted by Cov2( �β) and τ = (0, −1) indicating the second-order covariance matrix of the BCE β⋆ = �β denoted by Cov2( �β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 Wald-type test Let Cov−1 2 ( �β) and Cov−1 2 ( �β) the inverse of Cov2( �β) and Cov2( �β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' respectively and con- sidering also the partitions and the notation for the Fisher information matrix discussed in the introductory section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' we can propose two modification to the Wald test in (6): WMLE2 = � C �β − Cβ(0)�⊤ � C �Cov2( �β)C⊤�−1 � C �β − Cβ(0)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (14) WBCE = � C �β − Cβ(0)�⊤ � C � KC⊤�−1 � C �β − Cβ(0)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (15) WBCE2 = � C �β − Cβ(0)�⊤ � C �Cov2( �β)C⊤�−1 � C �β − Cβ(0)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (16) WFirth = � C ˇβ − Cβ(0)�⊤ � C ˇ KC⊤�−1 � C ˇβ − Cβ(0)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (17) where �Cov2( �β) is the matrix Cov2(β) evaluated at �β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' � K is the Fisher information evaluated at �β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' �Cov2( �β) is the matrix Cov2( �β) evaluated at �β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' ˇ K is the Fisher information evaluated at ˇβ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Under H, WMLE, WBCE, WBCE2, WFirth follow a χ2 p distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 Improved Estimator strategies In the supplemental material and the next section, we studied the statistical properties of improved estimators for β and its covariance matrix with the following strategies: the classi- cal ML estimator with the Fisher information as covariance matrix (MLE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' the classical ML estimator with the corrected covariance matrix defined in (13) (MLE2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' the bias corrected esti- mator (BCE) given in (11) with the Fisher information as covariance matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' the bias corrected estimator with the corrected covariance matrix (BCE2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' and the Firth estimator defined in (12) with its Fisher information as covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 10 5 Simulation study In our collaborative work, investigators wanted to establish a dose-response relationship between a new inhibitor agent for pancreatic cancer in combination with a given dose of gem- citabine in mouse models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Based on preliminary data, a survival median time of 4 months was estimated in control-treated KPC mice model such that previous studies showed no survival benefit with only gemcitabine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 Assuming a Weibull distribution with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5, we calculated the placebo effect equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='57 and maximum effect of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='96 considering a hazard ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Investigators were interested in the minimum effective dose yielding a minimum hazard ratio of 2 yielding ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Model Constraints Guess estimates/True parameters True MED θ0 θ1 θ2 Linear E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='569 δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0139 Emax 50% at x4 E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='569 EMax = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='079 ED50 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='000 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 Exponential 10% at x1 E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='569 E1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='017 δ = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='756 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='51 Logistic 10% at x3 80% at x4 E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='569 EMax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='391 ED50 = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='329 δ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='976 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='37 Beta 30% at x2 E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='569 EMax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='386 δ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='749 δ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='049 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='61 Table 1: Scenarios (Emax, Exponential, Logistic and Beta) and candidate models (Linear, Emax, Exponential, Logistic and Beta) defined based on, respectively, true parameters and guess estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' True parameters/guess estimates were calculated based on placebo effect of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='57, maximum effect of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='96 and constraints with scale parameter of 120 for Beta model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' True MED was calculated based on ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Five scenarios (constant, emax, exponential, logistic and beta model) presented in Table 1 were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' True parameters were defined based on the aforementioned placebo and maxi- mum effects, and the percent of maximum effect that is achieved at givens dose as discussed in Bornkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='13 For each scenario, the doses 0, 5, 25, 50 and 100 (mg/kg) were considered such that true MED was calculated as continuous dose given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Five candidate mod- els (linear, emax, exponential, logistic and beta model) were considered with guess estimates defined as the true parameters to calculate the optimal contrasts in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' For each model, the null hypothesis of non-signal of the new inhibitor agent was tested at 5% significance level, and we used Akaike Information Criteria (AIC) to choose the model to estimate MED when more than one model rejected the non-signal hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 11 For both steps of the MCP-Mod framework, the five strategies (MLE, MLE2, BCE, BCE2 and Firth) were evaluated based on a Monte Carlo simulation study with 100,000 replicates for sample sizes of 5, 10, 15, 20 and 25 mice per dose and censoring rate of 10%, 25% and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The following operating characteristics were studied: (i) convergence rate of GLS algorithm in the MCP-Mod General framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (ii) Probability of incorrectly detecting a dose-response signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=', type I error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (iii) Probability of correctly detecting a dose-response signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=', power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (iv) Probability of selecting the true model given that there is a signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (v) Bias of MED estimate and (vi) RMSE of MED estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='1 Results In Figure 1, convergence rates when calculating estimators and applying them as input for the MCP-Mod framework is presented for different censoring rates and true models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' When censoring is 10%, there is no difference among estimators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' when censoring is 25%, similar con- clusion can be drawn but the estimators Firth, MLE2 and BCE2 have lower convergence rates than MLE and BCE for the true model Exponential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' when censoring is 50%, the lower conver- gence rates of the estimators Firth, MLE2 and BCE2 are also observed in other scenarios with dose-relationship signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' These lower convergence rates are attained because small sample sizes in combination with high censoring rates result into doses with no events (in our case, deaths), therefore, very large estimates for the regression coefficients are obtained and, consequently, singular covariance matrix estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' For the MCP-step, type I error probability is displayed for different censoring rates and true models in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' When censoring is 10%, corrected estimators (MLE2, BCE, BCE2, Firth) show observed type I error probability closer to the nominal value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='05) than MLE for all sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In particular, MLE shows type I error probability more than twice than the nominal value for sample size of 5 and almost twice the nominal value for 10 mice per dose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' when censoring is 25%, corrected estimators are slightly conservative for small sample sizes of 5 and 10, while type I error probability of MLE is still inflated for all sample sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' when censoring is 50%, corrected estimators are overly conservative for all sample sizes and MLE produces the nominal type I probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' As sample size increases, it is expected that type I error probability converges to the nominal value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='05) for all estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 12 In Figure 3A, the probability of correctly detecting dose-response signal in the MCP-step is showed as function of censoring rates and true models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' It is expected that MLE show higher probability than corrected estimators given that the power function is inflated as seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' A fair comparison would require us to re-adjust critical values to reject the null hypothesis such that the observed type I probability was set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='05 for all estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Nonetheless, the probability of correctly detecting dose-response signal is similar among estimators when censoring rate is 10% because of the large effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' When censoring rate is 25%, all estimators are still comparable except when the true model is Exponential with BCE showing a similar or superior performance than MLE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' BCE2 having a higher probability than MLE for sample sizes from 10 to 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Firth estimator displaying comparable performance with MLE only when sample size is at least 15 mice per dose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' and MLE2 having the poorest performance for all sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' For 50% of censoring, there is no clear pattern, but MLE presents the highest probability when the sample size is 5 mice per dose in three out of four scenarios, while corrected estimators have poorer performance that becomes comparable with MLE as sample size increases, except MLE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In Figure 3B, the probability of correctly selecting dose-response model using AIC is cal- culated given that we selected at least one model in the MCP-step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=', it is a conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Conclusions are somewhat similar to 3A: when censoring is 10%, MLE shows a slight higher probability for Exponential, Logistic and Beta as true models such that this pat- tern becomes more prominent for censoring rates of 25% and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' For Emax as true model, BCE and BCE2 show consistently higher performance when censoring rates are 25% and 50% and no differences can be seen for censoring rate of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' For Mod-step, we calculated the relative bias and RMSE of � MED estimator in Figure 4A and B, respectively, for different censoring rates and true models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' When censoring is 10%, MLE and corrected estimators have negligible differences for bias and RMSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' when censoring is 25%, discrepancies can be see for bias and RMSE when true models are Emax and Exponential such that MLE consistently shows a better performance than corrected estimators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' when censoring is 50%, conclusions are hard to be drawn for sample sizes of 5 and 10 such that estimators perform similar - but MLE2 - for sample sizes greater than 15 mice per dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 13 6 Concluding Remarks We have derived improved inferences based on the Wald statistic for WRM particularizing general results from Cox and Snell7 and Magalhães et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=',8 which complements previous results for improved inference based on likelihood ratio, Rao score and gradient statistics discussed in Magalhães and Gallardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 Few authors have presented improved inference for survival models with small sample sizes under the classical approach: Cordeiro and Colosimo14,15 and Medeiros16 derived those statistics, respectively, for censored exponential regression models (ERM), a par- ticular case of censored WRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Also for ERM, Lemonte17 presented the second-order covariance matrix of the MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' We have also proposed to use bias-corrected (BCE, BCE2 and Firth) and second-order covariance matrices (MLE2, BCE2) as input for the general MCP-Mod framework introduced by Pinheiros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=',3 which addresses the issue of relying on the asymptotic properties of MLE, which might not be valid for small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' To the best of our knowledge, this work is the first attempt to apply refined estimators for small sample sizes in the general MCP- Mod framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Two simulations studies were performed to study the properties of refined estimators in an usual context of regression models and relevant operating characteristics in the MCP-Mod framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In the simulation study presented in the Supplementary material, we showed numerical evidences that BCE and Firth estimator have lower bias and RMSE than MLE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' second-order covariance matrices evaluated at MLE and BCE are closer to their respective empirical covari- ance matrices in comparison to the first-order covariance matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' and Wald statistics derived from the combination between bias-corrected estimators and second-order covariance matrices yielded type I error probability closer to the nominal value than the standard Wald statistic with no loss of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In the simulation study for the MCP-Mod framework, we have found that refined estimators and second-order covariance matrices approximate type I error probability of its nominal value in the MCP-step, while there are negligible differences between MLE and refined estimators in the probability of correctly detecting the dose-response signal, probability correctly selecting the dose-response model, bias and RMSE when censoring rates are up to 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' For censoring rate of 50%, we found convergence issues for the corrected estimators and second-order covariance 14 matrices and poorer performance in the assessed operating characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In conclusion, we recommend the use of refined estimators for small sample sizes when the expected censoring rate is at most 25% in the MCP-Mod framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In the context of basic science with limited sample sizes and large effect sizes, we do not expect large censoring rates in mouse experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In human trials, smaller effect sizes are pursued such that larger sample sizes are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' In this case, refined estimators are not needed because all estimators present comparable performance for large sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Nonetheless, we showed that type I error probability is still inflated even for sample sizes of 25 subjects per dose, therefore, the use of refined estimators would avoid to dedicate further efforts on non-promising drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' We hope that refined estimators allow statisticians to implement the MCP-Mod framework with small sample sizes accelerating the pre-clinical and clinical drug development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Similar ideas can be applied to other distributions such as Binomial, Negative Binomial and Poisson, and they are currently under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 15 10% 25% 50% Constant Emax Exponential Logistic Beta 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 Sample size per dose MCP Convergence Rate Method MLE MLE2 BCE BCE2 Firth Figure 1: Convergence rate when calculating MLE, MLE2, BCE, BCE2 and Firth estimators and applying them as input for the MCP-Mod framework as function of censoring rate and true model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 10% 25% 50% 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='125 Sample size per dose Type I error probability Method MLE MLE2 BCE BCE2 Firth Figure 2: Type I error probability when in the MCP-step using MLE, MLE2, BCE, BCE2 and Firth estimators as function of censoring rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 16 10% 25% 50% Emax Exponential Logistic Beta 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 Sample size per dose Probability of Correctly Detecting Dose Response Signal A 10% 25% 50% Emax Exponential Logistic Beta 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 Sample size per dose Probability of Correctly Selecting Dose Response Model B Method MLE MLE2 BCE BCE2 Firth Figure 3: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Probability of correctly detecting dose-response signal when in the MCP-step using MLE, MLE2, BCE, BCE2 and Firth estimators as function of censoring rate and true model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Probability of correctly selecting dose-response model using AIC in the Mod-step using MLE, MLE2, BCE, BCE2 and Firth estimators as function of censoring rate and true model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 17 10% 25% 50% Emax Exponential Logistic Beta 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 −50% −25% 0% 25% −75% −50% −25% 0% −80% −60% −40% −20% 0% −50% −25% 0% Sample size per dose Relative Bias (%) A 10% 25% 50% Emax Exponential Logistic Beta 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 10 15 20 25 30 0 50 100 150 200 10 20 30 5 10 15 Sample size per dose RMSE B Method MLE MLE2 BCE BCE2 Firth Figure 4: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Bias of � MED derived from the MCP-Mod framework when using MLE, MLE2, BCE, BCE2 and Firth estimators as function of censoring rate and true model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' B.' metadata={'source': 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Covariance matrix of maximum likelihood estimators in censored exponen- tial regression models Communications in Statistics - Theory and Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='51:1765– 1777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Improved inference for MCP-Mod approach for time-to-event endpoints with small sample sizes - Supplementary material Márcio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Diniz∗ Diego I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Gallardo† Tiago M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Magalhães‡ July 2, 2022 We performed a simulation study with 10,000 Monte Carlo replicates where censored Weibull data was generated with the regression structure established in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' We considered scenarios with p = 3, 5, 7 covariates following the standard normal distribution associated with the first p components of the regression coefficient vector β = (−2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5, −1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Censoring was assumed as 10%, 25% and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Bias and RMSE for MLE, Firth and BCE estimators are presented in Supplemental Figures S1-S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Furthermore, we calculated matrix distances between the sampling covariance matrix - Cov(β∗) - and the Fisher Information - K(β∗) - and the corrected covariance matrix Cov2(β∗) from 13 evaluated at MLE (�β) and BCE (�β) in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The following distances between matrices A and B were considered: d1 = max diag|A − B|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' d2 = � tr((A − B)⊤(A − B));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' d3 = � i � j |aij − bij|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Finally, we evaluated type I error testing the composite null hypothesis H : β1 = β(0) 1 against a composite alternative hypothesis A : H is false, where β(0) 1 is a specified vector, β1 is a q-dimensional vector and β2 contains the remaining p − q parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' This partition ∗Biostatistics and Bioinformatics Research Center, Samuel Oschin Cancer Center, Cedars Sinai Medical Center, Los Angeles, California, USA, e-mail to marcio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='diniz@cshs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' †Department of Mathematics, Engineering School, University of Atacama, Copiapó, Chile, e-mail to diego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='gallardo@uda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' ‡Department of Statistics, Institute of Exact Sciences, Federal University of Juiz de Fora, Juiz de Fora, Brazil, e-mail to tiago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='magalhaes@ufjf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00325v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='ME] 1 Jan 2023 2 β = (β⊤ 1 , β⊤ 2 )⊤ induces the corresponding partitions K = � � � K11 K12 K21 K22 � � � = σ−2 � � � X⊤ 1 W X1 X⊤ 1 W X2 X⊤ 2 W X1 X⊤ 2 W X2 � � � and K−1 = � � � K11 K12 K21 K22 � � � , where X = [X1 X2], X1, X2 being n × q and n × (p − q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' We consider the following Wald test[1] statistics: WMLE = � �β1 − β(0) 1 �⊤ � � K11�−1 � �β1 − β(0) 1 � , (1) WMLE2 = � �β1 − β(0) 1 �⊤ � �Cov11 2 ( �β) �−1 � �β1 − β(0) 1 � , (2) WBCE = � �β1 − β(0) 1 �⊤ � � K11�−1 � �β1 − β(0) 1 � , (3) WBCE2 = � �β1 − β(0) 1 �⊤ � �Cov11 2 ( �β) �−1 � �β1 − β(0) 1 � , (4) WFirth = � ˇβ1 − β(0) 1 �⊤ � ˇ K11�−1 � ˇβ1 − β(0) 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' (5) In the censored Weibull regression model the statistic (1)-(5) can be rewritten as W = � β1 − β(0) 1 �⊤ � R⊤W R � � β1 − β(0) 1 � , (6) with R = X1 − X2C, C = � X⊤ 2 W X2 �−1 X⊤ 2 W X1 represents a (p − q) × q matrix whose columns are the vectors of regression coefficients obtained in the weighted normal linear regres- sion of the columns of X1 on the model matrix X2 with W as a weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Under the null hypothesis H, W has a χ2 q distribution up to an error of order n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' The null hypothesis is rejected for a given nominal level, α say, if the test statistic exceeds the upper 100(1 − α)% quantile of the χ2 q distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Similarly, we evaluated power with the composite alternative hypothesis A : β⊤ = (ψ1q, 0p−q)⊤ for ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='50, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Results are presented in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 1 Results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='1 Assessing the bias for different estimation procedures 3 p = 3 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='14 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 Sample size Bias β2 Method MLE BCE Firth p = 5 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='16 Sample size Bias β3 Method MLE BCE Firth p = 7 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='14 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='14 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 Sample size Bias β6 Method MLE BCE Firth Figure S1: Empirical bias for different estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 4 p = 3 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='150 Sample size RMSE β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='150 Sample size RMSE β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='150 Sample size RMSE β2 Method MLE BCE Firth p = 5 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 Sample size RMSE β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 Sample size RMSE β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='200 Sample size RMSE β3 Method MLE BCE Firth p = 7 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='175 Sample size RMSE β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='175 Sample size RMSE β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='150 Sample size RMSE β6 Method MLE BCE Firth Figure S2: Empirical RMSE for different estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 5 p = 3 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='28 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='24 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='24 Sample size Bias β2 Method MLE BCE Firth p = 5 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 Sample size Bias β3 Method MLE BCE Firth p = 7 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 Sample size Bias β6 Method MLE BCE Firth Figure S3: Empirical bias for different estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Case σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 6 p = 3 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 Sample size RMSE β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 Sample size RMSE β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 Sample size RMSE β2 Method MLE BCE Firth p = 5 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='40 Sample size RMSE β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='40 Sample size RMSE β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='35 Sample size RMSE β3 Method MLE BCE Firth p = 7 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 Sample size RMSE β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 Sample size RMSE β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='45 Sample size RMSE β6 Method MLE BCE Firth Figure S4: Empirical RMSE for different estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Case σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 7 p = 3 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='7 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='7 Sample size Bias β2 Method MLE BCE Firth p = 5 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 Sample size Bias β3 Method MLE BCE Firth p = 7 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 Sample size Bias β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 Sample size Bias β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='9 Sample size Bias β6 Method MLE BCE Firth Figure S5: Empirical bias for different estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Case σ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 8 p = 3 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 Sample size RMSE β2 Method MLE BCE Firth p = 5 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 Sample size RMSE β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 Sample size RMSE β1 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 Sample size RMSE β3 Method MLE BCE Firth p = 7 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 Sample size RMSE β0 Method MLE BCE Firth 10% 25% 50% 20 30 40 20 30 40 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='0 Sample size RMSE β6 Method MLE BCE Firth Figure S6: Empirical RMSE for different estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Case σ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='2 Assessing the variance estimation Table S1: Distance between different estimated covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' n = 20 n = 30 n = 40 Cov(�β) and Cov(�β) and Cov(�β) and Cov(�β) and Cov(�β) and Cov(�β) and c σ p d �K Cov2(�β) �K Cov2(�β) �K Cov2(�β) 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='9876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='9759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='9876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='9839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='9876 29 References [1] Wald Abraham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' Test of statistical hypotheses concerning several parameter when the number of observations is large Transactions of the American Mathematical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content=' 1943;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} +page_content='54:426–482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfevgJ/content/2301.00325v1.pdf'} diff --git a/DNAyT4oBgHgl3EQf4frj/content/tmp_files/2301.00789v1.pdf.txt b/DNAyT4oBgHgl3EQf4frj/content/tmp_files/2301.00789v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3f6957e2aef32258d6960223413e3ae128b9c47 --- /dev/null +++ b/DNAyT4oBgHgl3EQf4frj/content/tmp_files/2301.00789v1.pdf.txt @@ -0,0 +1,1166 @@ +Inertial migration in pressure-driven channel flow: beyond the Segre-Silberberg pinch +Prateek Anand1 and Ganesh Subramanian1, ∗ +1Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India +We examine theoretically the inertial migration of a neutrally buoyant rigid sphere in pressure- +driven channel flow, accounting for its finite size relative to the channel width (the confinement +ratio). For sufficiently large channel Reynolds numbers (Rec), a small but finite confinement ratio +qualitatively alters the inertial lift velocity profiles obtained using a point-particle formulation. Fi- +nite size effects are shown to lead to new equilibria, in addition to the well known Segre-Silberberg +pinch locations. Consequently, a sphere can migrate to either the near-wall Segre-Silberberg equi- +libria, or the new stable equilibria located closer to the channel centerline, depending on Rec and +its initial position. Our findings are in accord with recent experiments and simulations, and have +implications for passive sorting of particles based on size, shape and other physical characteristics, +in microfluidic applications. +Inertia-driven +cross-stream +migration +of +neutrally +buoyant spheres in pipe flow, to an annular location +between the centerline and walls, was first observed by +Segre and Silberberg[1–3], the location termed the Segre- +Silberberg annulus (henceforth, SS-annulus or equilib- +ria). Equilibria arising from inertial lift forces have since +been exploited in a range of microfluidic applications[4– +8]. The first theoretical explanations of the phenomenon +were for pressure-driven channel flow (the plane Poiseuille +profile)[9, 10], and involved determining the inertial lift +on a sphere for Rep, Rec ≪ 1, Rep and Rec being the +particle and channel Reynolds numbers, respectively [11]. +The pair of zero-crossings of the O(Rep) lift profile, sym- +metrically located about the centerline, corresponded to +the SS-equilibria. The calculations were later extended +to Rec ∼ O(1) and larger[12, 13], with the SS equilibria +starting at a location intermediate between the walls and +centerline for Rec ≪ 1, and moving wallward with in- +creasing Rec. An analogous dependence on the Reynolds +number was predicted for the SS-annulus in pipe flow[14], +pointing to the similar physics governing inertial migra- +tion in the two configurations. +Later experiments[15], while confirming the original +observations[1–3], revealed an additional inner annulus, +this being the only equilibrium location beyond a cer- +tain Rec[40]. The calculations above[9, 10, 12, 13] use a +point-particle approximation, and predict only the pair of +SS-equilibria in plane Poiseuille flow, and the SS-annulus +alone in pipe flow[14], regardless of Rec. The inner an- +nulus was therefore speculated to arise from finite-size +effects[15]. +Although initially regarded as a transient +feature[17], recent experiments[18] have confirmed the in- +ner annulus to be a stable equilibrium, leading to the +following migration scenario: all spherical particles fo- +cus onto the SS-annulus at low Rec (Regime A); for Rec +greater than a threshold, particles focus onto either the +SS-annulus or the aforementioned inner annulus depend- +ing on their radial distance from the centerline (Regime +B); particles focus solely onto the inner annulus beyond +a second threshold (Regime C). The threshold Rec’s de- +marcating different regimes are observed to decrease with +increasing λ, λ being the confinement ratio defined as the +ratio of the sphere radius a to channel width H (or pipe +radius). This scenario has been qualitatively confirmed +by simulations[19, 20], although the parameter ranges ex- +plored in the above studies are necessarily restricted. +In this letter, for the first time, we move beyond earlier +point-particle formulations, and theoretically examine in- +ertial migration in plane Poiseuille flow for small but fi- +nite λ, with Rec = VmaxH/ν being arbitrary; Vmax here +is the centerline speed of plane Poiseuille flow, and ν the +kinematic viscosity of the suspending fluid. Rep = λ2Rec +is assumed small, allowing analytical progress based on +a leading order Stokesian approximation. The O(λRep) +finite-size contribution is shown to qualitatively alter the +inertial lift profiles obtained from a point-particle for- +mulation (λ = 0), for large Rec, in a manner consistent +with the recent studies above. Our calculations show that +a new pair of stable equilibria, closer to the centerline, +emerges beyond a threshold Rec, even for Rep ≪ 1. We +provide a complete characterization of migration scenar- +ios in the λ−Rec plane for plane Poiseuille flow. +FIG. 1: Neutrally buoyant sphere of radius a in +pressure-driven flow through a channel of width H. +For a neutrally buoyant rigid sphere in plane Poiseuille +flow, at a non-dimensional distance of λ−1s from the +lower wall (see Fig.1), use of the generalized reciprocal +theorem leads to the following expression for the inertial +arXiv:2301.00789v1 [physics.flu-dyn] 2 Jan 2023 + +12 +d +H +s = +H +a +r +d +r32 +lift velocity[21]: +Vp(s) = −Rep +� +V F +V P +� +uSt · +� +ustr · ∇U ∞ + U ∞ · ∇ustr +� +dV + λRep +� +− +� +V ∞ ui · +� +us,i · ∇us,i +� +dV − +� +V ∞ +� +ui · +� +us,i · ∇u∞ ++ u∞ · ∇us,i +� +− uSt,i · +� +ustr,i · ∇U ∞ + U ∞ · ∇ustr,i +�� +dV + +� +V P uSt,i · +� +ustr,i · ∇U ∞ + U ∞ · ∇ustr,i +� +dV +� +, +(1) +for Rep small but finite, Vp in being scaled with Vmaxλ. +The first integral in (1) is the point-particle contribu- +tion examined earlier[9, 10, 12, 13], the domain of in- +tegration (V F + V P ) being the total volume contained +between the channel walls. +The dominant contribu- +tion to this integral comes from scales of O(H), whence +the finite sphere size may be neglected. +Thus, uSt +and ustr in the integrand are, respectively, the veloc- +ity fields due to a Stokeslet and a stresslet confined +between the channel walls. +The Stokeslet is oriented +perpendicular to the walls, while the stresslet is pro- +portional to the rate of strain tensor, associated with +the plane Poiseuille flow, evaluated at the sphere loca- +tion. This tensor is 1 +2β(1112 + 1211), with the Poiseuille +flow given by U ∞ = (βr2 + γλr2 +2)11 in a reference +frame moving with the fluid velocity at the sphere cen- +ter; here, 11 is a unit vector along the flow direction, r2 +the gradient coordinate relative to the sphere center, and +β = 4(1 − 2s) and γ = −4 denote the shear rate and cur- +vature of the plane Poiseuille profile. The dependence of +the point-particle contribution on H amounts to an Rec- +dependence in non-dimensional terms, so the first term +in (1) is of the form RepF1(s, Rec), with F1 determined +semi-analytically for Rec ≪ 1[9, 10, 16], and numerically +for Rec ≳ O(1)[12, 13, 16]. +The integrals within square brackets, in (1), are the +O(λRep) contributions. +The dominant contributions +to the first two integrals arise from scales of O(a), so +the channel walls may be neglected, the integration be- +ing over an unbounded fluid domain (V ∞) outside the +sphere. Thus, ui is the Stokesian velocity disturbance +due to a sphere translating under a constant force nor- +mal to the walls, and us,i is the Stokesian disturbance +due to a force-free torque-free sphere in an ambient +plane Poiseuille flow, both in an unbounded fluid do- +main; u∞ = (βr2 + λγr2 +2 − λγ/3)11 is the Poiseuille +flow in a reference frame translating with the sphere, +and differs from U ∞ above since the sphere transla- +tion speed includes a contribution (λγ/3) from the profile +curvature[21, 22]. uSt,i and ustr,i denote the unbounded- +domain Stokeslet and the stresslet, respectively, the for- +mer given by the Oseen-Burgers tensor[22]; they differ +from uSt and ustr above in not including the wall-image +contributions [22, 23]. The third integral within brackets +corrects for the inclusion of the sphere volume (V P ) in +the domain of integration of the point-particle integral. +The irrelevance of H for the finite-size integrals im- +plies that the expression within square brackets, in (1), +is only a function of s. Further, the simple domain of +integration (V ∞ or V P ) leads to this s-dependence being +evaluable in closed form[21], and (1) reduces to: +Vp(s) = Rep +� +F1(s, Rec) + λ1141(1 − 2s) +216 +� +, +(2) +F1(s, Rec) in (2) can be computed numerically for any +Rec using a shooting method[12, 13, 16]. +In Fig. 2a, +the resulting (scaled) lift profiles are shown, for different +Rec’s, over the lower half-channel with s ∈ [0, 0.5] (due +to anti-symmetry about the centerline). In addition to +the wallward movement of the lone zero-crossing (the SS- +equilibrium), the magnitude of the lift at any fixed loca- +tion, not in the neighbourhood of the wall[24], decreases +sharply with increasing Rec[13]. This reflects the weak- +ened particle-wall interaction when the walls recede be- +yond the inertial screening length of O(HRe +− 1 +2 +c +), owing +to a more rapid decay of the disturbance velocity field +at these distances. Apart from the overall decrease in +magnitude, the shape of the profile also changes, with +an intermediate concave-downward portion emerging for +Rec ≳ 296. An analogous scenario prevails for pipe flow, +although the lift is smaller than that for channel flow at +the same Rec[14, 18]. +The changes in the point-particle contribution above +imply that the finite-size term in (2), although O(λ) +smaller for Rec ≪ 1, becomes comparable for sufficiently +large Rec. This is seen in Fig. 2b which shows the lift +profiles, for λ = 0.01, for the same Rec’s as in Fig. 2a. For +Rec = 50, the lift profile and the SS-equilibrium are only +marginally affected. In contrast, for Rec = Rethreshold +c1 +(≈ 665 for λ = 0.01), while the SS-equilibrium (expect- +edly) has moved closer to the walls, a pair of stable and +unstable equilibria appear between it and the centerline +via a saddle-node bifurcation; the unstable equilibrium +demarcating the basins of attraction of the SS equilib- +rium and the inner stable equilibrium. The bifurcation +arises due to finite-size effects causing the region of neg- +ative curvature, in the point-particle profile, to cross the + +3 +(a) λ = 0 (point-particle) +(b) λ = 0.01 +FIG. 2: Inertial lift profiles in the lower half-channel for Rec ∈(50, 2000): (a) λ = 0 (point-particle); (b) λ = 0.01; the +two insets for λ = 0.01 provide a magnified view of the saddle-node bifurcations at Rethreshold +c1 +≈ 665 and +Rethreshold +c2 +≈ 1500. The black, red and blue symbols denote the SS, unstable and stable equilibria, respectively. The +arrow in (a) shows the movement of the SS equilibria with increasing Rec. +zero-lift line (upper inset in Fig. 2b). As Rec increases to +800, the unstable equilibrium moves towards the SS equi- +librium even as both move wallward, while the inner sta- +ble equilibrium moves towards the centerline. A second +saddle-node bifurcation at Rec = Rethreshold +c2 +(≈ 1500 for +λ = 0.01) leads to the near-center stable equilibrium be- +ing the only one in the half-channel for larger Rec (lower +inset in Fig. 2b). Note that for Rec ∈ (50, 2000) as in +Fig. 2b, and λ = 0.01, Rep ∈ (0.005,0.2), consistent with +the theoretical assumption of weak fluid inertial effects +on scales of O(a). +Fig. 3a plots the equilibrium loci identified above, for +λ = 0.01, as a function of Rec. The SS-branch is seen to +start at s ≈ 0.182 for Rec ≪ 1, moving to smaller s with +increasing Rec. +The inner stable equilibrium emerges +discontinuously at s ≈ 0.18 for Rec ≈ 665, moving to +larger s thereafter (the SS-equilibrium is at s ≈ 0.09 for +this Rec). The loci of the SS and the inner equilibria are +shown as sequences of black and blue dots, respectively, +with the unstable equilibrium locus connecting the two +shown as a sequence of red dots. The fold that devel- +ops in the interval Rec ∈ (665, 1500), bracketed by the +two saddle-node bifurcations, implies a hysteretic behav- +ior in an experiment [29]. A quasi-static protocol of in- +creasing flow rate will lead to spheres remaining at the +SS-equilibrium until Rec ≈ 1500, at which point they +will jump onto the new stable branch closer to the cen- +terline. In contrast, along a path of decreasing flow rate, +spheres will remain at the inner stable equilibrium down +to Rec ≈ 665, before jumping back to the SS-branch. +A behavior analogous to that in Fig. 3a occurs for +λ less than 0.01, with the pair of Rec-thresholds in- +creasing with decreasing λ. +However, the equilibrium +loci undergo a qualitative change as λ increases. +To +see this, note that the SS-equilibrium, in the point- +particle framework, emerges from a balance between an +O(βγ) curvature-induced contribution driving migration +towards higher shear rates (that is, away from the cen- +terline), and an O(β2) wall-induced repulsion. Both con- +tributions arise due to inertial forces acting on scales +of O(H) for Rec ≪ 1[16], and decrease with increasing +Rec. The O(β2) contribution decreases faster, leading to +the wallward movement of the SS-equilibrium. At O(λ), +there arises an additional curvature-induced contribution +on scales of O(a), and that drives migration towards the +centerline [32]. The opposing signs of the point-particle +and finite-size curvature-induced contributions weakens +the wallward movement (with increasing Rec) of the SS- +equilibrium for larger λ. The profound effect of this weak- +ening may be seen from Fig. 3b which shows the equilib- +rium locus for λ = 0.025. The region of multiple equi- +libria is now absent, with the SS-equilibrium smoothly +transitioning from an initial wallward movement, to a +movement towards the centerline, across Rec ≈ 200. +By identifying the equilibrium loci as a function of +Rec, for different λ, a ‘phase diagram’ of migration sce- +narios in the λ − Rec plane may be constructed as in +Fig. 4. The figure highlights the existence of three dis- +tinct regions. +Region +1 +○, corresponding to the area +below the red curve and outside the (gray) shaded re- +gion, contains lift profiles with a single stable off-center +equilibrium in the half-channel. Region 2 +○, correspond- +ing to the shaded region, contains lift profiles with two +stable equilibria, separated by an intervening unstable +one, in the half-channel. The upper and lower bound- +aries of this region are determined by the pair of turning +points on the equilibrium locus, corresponding to saddle- +node bifurcations - these were identified in Fig 3a for +λ = 0.01. The two boundaries end in a cusp for the fold +bifurcation under consideration [33, 34], corresponding to + +Rec=50 +0.2 +Rec=100 +Rec=300 +- Rec=665 +0.1 +Rec=800 +— Rec=1500 + Rec=1700 +0.0 +-0.1 +-0.2 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +S0.0005 +Rec=50 +Rec=660 +0.2 +Rec=670 +- Rec=100 +Vp/Rep +0.0000 +Rec=300 + Rec=665 +0.0005 +0.1 +- Rec=800 +- Rec=1500 +0.0010 +0.16 +0.17 +0.18 +0.19 +0.20 +0.21 +— Rec=2000 +S +0.0 +0.010 +-0.1 +Vp/Rep +0.005 +Rec=1400 +Rec=1700 +0.000 +-0.2 +0.005 +0.06 +80'0 +0.10 +0.12 +0.14 +s +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +s4 +(a) λ = 0.01 +(b) λ = 0.025 +FIG. 3: Inertial equilibrium loci for (a)λ = 0.01 and (b)λ = 0.025; black, blue and red dots denote the SS, and the +inner stable and unstable equilibria, respectively. The region of multiple equilibria in (a), for Rec ∈ (665, 1500), +leads to hysteretic jumps in the equilibrium location marked by the vertical arrows A1 (s = 0.179 → 0.09) and +A2(s = 0.08 → 0.31). Vertical dashed lines in (a) and (b) denote the laminar-turbulent transition. +(λcritical, Recritical +c +) ≡ (0.0216, 296) in Fig. 4; see top right +inset. Along either a vertical or a horizontal line, the lat- +ter corresponding to an experimental path of changing +flow rate, Region 2 +○ mediates a discontinuous transition +from the SS-equilibrium to the inner stable equilibrium +closer to the centerline. Region 3 +○, above the red curve, +includes lift profiles with the centerline as the only sta- +ble equilibrium. +Note that the centerline is always an +equilibrium by symmetry, albeit an unstable one in Re- +gions 1 +○ and 2 +○. Insets in Fig. 4 show lift profiles for +the following (λ, Rec) pairs: (0.3, 5); (b) (0.05, 10); (c) +(0.01, 1000); (d) (0.015, 1500), which are consistent with +the aforementioned features of Regions 1 +○- 3 +○. +The black dot-dashed line in Fig. 4 corresponds to +Rep = 1, and may be regarded as a rough threshold above +which the present theoretical results may no longer be +valid. This implies, for instance, that our results may not +be quantitatively accurate beyond Rec = 100 at λ = 0.1. +Importantly, the region of multiple equilibria and the +associated hysteretic transitions, predicted here for the +first time, lie well within this threshold. A second factor +that limits the observability of Region 2 +○ is the laminar- +turbulence transition. Although plane Poiseuille flow is +predicted to become linearly unstable at Rec = 11544 +[35], experiments show a nonlinear subcritical transition +to turbulence at a much lower Rec ∼ O(2000)[36, 37]. +This subcritical transition is shown as vertical dashed +lines in both Figs. 3 and 4. The emergence of the region +of multiple equilibria in the latter figure clearly occurs +well before the transition threshold. +While the migration pattern for a fixed λ, implied by +Fig. 4, is in qualitative agreement with studies quoted at +the beginning[15, 17–20], the inner annulus in these stud- +ies is observed for higher λ (≳ 0.05) and for Rep ≳ O(1) +- see hatched region in Fig. 4. The absence of multiple +equilibria for smaller λ is very likely due to the develop- +ment length, needed for a steady particle distribution, +being larger than the pipe length used in the experi- +ments. For Rec fixed, this length scales as O(λ−3)[15], +increasing rapidly with decreasing particle size. Notwith- +standing differences between the pipe and channel ge- +ometries, experiments with longer pipes should lead to +the hatched region in Fig. 4 extending down to smaller +λ. There remains the provocative question of how the +secondary finite-Rep region of multiple equilibria, iden- +tified in the said studies, connects to the theoretically +identified small-Rep region (Region 2 +○ in Fig. 4). +FIG. 4: Migration scenarios in the λ − Rec plane. +Inertial lift profiles for canonical (λ, Rec) pairs +(highlighted by black dots) are shown (see text for +details). The hatched region shows parameter ranges +covered in earlier studies [15, 17–20]. +Apart from the fundamental significance of our find- +ings, in terms of enriching the inertial migration land- +scape, and providing an explanation for recent exper- +iments and computations, Fig. 4 may be leveraged to- +wards passive sorting in microfluidic applications. The +simplest scenario pertains to separating spheres of two + +0.4 +Critical Re. for transition +0.3 +to turbulence ～ 2000 +location +A2 +0.1 +Lower wall +(s = 0) +～ 665 +～ 1500 +5 +10 +50 +100 +500 +1000 +Rec0.45F +0.40 +0.35 +cation +0.30 +O1 +Critical Rec for transition +to turbulence ～ 2000 +0.25 +0.20 +Lower wall +(s = 0) +5 +10 +50 +100 +500 +1000 +Rec(a) +(c) +3 +0.500 +d +(b) +0.100 +0.050 +0.0220 +0.0215 +0.010 +0.0210 +2 +(0.02156,296) +0.005 +0.0205 +296 +298 +300 +302 +304 +Rec +1 +10 +100 +1000 +Critical Re.fortransition +Rec +toturbulence20005 +different sizes, corresponding to confinement ratios λ1 +and λ2 (λ2 > λ1). An experimental protocol of changing +flow rate (Rec) for a bi-disperse suspension, with parti- +cles of the aforementioned sizes, would appear as a pair +of horizontal lines in Fig. 4, the upper one correspond- +ing to λ2. With increasing flow rate, separation would +be achieved at an Rec when the point on the λ2-line is +above Region 2 +○ (after crossing it to the right), with the +one on the λ1-line directly below. +At this Rec, larger +spheres would focus onto the pair of near-centerline sta- +ble equilibria, with the smaller ones focusing onto the +near-wall SS-equilibria. If the point on the λ1-line lies +within Region 2 +○, rather than below it, as would be the +case when λ2/λ1 is not far from unity, partial separation +will be achieved owing to smaller spheres focusing onto +both the SS and inner equilibria (the relative fractions +determined by the pair of unstable equilibria). +The implications of the near-centerline stable equilib- +ria found here go well beyond the size-sorting scenario +above. +The dependence of the interval of existence of +these equilibria, on inertial forces in a region of order the +sphere size, implies a generic sensitivity of the finite-size +contributions to the detailed characteristics of the sus- +pended microstructure. +Thus, for anisotropic particles +such as spheroids or ellipsoids, the threshold Reynolds +numbers (Rethreshold +c1 +, Rethreshold +c2 +) that characterize the +region of multiple equilibria are expected to be functions +of the particle aspect ratio(s). +In contrast, the time- +averaged inertial lift for spheroids with order unity aspect +ratios, calculated within a point-particle framework for +plane Poiseuille flow, may be shown to yield SS-equilibria +identical to those for spheres[16]; the spheroid aspect +ratio only affecting the magnitude of the point-particle +lift profile, not the equilibrium locations. +The aspect- +ratio-dependence expected for the near-centerline equi- +libria will therefore be crucial for shape-sorting proto- +cols in microfluidic applications[38]. Along similar lines, +there will be a dependence on the viscosity ratio for +drops, allowing, in principle, for separation of weakly +deformed drops based on both size and viscosity ratio +differences[39]. Analogous remarks apply to other elastic +microstructures such as vesicles, capsules or red blood +cells. Migration phase diagrams for these cases will have +at least one additional axis - this axis could be the appro- +priate shape parameter for anisotropic particles (aspect +ratio for spheroids) or the viscosity ratio for drops. In +the latter case, the degree of deformability, as character- +ized by the Capillary number, offers an additional degree +of freedom. It would be of interest, in future, to quan- +titatively determine these phase diagrams, allowing for +rational design of passive sorting protocols. +∗ sganesh@jncasr.ac.in +[1] Segre, G., & A. Silberberg. Nature 189.4760 (1961): 209- +210. +[2] Segre, G., & Silberberg, A. J. (1962). Journal of fluid me- +chanics, 14(1), 115-135. +[3] Segre, G., & Silberberg, A. (1962). Journal of fluid me- +chanics, 14(1), 136-157. +[4] Di Carlo, D., Irimia, D., Tompkins, R. G., & Toner, M. +(2007). Proceedings of the National Academy of Sciences, +104(48), 18892-18897. +[5] Di Carlo, D. (2009). Lab on a Chip, 9(21), 3038-3046. +[6] Amini, H., Lee, W., & Di Carlo, D. (2014). Lab on a Chip, +14(15), 2739-2761. +[7] Paie, P., Bragheri, F., Di Carlo, D., & Osellame, R. (2017). +Microsystems & nanoengineering, 3(1), 1-8. +[8] Li, M., van Zee, M., Goda, K., & Di Carlo, D. (2018). Lab +on a Chip, 18(17), 2575-2582. +[9] Ho, B. P., & Leal, L. G. (1974). Journal of fluid mechanics, +65(2), 365-400. +[10] Vasseur, P., & Cox, R. G. (1976). Journal of Fluid Me- +chanics, 78(2), 385-413. +[11] Rec will be used here to denote the Reynolds number +based on the relevant macroscopic scale, either the channel +width or the pipe diameter. +[12] Schonberg, J. A., & Hinch, E. J. (1989). Journal of Fluid +Mechanics, 203, 517-524. +[13] Asmolov, E. S. (1999). Journal of fluid mechanics, 381, +63-87. +[14] Matas, J. P., Morris, J. F., & Guazzelli, E. (2009). Jour- +nal of Fluid Mechanics, 621, 59-67. +[15] Matas, J. P., Morris, J. F., & Guazzelli, ´E. (2004). Jour- +nal of fluid mechanics, 515, 171-195. +[16] Anand, P., & Subramanian, G. (2022). To be submitted +to JFM. +[17] Morita, Y., Itano, T., & Sugihara-Seki, M. (2017). Jour- +nal of Fluid Mechanics, 813, 750-767. +[18] Nakayama, S., Yamashita, H., Yabu, T., Itano, T., & +Sugihara-Seki, M. (2019). Journal of Fluid Mechanics, 871, +952-969. +[19] Shao, X., Yu, Z., & Sun, B. (2008). Physics of Fluids, +20(10), 103307. +[20] Pan, T. W., Li, A., & Glowinski, R. (2021). Physics of +Fluids, 33(3), 033301. +[21] Supplemental material. +[22] Kim, S., & Karrila, S. J. (2013). Microhydrodynamics: +principles and selected applications. Courier Corporation. +[23] Leal, L. G. (2007). Advanced transport phenomena: fluid +mechanics and convective transport processes (Vol. 7). +Cambridge University Press. +[24] Within a point-particle framework, the inertial lift veloc- +ity increases to approximately 1.52, independent of Rec, +on approach towards either wall[16]. +[25] Leal, L. G., & Hinch, E. J. (1971). The effect of weak +Brownian rotations on particles in shear flow. Journal of +Fluid Mechanics, 46(4), 685-703. +[26] Dabade, V., Marath, N. K., & Subramanian, G. (2016). +The effect of inertia on the orientation dynamics of +anisotropic particles in simple shear flow. Journal of Fluid +Mechanics, 791, 631-703. +[27] Marath, N. K., & Subramanian, G. (2017). The effect +of inertia on the time period of rotation of an anisotropic +particle in simple shear flow. Journal of Fluid Mechanics, +830, 165-210. +[28] Okagawa, A., Cox, R. G., & Mason, S. G. (1973). The +kinetics of flowing dispersions. VI. Transient orientation + +6 +and rheological phenomena of rods and discs in shear flow. +Journal of Colloid and Interface Science, 45(2), 303-329. +[29] Such a hysteresis will arise only in the absence of stochas- +tic fluctuations, or for short channels. Positional fluctua- +tions arising either from weak Brownian motion [25–27], +or from pair-hydrodynamic interactions[28], will eliminate +hysteretic behavior in sufficiently long channels. +[30] Ho, B. P., & Leal, L. G. (1976). Migration of rigid spheres +in a two-dimensional unidirectional shear flow of a second- +order fluid. Journal of Fluid Mechanics, 76(4), 783-799. +[31] Chan, P. H., & Leal, L. (1979). The motion of a de- +formable drop in a second-order fluid. Journal of fluid me- +chanics, 92(1), 131-170. +[32] For cross-stream migration in plane Poiseuille flow +arising from mechanisms other than inertia such as +viscoelasticity[30] or drop deformation[31], the lift at lead- +ing order arises from this curvature-induced finite size con- +tribution, and drives migration towards the centerline. +[33] Zeeman, E. C. (1976). Catastrophe theory. Scientific +American, 234(4), 65-83. +[34] Dubey, P., Roy, A., & Subramanian, G. (2022). Linear +stability of a rotating liquid column revisited. Journal of +Fluid Mechanics, 933. +[35] Orszag, S. A. (1971). Journal of Fluid Mechanics, 50(4), +689-703. +[36] Carlson, D. R., Widnall, S. E., & Peeters, M. F. (1982). +Journal of Fluid Mechanics, 121, 487-505. +[37] Nishioka, M., & Asai, M. (1985). Journal of Fluid Me- +chanics, 150, 441-450. +[38] Masaeli, M., Sollier, E., Amini, H., Mao, W., Camacho, +K., Doshi, N., ... & Di Carlo, D. (2012). Continuous iner- +tial focusing and separation of particles by shape. Physical +Review X, 2(3), 031017. +[39] The SS-equilibria for drops, within the framework of a +point-particle formulation, may be shown to be identical +to those for rigid spheres. The point-particle disturbance +velocity field depends only on the strength of the induced +stresslet, and as a result, the inertial lift for a spherical +drop may be obtained from that for a rigid sphere by mul- +tiplication by the ratio of the corresponding stresslet co- +efficients. +[40] Wherever appropriate, +Rec +is taken to denote the +Reynolds number based on the pipe diameter and the +mean velocity of the flow through the pipe. + +1 +Supplemental material +Following [S1] and [S2], one may use a generalized reciprocal theorem formulation to derive a formal expression for +the inertial lift velocity of a neutrally buoyant sphere, in an ambient plane Poiseuille flow, for arbitrary Rep. In the +limit Rep ≪ 1, the non-dimensional inertial lift is O(Rep), being given by: +Vp = −Rep +� +V F u · (us · ∇us + us · ∇u∞ + u∞ · ∇us) dV. +(S1) +where Vp is scaled by Vmaxa/H or Vmaxλ. +The actual problem in the reciprocal theorem framework corresponds to the one of interest, that is, a neutrally +buoyant sphere freely moving in wall-bounded plane Poiseuille flow for Rep and λ small but finite, with Rec = λ−2Rep +being arbitrary. For purposes of calculating the inertial lift to O(Rep), the disturbance velocity field in the actual +problem may be replaced by its Stokesian approximation. Thus, us, in the inertial acceleration terms in (S1), is the +Stokesian disturbance field due to a force-free and torque-free sphere translating with Up in an ambient plane Poiseuille +flow. In a reference frame moving with the sphere center, the latter flow is given by u∞ = (α + βr2 + λγr2 +2)11 − Up +where, with the sphere at a distance λ−1s (in units of a) from the lower wall, one has α = 4λ−1s(1 − s), β = 4(1 − 2s) +and γ = −4. Here, α11 −Up and β are the ambient slip and shear rate at the sphere center, the latter varying linearly +across the channel and equalling zero at the centerline (s = 0.5); γ denotes the constant curvature of the Poiseuille +profile. Up is determined using the force-free constraint in Faxen’s law for translation. This leads to Up = (α+λγ/3)11 +and, as a result, the ambient flow in the said reference frame takes the form u∞ = (βr2 + λγr2 +2 − λγ/3)11. The test +problem in the reciprocal theorem framework corresponds to the Stokesian translation of a sphere in an otherwise +quiescent fluid confined between parallel walls (ones that bound the Poiseuille flow in the actual problem), under a +constant force acting along the gradient direction. The test disturbance field u multiplies the inertial acceleration +terms involving us in (S1). +We now examine the length scales that contribute dominantly to the integral in (S1), beginning with the limit Rec ≪ +1, when the inertial screening length (HRe−1/2 +c +) is much larger than the channel width, and therefore, irrelevant. The +dominant contributions to the volume integral in this limit may arise from either scales of O(a) (the inner region) or +those of O(H) (the outer region). In order to assess their relative magnitudes, we consider the intermediate asymptotic +interval 1 ≪ r ≪ λ−1 (r is measured in units of a) where both the finite size of the sphere and wall-induced image +contributions may be neglected at leading order. For r in this interval, u ∼ 1/r corresponding to the farfield Stokeslet, +and us ∼ β/r2 + γλ/r3 corresponding to the farfield stresslet and force-quadrupole contributions associated with the +linear and quadratic ambient flow components, respectively. Using these forms along with u∞ ∼ βr + γλr2, one +obtains the following estimates for parts of the integrand involving the linear and nonlinear components of the inertial +acceleration: +• u · (us · ∇u∞ + u∞ · ∇us) ∼ β2 +r3 + λβγ +r2 ++ λγβ +r4 ++ γ2 +r3 +(linear), +• u · (us · ∇us) ∼ β2 +r6 + λβγ +r7 ++ λ2γ2 +r8 +(nonlinear). +Using dV ∼ O(r2dr), one obtains the following estimates for contributions to the lift velocity integral: +V linear +p +∼ Rep +� r +dr′ +� β2 +r′ + λβγ + λγβ +r′2 + λ2γ2 +r′ +� +, +∼ Rep +� +β2 ln r + λβγr + λγβ +r ++ λ2γ2 ln r +� +, +(S2a) +V non-linear +p +∼ Rep +� r +dr′ +� β2 +r′4 + λβγ +r′5 + λ2γ2 +r′6 +� +, +∼ Rep +� β2 +r3 + λβγ +r4 ++ λ2γ2 +r5 +� +. +(S2b) +The algebraically growing terms in (S2a) will be dominated by scales of O(H), and the resulting contributions to the +lift velocity are obtained by cutting off the divergence (for r → ∞) at r ∼ O(λ−1); this cut-off recognizes the wall- +induced screening of the unbounded-domain behavior that eventually leads to a more rapid decay for r ≫ O(λ−1), +and thence, convergence. The algebraically decaying terms in both (S2a) and (S2b) will be dominated by scales of + +2 +O(a), with the lift velocity contributions now obtained by cutting off the divergence (for r → 0) at r ∼ O(1). The ln r +terms in (S2a) imply the dominance of the intermediate (matching) interval 1 ≪ r ≪ λ−1, leading, in principle, to +contributions of O(β2 ln λ−1) and O(λ2γ2 ln λ−1) to the lift velocity; logarithmically smaller contributions of O(β2) +and O(λ2γ2) must arise from both scales of O(a) (r ∼ O(1)) and O(H) (r ∼ O(λ−1)). However, contributions from +the inner and matching regions turn out to be zero by symmetry. Owing to the absence of walls at leading order, the +O(β2) and O(β2 ln λ−1) inner and matching-region contributions must correspond to the lift on a neutrally buoyant +sphere (or the equivalent point-particle singularity) in an unbounded simple shear flow; likewise, the O(λ2γ2) and +O(λ2γ2 ln λ−1) inner and matching-region contributions must correspond to the lift on a sphere at the origin of an +unbounded quadratic flow. In both these scenarios, however, the two lateral directions are equivalent, and there can +be no lift. That these contributions are zero may also be seen from the fact that one cannot construct a true vector, +the inertial lift velocity, from any quadratic combination of the velocity gradient tensor associated with an ambient +linear flow (simple shear for the present case of an ambient Poiseuille flow), or from any quadratic combination of the +third order tensor that would characterize a generic quadratic flow. Crucially, the O(β2) and O(λ2γ2) outer-region +contributions are not subject to the above symmetry-argument-based limitation. This is due to the importance of +walls at leading order, and the implied availability of an additional vector (the unit normal characterizing the wall +orientations) to construct the lift velocity vector. +Based on the above arguments, one is led to the following lift velocity contributions from the linear and nonlinear +inertial terms in the integrand: +V linear +p +∼ Rep +� +β2(outer) + βγ(outer) + λγβ(inner) + λ2γ2(outer) +� +, +(S3a) +V non-linear +p +∼ Rep λβγ(inner). +(S3b) +From (S3a) and (S3b), the leading order contribution to the lift velocity is seen to come from the linearized inertial +terms, and from scales of O(H), with there being two such contributions: one proportional to β2 that characterizes +wall-induced repulsion in an ambient linear flow, and the other proportional to βγ that denotes the contribution due +to the ambient profile curvature. The dominance of scales of O(H) implies that, for purposes of evaluating the above +contributions, the sphere in both the actual and test problems can be replaced by the corresponding point singularity. +Thus, at leading order, one only need consider the terms in (S1) that are linear in us, and further, us and u may be +approximated as the disturbance fields due to a stresslet (ustr) and Stokeslet (uSt), respectively. Note that uSt and +ustr are a combination of the unbounded domain components and additional wall-image contributions, both of which +are of comparable importance on scales of O(H); detailed expressions are given in [S2]. Thus, the inertial lift velocity +at leading order in Rep and λ reduces to: +Vp = −Rep +� +V F +V P uSt · (ustr · ∇U ∞ + U ∞ · ∇ustr) dV, +(S4) +where the O(λγ) term in u∞ has been neglected, so U ∞ = (βr2 + λγr2 +2)11 in (S4). Further, on account of the +subdominance of scales of O(a), the domain of integration has been extended to include the particle volume (V P ), +with an accompanying change in the integration variable that is now the position vector scaled by H (rather than a as +in (S1)). (S4) is the point-particle approximation for the lift velocity for Rec ≪ 1, and corresponds to a dimensional +lift velocity of O(V 2 +maxλ2a/ν). The use of a rescaled (with λ) integration variable, as mentioned above, leads to the +integral in (S4) being only a function of s; the detailed evaluation of this integral, via a partial Fourier transform, is +discussed in [S2]. +For Rec ≳ O(1), the scaling arguments used above to establish the dominance of the outer region contributions +still hold. The outer region now corresponds to scales of order the inertial screening length or larger, so that the +farfield Stokesian estimate for us, used above to establish outer-region dominance, remains valid only for 1 ≪ r ≪ +λ−1Re +− 1 +2 +c +, with the disturbance velocity field in the actual problem decaying more rapidly for larger r. The outer- +region dominance for Rec ≳ O(1) implies that this disturbance field may still be approximated as being driven by +a stresslet forcing, although the forcing appears in the linearized Navier-Stokes equations. While there still exist +physically distinct contributions arising from profile curvature and wall-induced repulsion, the lift velocity can no +longer be written as an additive superposition of the two. Furthermore, despite the relevance of the inertial screening +length, asymptotically larger scales of O(H) continue to be relevant. The ratio of these two outer scales involves Rec +which appears in the governing linearized equations of motion. Thus, the version of the reciprocal theorem integral +in (S4) for Rec ≳ O(1), with us replaced by its finite-Rec analog, will be a function of both s and Rec. Although +the actual calculation is more easily accomplished via a direct solution of the partially Fourier transformed ODE’s +using a shooting method [S2, S3], one may nevertheless write the point-particle lift velocity contribution formally in +the form (V 2 +maxλ2a/ν)F1(s; Rec). + +3 +The modification of the leading order lift velocity, for λ small but finite, arises from contributions in (S3) that +are of a smaller order in λ than the outer-region contributions included in (S4). The largest such contributions are +proportional to λ(βγ), and pertain to the inner region, arising from both the linear and nonlinear inertial terms +in (S3a) and (S3b). +The βγ-dependence implies that these contributions arise solely due to the coupling of the +shear and curvature of the ambient profile, consistent with earlier symmetry arguments. Further, scales of O(H) are +irrelevant, implying that the λ(βγ) terms will lead to contributions independent of Rec, with the dimensional lift +velocity being of the form (V 2 +maxλ3a/ν)F2(s), as in equation (2) of the main manuscript. A second implication of the +O(λβγ) contributions being from the inner region is that they arise independently of the outer-region point-particle +contribution. +Said differently, the (V 2 +maxλ2a/ν)F1(s; Rec) and (V 2 +maxλ3a/ν)F2(s) contributions to the inertial lift +correspond to the leading order terms of the underlying asymptotic expansions of the integrand in the outer and inner +regions, respectively. This implies that the O(V 2 +maxλ3a/ν)F2(s) contribution is not a correction to the leading order +point-particle result, and thereby, not constrained to be small in comparison. This feature is especially significant +since the emergence of multiple equilibria in the lift profiles (Region 2 +○ in Fig 4 of the main manuscript) is only made +possible by allowing the finite-size contribution to be comparable to the leading point-particle one. +Note that there are other corrections for finite λ: for instance, the O(λ2γ2) outer-region contribution in (S3), the +correction to us arising from λ-dependent corrections to the stresslet coefficient, etc. While both of these may be +shown to be of a smaller order in λ for Rec ≪ 1, they will be far smaller for large Rec owing to an overall weakening +of wall-induced corrections. This weakening arises from the more rapid decay of the disturbance velocity field for +distances larger than O(HRe +− 1 +2 +c +). This Rec-dependence is important since, as explained in the main manuscript, the +finite-size contributions become significant only at large Rec owing to the reduction in the magnitude of the point- +particle contribution with increasing Rec. Clearly, the only finite-size contributions of relevance correspond to the +O(λβγ) terms in (S3a) and (S3b). To calculate these, we return to the original reciprocal theorem result, viz. (S1), +and subtract and add back the leading point-particle contribution given by (S4): +Vp = − Rep +� +V F +V P uSt · +� +ustr · ∇U ∞ + U ∞ · ∇ustr +� +dV +− Rep +�� +V F u · +� +us · ∇us + us · ∇u∞ + u∞ · ∇us +� +dV +− +� +V F +V P uSt · +� +ustr · ∇U ∞ + U ∞ · ∇ustr +� +dV +� +. +(S5) +Splitting the point-particle contribution within brackets into separate integrals over the fluid (V F ) and particle (V P ) +domains, separating the nonlinear and linear inertial terms in the original reciprocal theorem integral, and then +combining the integrals involving the linear inertial terms, one obtains: +Vp = − Rep +� +V F +V P uSt · +� +ustr · ∇U ∞ + U ∞ · ∇ustr +� +dV +− Rep +�� +V F u · +� +us · ∇us +� +dV + +� +V F +� +u · +� +us · ∇u∞ + u∞ · ∇us +� +− uSt · +� +ustr · ∇U ∞ + U ∞ · ∇ustr +�� +dV − +� +V P uSt · +� +ustr · ∇U ∞ ++ U ∞ · ∇ustr +� +dV +� +. +(S6) +The last integral within brackets, over V P , evidently involves scales of O(a), and we therefore focus on the remaining +two integrals. The first integral within brackets contains the nonlinear inertial term, and as already seen in (S2b), the +integrand exhibits an algebraic decay for r ≫ 1. The second integral involves the difference between the linearized +inertial terms, and their approximations based on point-particle representations of the corresponding velocity fields, +as a result of which growing terms cancel out, and only the algebraically decaying one in (S3a) survives. The algebraic +decay implies the dominance of scales of O(a), and therefore, that wall-induced image contributions do not contribute +at leading order. Thus, the leading approximations of all of the integrals within brackets, in (S6), may be obtained by + +4 +replacing the original fluid domain (V F ) by a completely unbounded one (V ∞), and (S6) may be written in the form: +Vp = − Rep +� +V F +V P uSt · +� +ustr · ∇U ∞ + U ∞ · ∇ustr +� +dV. ++ λRep +� +− +� +V ∞ ui · +� +us,i · ∇us,i +� +dV − +� +V ∞ +� +ui · +� +us,i · ∇u∞ + u∞ · ∇us,i +� +− uSt,i · +� +ustr,i · ∇U ∞ + U ∞ · ∇u(1) +str,i +�� +dV + +� +V P uSt,i · +� +ustr,i · ∇U ∞ ++ U ∞ · ∇ustr,i +� +dV +� +, +(S7) +where both the actual velocity fields and their point-particle approximations, that appear in the three bracketed +integrals, are now approximated by simpler expressions pertaining to an unbounded fluid domain; these are indicated +by the additional subscript ‘i’, and are given below:: +ui = 12 +8π · +�I +r + rr +r3 +� ++ 12 +8π · +� I +3r3 − rr +r5 +� +, +(S8a) +us,i = −5 +2 +(E : rr)r +r5 +− +�E · r +r5 +− 5(E : rr)r +2r7 +� ++ λγ +� +− +1 +24r9 +� +3r6 + r4(−23r2 +1 + r2 +2 − 8r2 +3 − 3) + 15r2� +r2 +1(7r2 +2 + 1) + r2 +2 +� +− 105r2 +1r2 +2 +� +11 + +5 +8r9 (r2 − 1)r1r2(3r2 − 7r2 +2)12 + +5 +8r9 (r2 − 1)r1r3(r2 − 7r2 +2)13 +� +, +(S8b) +uSt,i = 12 +8π · +�I +r + rr +r3 +� +, +(S8c) +ustr,i = −5 +2 +(E : rr)r +r5 +. +(S8d) +(S8a) is the disturbance velocity field induced by a translating sphere. (S8b) is the disturbance field due to a force-free +sphere in an ambient flow with both linear and quadratic components: the part involving E (= β +2 (1112 + 1211), the +rate of strain tensor of the ambient Poiseuille flow), constitutes the disturbance in an ambient linear flow, while that +proportional to γ constitutes the response to the quadratic component. As originally shown (but not used) by [S1], +the latter may be obtained using an expansion in spherical harmonics [S4]. (S8c) and (S8d) are the usual velocity +fields for a Stokeslet and the stresslet in an unbounded domain. +Finally, note that the integrals over V P and V ∞ extend down to the origin (the sphere center) with both integrands +including an O(β2) contribution arising from the coupling of the linear ambient flow component with the stresslet field, +that is O(1/r3) for r → 0 (see initial integrand estimates); while this leads to a conditionally convergent behavior, +doing the angular integration first leads to a trivial answer, as must be the case based on symmetry arguments +above. For the integral over V ∞, a similar conditionally convergent behavior arises at infinity from the leading O(γ2) +contribution, that is again resolved by appropriate choice of the order of integration. Even the leading point-particle +integral is conditionally convergent at the origin, an issue (implicitly) addressed by a particular order of integration +in the calculation procedure[S2]. +The volume integrals, within brackets in (S8), are readily calculated analytically: +� +V ∞ ui · +� +us,i · ∇us,i +� +dV =6143 βγ +120960 , +(S9) +� +V ∞ +� +ui· +� +us,i·∇u∞+u∞·∇us,i +� +−uSt,i· +� +ustr,i·∇U ∞+U ∞·∇ustr,i +�� +dV =37 βγ +1260 , +(S10) +� +V P uSt,i · +� +ustr,i · ∇U ∞ + U ∞ · ∇ustr,i +� +dV =− βγ +4 . +(S11) +Substituting (S9-S11) in (S6), with β = 4(1 − 2s) and γ = −4, gives: +Vp = Rep +� +F1(s, Rec) + λF2(s) +� +, +(S12) +with F2(s) = 1141(1−2s) +216 +. + +5 +∗ sganesh@jncasr.ac.in +[S1] Ho, B. P., & Leal, L. G. (1974). Journal of fluid mechanics, 65(2), 365-400. +[S2] Anand, P., & Subramanian, G. (2022). To be submitted to JFM. +[S3] Schonberg, J. A., & Hinch, E. J. (1989). Journal of Fluid Mechanics, 203, 517-524. +[S4] Kim, S., & Karrila, S. J. (2013). Microhydrodynamics: principles and selected applications. Courier Corporation. + diff --git a/DNAyT4oBgHgl3EQf4frj/content/tmp_files/load_file.txt b/DNAyT4oBgHgl3EQf4frj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0ea6561ab7cd81160a323d4d775234fef594e09 --- /dev/null +++ b/DNAyT4oBgHgl3EQf4frj/content/tmp_files/load_file.txt @@ -0,0 +1,578 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf,len=577 +page_content='Inertial migration in pressure-driven channel flow: beyond the Segre-Silberberg pinch Prateek Anand1 and Ganesh Subramanian1, ∗ 1Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India We examine theoretically the inertial migration of a neutrally buoyant rigid sphere in pressure- driven channel flow, accounting for its finite size relative to the channel width (the confinement ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' For sufficiently large channel Reynolds numbers (Rec), a small but finite confinement ratio qualitatively alters the inertial lift velocity profiles obtained using a point-particle formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Fi- nite size effects are shown to lead to new equilibria, in addition to the well known Segre-Silberberg pinch locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Consequently, a sphere can migrate to either the near-wall Segre-Silberberg equi- libria, or the new stable equilibria located closer to the channel centerline, depending on Rec and its initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Our findings are in accord with recent experiments and simulations, and have implications for passive sorting of particles based on size, shape and other physical characteristics, in microfluidic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Inertia-driven cross-stream migration of neutrally buoyant spheres in pipe flow, to an annular location between the centerline and walls, was first observed by Segre and Silberberg[1–3], the location termed the Segre- Silberberg annulus (henceforth, SS-annulus or equilib- ria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Equilibria arising from inertial lift forces have since been exploited in a range of microfluidic applications[4– 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The first theoretical explanations of the phenomenon were for pressure-driven channel flow (the plane Poiseuille profile)[9, 10], and involved determining the inertial lift on a sphere for Rep, Rec ≪ 1, Rep and Rec being the particle and channel Reynolds numbers, respectively [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The pair of zero-crossings of the O(Rep) lift profile, sym- metrically located about the centerline, corresponded to the SS-equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The calculations were later extended to Rec ∼ O(1) and larger[12, 13], with the SS equilibria starting at a location intermediate between the walls and centerline for Rec ≪ 1, and moving wallward with in- creasing Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' An analogous dependence on the Reynolds number was predicted for the SS-annulus in pipe flow[14], pointing to the similar physics governing inertial migra- tion in the two configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Later experiments[15], while confirming the original observations[1–3], revealed an additional inner annulus, this being the only equilibrium location beyond a cer- tain Rec[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The calculations above[9, 10, 12, 13] use a point-particle approximation, and predict only the pair of SS-equilibria in plane Poiseuille flow, and the SS-annulus alone in pipe flow[14], regardless of Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The inner an- nulus was therefore speculated to arise from finite-size effects[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Although initially regarded as a transient feature[17], recent experiments[18] have confirmed the in- ner annulus to be a stable equilibrium, leading to the following migration scenario: all spherical particles fo- cus onto the SS-annulus at low Rec (Regime A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' for Rec greater than a threshold, particles focus onto either the SS-annulus or the aforementioned inner annulus depend- ing on their radial distance from the centerline (Regime B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' particles focus solely onto the inner annulus beyond a second threshold (Regime C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The threshold Rec’s de- marcating different regimes are observed to decrease with increasing λ, λ being the confinement ratio defined as the ratio of the sphere radius a to channel width H (or pipe radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This scenario has been qualitatively confirmed by simulations[19, 20], although the parameter ranges ex- plored in the above studies are necessarily restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In this letter, for the first time, we move beyond earlier point-particle formulations, and theoretically examine in- ertial migration in plane Poiseuille flow for small but fi- nite λ, with Rec = VmaxH/ν being arbitrary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Vmax here is the centerline speed of plane Poiseuille flow, and ν the kinematic viscosity of the suspending fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Rep = λ2Rec is assumed small, allowing analytical progress based on a leading order Stokesian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The O(λRep) finite-size contribution is shown to qualitatively alter the inertial lift profiles obtained from a point-particle for- mulation (λ = 0), for large Rec, in a manner consistent with the recent studies above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Our calculations show that a new pair of stable equilibria, closer to the centerline, emerges beyond a threshold Rec, even for Rep ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' We provide a complete characterization of migration scenar- ios in the λ−Rec plane for plane Poiseuille flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 1: Neutrally buoyant sphere of radius a in pressure-driven flow through a channel of width H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' For a neutrally buoyant rigid sphere in plane Poiseuille flow, at a non-dimensional distance of λ−1s from the lower wall (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1), use of the generalized reciprocal theorem leads to the following expression for the inertial arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='00789v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='flu-dyn] 2 Jan 2023 12 d H s = H a r d r32 lift velocity[21]: Vp(s) = −Rep � V F +V P � uSt · � ustr · ∇U ∞ + U ∞ · ∇ustr � dV + λRep � − � V ∞ ui · � us,i · ∇us,i � dV − � V ∞ � ui · � us,i · ∇u∞ + u∞ · ∇us,i � − uSt,i · � ustr,i · ∇U ∞ + U ∞ · ∇ustr,i �� dV + � V P uSt,i · � ustr,i · ∇U ∞ + U ∞ · ∇ustr,i � dV � , (1) for Rep small but finite, Vp in being scaled with Vmaxλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The first integral in (1) is the point-particle contribu- tion examined earlier[9, 10, 12, 13], the domain of in- tegration (V F + V P ) being the total volume contained between the channel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The dominant contribu- tion to this integral comes from scales of O(H), whence the finite sphere size may be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Thus, uSt and ustr in the integrand are, respectively, the veloc- ity fields due to a Stokeslet and a stresslet confined between the channel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The Stokeslet is oriented perpendicular to the walls, while the stresslet is pro- portional to the rate of strain tensor, associated with the plane Poiseuille flow, evaluated at the sphere loca- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This tensor is 1 2β(1112 + 1211), with the Poiseuille flow given by U ∞ = (βr2 + γλr2 2)11 in a reference frame moving with the fluid velocity at the sphere cen- ter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' here, 11 is a unit vector along the flow direction, r2 the gradient coordinate relative to the sphere center, and β = 4(1 − 2s) and γ = −4 denote the shear rate and cur- vature of the plane Poiseuille profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The dependence of the point-particle contribution on H amounts to an Rec- dependence in non-dimensional terms, so the first term in (1) is of the form RepF1(s, Rec), with F1 determined semi-analytically for Rec ≪ 1[9, 10, 16], and numerically for Rec ≳ O(1)[12, 13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The integrals within square brackets, in (1), are the O(λRep) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The dominant contributions to the first two integrals arise from scales of O(a), so the channel walls may be neglected, the integration be- ing over an unbounded fluid domain (V ∞) outside the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Thus, ui is the Stokesian velocity disturbance due to a sphere translating under a constant force nor- mal to the walls, and us,i is the Stokesian disturbance due to a force-free torque-free sphere in an ambient plane Poiseuille flow, both in an unbounded fluid do- main;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' u∞ = (βr2 + λγr2 2 − λγ/3)11 is the Poiseuille flow in a reference frame translating with the sphere, and differs from U ∞ above since the sphere transla- tion speed includes a contribution (λγ/3) from the profile curvature[21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' uSt,i and ustr,i denote the unbounded- domain Stokeslet and the stresslet, respectively, the for- mer given by the Oseen-Burgers tensor[22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' they differ from uSt and ustr above in not including the wall-image contributions [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The third integral within brackets corrects for the inclusion of the sphere volume (V P ) in the domain of integration of the point-particle integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The irrelevance of H for the finite-size integrals im- plies that the expression within square brackets, in (1), is only a function of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Further, the simple domain of integration (V ∞ or V P ) leads to this s-dependence being evaluable in closed form[21], and (1) reduces to: Vp(s) = Rep � F1(s, Rec) + λ1141(1 − 2s) 216 � , (2) F1(s, Rec) in (2) can be computed numerically for any Rec using a shooting method[12, 13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 2a, the resulting (scaled) lift profiles are shown, for different Rec’s, over the lower half-channel with s ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='5] (due to anti-symmetry about the centerline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In addition to the wallward movement of the lone zero-crossing (the SS- equilibrium), the magnitude of the lift at any fixed loca- tion, not in the neighbourhood of the wall[24], decreases sharply with increasing Rec[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This reflects the weak- ened particle-wall interaction when the walls recede be- yond the inertial screening length of O(HRe − 1 2 c ), owing to a more rapid decay of the disturbance velocity field at these distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Apart from the overall decrease in magnitude, the shape of the profile also changes, with an intermediate concave-downward portion emerging for Rec ≳ 296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' An analogous scenario prevails for pipe flow, although the lift is smaller than that for channel flow at the same Rec[14, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The changes in the point-particle contribution above imply that the finite-size term in (2), although O(λ) smaller for Rec ≪ 1, becomes comparable for sufficiently large Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 2b which shows the lift profiles, for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01, for the same Rec’s as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' For Rec = 50, the lift profile and the SS-equilibrium are only marginally affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In contrast, for Rec = Rethreshold c1 (≈ 665 for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01), while the SS-equilibrium (expect- edly) has moved closer to the walls, a pair of stable and unstable equilibria appear between it and the centerline via a saddle-node bifurcation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' the unstable equilibrium demarcating the basins of attraction of the SS equilib- rium and the inner stable equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The bifurcation arises due to finite-size effects causing the region of neg- ative curvature, in the point-particle profile, to cross the 3 (a) λ = 0 (point-particle) (b) λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 2: Inertial lift profiles in the lower half-channel for Rec ∈(50, 2000): (a) λ = 0 (point-particle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (b) λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' the two insets for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01 provide a magnified view of the saddle-node bifurcations at Rethreshold c1 ≈ 665 and Rethreshold c2 ≈ 1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The black, red and blue symbols denote the SS, unstable and stable equilibria, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The arrow in (a) shows the movement of the SS equilibria with increasing Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' zero-lift line (upper inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' As Rec increases to 800, the unstable equilibrium moves towards the SS equi- librium even as both move wallward, while the inner sta- ble equilibrium moves towards the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' A second saddle-node bifurcation at Rec = Rethreshold c2 (≈ 1500 for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01) leads to the near-center stable equilibrium be- ing the only one in the half-channel for larger Rec (lower inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Note that for Rec ∈ (50, 2000) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 2b, and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01, Rep ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='005,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='2), consistent with the theoretical assumption of weak fluid inertial effects on scales of O(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 3a plots the equilibrium loci identified above, for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01, as a function of Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The SS-branch is seen to start at s ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='182 for Rec ≪ 1, moving to smaller s with increasing Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The inner stable equilibrium emerges discontinuously at s ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='18 for Rec ≈ 665, moving to larger s thereafter (the SS-equilibrium is at s ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='09 for this Rec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The loci of the SS and the inner equilibria are shown as sequences of black and blue dots, respectively, with the unstable equilibrium locus connecting the two shown as a sequence of red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The fold that devel- ops in the interval Rec ∈ (665, 1500), bracketed by the two saddle-node bifurcations, implies a hysteretic behav- ior in an experiment [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' A quasi-static protocol of in- creasing flow rate will lead to spheres remaining at the SS-equilibrium until Rec ≈ 1500, at which point they will jump onto the new stable branch closer to the cen- terline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In contrast, along a path of decreasing flow rate, spheres will remain at the inner stable equilibrium down to Rec ≈ 665, before jumping back to the SS-branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' A behavior analogous to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 3a occurs for λ less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01, with the pair of Rec-thresholds in- creasing with decreasing λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' However, the equilibrium loci undergo a qualitative change as λ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' To see this, note that the SS-equilibrium, in the point- particle framework, emerges from a balance between an O(βγ) curvature-induced contribution driving migration towards higher shear rates (that is, away from the cen- terline), and an O(β2) wall-induced repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Both con- tributions arise due to inertial forces acting on scales of O(H) for Rec ≪ 1[16], and decrease with increasing Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The O(β2) contribution decreases faster, leading to the wallward movement of the SS-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' At O(λ), there arises an additional curvature-induced contribution on scales of O(a), and that drives migration towards the centerline [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The opposing signs of the point-particle and finite-size curvature-induced contributions weakens the wallward movement (with increasing Rec) of the SS- equilibrium for larger λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The profound effect of this weak- ening may be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 3b which shows the equilib- rium locus for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The region of multiple equi- libria is now absent, with the SS-equilibrium smoothly transitioning from an initial wallward movement, to a movement towards the centerline, across Rec ≈ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' By identifying the equilibrium loci as a function of Rec, for different λ, a ‘phase diagram’ of migration sce- narios in the λ − Rec plane may be constructed as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The figure highlights the existence of three dis- tinct regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Region 1 , corresponding to the area below the red curve and outside the (gray) shaded re- gion, contains lift profiles with a single stable off-center equilibrium in the half-channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Region 2 , correspond- ing to the shaded region, contains lift profiles with two stable equilibria, separated by an intervening unstable one, in the half-channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The upper and lower bound- aries of this region are determined by the pair of turning points on the equilibrium locus, corresponding to saddle- node bifurcations - these were identified in Fig 3a for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The two boundaries end in a cusp for the fold bifurcation under consideration [33, 34], corresponding to Rec=50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='2 Rec=100 Rec=300 Rec=665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1 Rec=800 — Rec=1500 Rec=1700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='5 S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0005 Rec=50 Rec=660 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='2 Rec=670 Rec=100 Vp/Rep 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0000 Rec=300 Rec=665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1 Rec=800 Rec=1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='21 — Rec=2000 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1 Vp/Rep 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='005 Rec=1400 Rec=1700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content="06 80'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='14 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='5 s4 (a) λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01 (b) λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='025 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 3: Inertial equilibrium loci for (a)λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01 and (b)λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='025;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' black, blue and red dots denote the SS, and the inner stable and unstable equilibria, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The region of multiple equilibria in (a), for Rec ∈ (665, 1500), leads to hysteretic jumps in the equilibrium location marked by the vertical arrows A1 (s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='179 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='09) and A2(s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='08 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Vertical dashed lines in (a) and (b) denote the laminar-turbulent transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (λcritical, Recritical c ) ≡ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0216, 296) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' see top right inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Along either a vertical or a horizontal line, the lat- ter corresponding to an experimental path of changing flow rate, Region 2 mediates a discontinuous transition from the SS-equilibrium to the inner stable equilibrium closer to the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Region 3 , above the red curve, includes lift profiles with the centerline as the only sta- ble equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Note that the centerline is always an equilibrium by symmetry, albeit an unstable one in Re- gions 1 and 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Insets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4 show lift profiles for the following (λ, Rec) pairs: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='3, 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (b) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='05, 10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='01, 1000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='015, 1500), which are consistent with the aforementioned features of Regions 1 ○- 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The black dot-dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4 corresponds to Rep = 1, and may be regarded as a rough threshold above which the present theoretical results may no longer be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This implies, for instance, that our results may not be quantitatively accurate beyond Rec = 100 at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Importantly, the region of multiple equilibria and the associated hysteretic transitions, predicted here for the first time, lie well within this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' A second factor that limits the observability of Region 2 is the laminar- turbulence transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Although plane Poiseuille flow is predicted to become linearly unstable at Rec = 11544 [35], experiments show a nonlinear subcritical transition to turbulence at a much lower Rec ∼ O(2000)[36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This subcritical transition is shown as vertical dashed lines in both Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The emergence of the region of multiple equilibria in the latter figure clearly occurs well before the transition threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' While the migration pattern for a fixed λ, implied by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4, is in qualitative agreement with studies quoted at the beginning[15, 17–20], the inner annulus in these stud- ies is observed for higher λ (≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='05) and for Rep ≳ O(1) see hatched region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The absence of multiple equilibria for smaller λ is very likely due to the develop- ment length, needed for a steady particle distribution, being larger than the pipe length used in the experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' For Rec fixed, this length scales as O(λ−3)[15], increasing rapidly with decreasing particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Notwith- standing differences between the pipe and channel ge- ometries, experiments with longer pipes should lead to the hatched region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4 extending down to smaller λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' There remains the provocative question of how the secondary finite-Rep region of multiple equilibria, iden- tified in the said studies, connects to the theoretically identified small-Rep region (Region 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4: Migration scenarios in the λ − Rec plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Inertial lift profiles for canonical (λ, Rec) pairs (highlighted by black dots) are shown (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The hatched region shows parameter ranges covered in earlier studies [15, 17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Apart from the fundamental significance of our find- ings, in terms of enriching the inertial migration land- scape, and providing an explanation for recent exper- iments and computations, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4 may be leveraged to- wards passive sorting in microfluidic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The simplest scenario pertains to separating spheres of two 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='4 Critical Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' for transition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='3 to turbulence ～ 2000 location A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='1 Lower wall (s = 0) ～ 665 ～ 1500 5 10 50 100 500 1000 Rec0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='45F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='35 cation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='30 O1 Critical Rec for transition to turbulence ～ 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='20 Lower wall (s = 0) 5 10 50 100 500 1000 Rec(a) (c) 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='500 d (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0210 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='02156,296) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='0205 296 298 300 302 304 Rec 1 10 100 1000 Critical Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='fortransition Rec toturbulence20005 different sizes, corresponding to confinement ratios λ1 and λ2 (λ2 > λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' An experimental protocol of changing flow rate (Rec) for a bi-disperse suspension, with parti- cles of the aforementioned sizes, would appear as a pair of horizontal lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 4, the upper one correspond- ing to λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' With increasing flow rate, separation would be achieved at an Rec when the point on the λ2-line is above Region 2 (after crossing it to the right), with the one on the λ1-line directly below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' At this Rec, larger spheres would focus onto the pair of near-centerline sta- ble equilibria, with the smaller ones focusing onto the near-wall SS-equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' If the point on the λ1-line lies within Region 2 , rather than below it, as would be the case when λ2/λ1 is not far from unity, partial separation will be achieved owing to smaller spheres focusing onto both the SS and inner equilibria (the relative fractions determined by the pair of unstable equilibria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The implications of the near-centerline stable equilib- ria found here go well beyond the size-sorting scenario above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The dependence of the interval of existence of these equilibria, on inertial forces in a region of order the sphere size, implies a generic sensitivity of the finite-size contributions to the detailed characteristics of the sus- pended microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Thus, for anisotropic particles such as spheroids or ellipsoids, the threshold Reynolds numbers (Rethreshold c1 , Rethreshold c2 ) that characterize the region of multiple equilibria are expected to be functions of the particle aspect ratio(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In contrast, the time- averaged inertial lift for spheroids with order unity aspect ratios, calculated within a point-particle framework for plane Poiseuille flow, may be shown to yield SS-equilibria identical to those for spheres[16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' the spheroid aspect ratio only affecting the magnitude of the point-particle lift profile, not the equilibrium locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The aspect- ratio-dependence expected for the near-centerline equi- libria will therefore be crucial for shape-sorting proto- cols in microfluidic applications[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Along similar lines, there will be a dependence on the viscosity ratio for drops, allowing, in principle, for separation of weakly deformed drops based on both size and viscosity ratio differences[39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Analogous remarks apply to other elastic microstructures such as vesicles, capsules or red blood cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Migration phase diagrams for these cases will have at least one additional axis - this axis could be the appro- priate shape parameter for anisotropic particles (aspect ratio for spheroids) or the viscosity ratio for drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In the latter case, the degree of deformability, as character- ized by the Capillary number, offers an additional degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' It would be of interest, in future, to quan- titatively determine these phase diagrams, allowing for rational design of passive sorting protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' ∗ sganesh@jncasr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='in [1] Segre, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Silberberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Nature 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='4760 (1961): 209- 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [2] Segre, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Silberberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [21] Supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [22] Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Karrila, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Microhydrodynamics: principles and selected applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Courier Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [23] Leal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Advanced transport phenomena: fluid mechanics and convective transport processes (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [24] Within a point-particle framework, the inertial lift veloc- ity increases to approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='52, independent of Rec, on approach towards either wall[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [25] Leal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Hinch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The effect of weak Brownian rotations on particles in shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Mechanics, 46(4), 685-703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [26] Dabade, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Marath, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Subramanian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Mechanics, 791, 631-703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [27] Marath, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Subramanian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The effect of inertia on the time period of rotation of an anisotropic particle in simple shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Mechanics, 830, 165-210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [28] Okagawa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Cox, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Mason, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The kinetics of flowing dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Transient orientation 6 and rheological phenomena of rods and discs in shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Colloid and Interface Science, 45(2), 303-329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [29] Such a hysteresis will arise only in the absence of stochas- tic fluctuations, or for short channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Positional fluctua- tions arising either from weak Brownian motion [25–27], or from pair-hydrodynamic interactions[28], will eliminate hysteretic behavior in sufficiently long channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [30] Ho, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Leal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second- order fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Mechanics, 76(4), 783-799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [31] Chan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Leal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The motion of a de- formable drop in a second-order fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of fluid me- chanics, 92(1), 131-170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [32] For cross-stream migration in plane Poiseuille flow arising from mechanisms other than inertia such as viscoelasticity[30] or drop deformation[31], the lift at lead- ing order arises from this curvature-induced finite size con- tribution, and drives migration towards the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [33] Zeeman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Catastrophe theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Scientific American, 234(4), 65-83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [34] Dubey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Subramanian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Linear stability of a rotating liquid column revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Mechanics, 933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [35] Orszag, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Mechanics, 50(4), 689-703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [36] Carlson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Widnall, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Peeters, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Mechanics, 121, 487-505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [37] Nishioka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Asai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Me- chanics, 150, 441-450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [38] Masaeli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Sollier, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Amini, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Mao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Camacho, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', Doshi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' & Di Carlo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Continuous iner- tial focusing and separation of particles by shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Physical Review X, 2(3), 031017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [39] The SS-equilibria for drops, within the framework of a point-particle formulation, may be shown to be identical to those for rigid spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The point-particle disturbance velocity field depends only on the strength of the induced stresslet, and as a result, the inertial lift for a spherical drop may be obtained from that for a rigid sphere by mul- tiplication by the ratio of the corresponding stresslet co- efficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [40] Wherever appropriate, Rec is taken to denote the Reynolds number based on the pipe diameter and the mean velocity of the flow through the pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 1 Supplemental material Following [S1] and [S2], one may use a generalized reciprocal theorem formulation to derive a formal expression for the inertial lift velocity of a neutrally buoyant sphere, in an ambient plane Poiseuille flow, for arbitrary Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In the limit Rep ≪ 1, the non-dimensional inertial lift is O(Rep), being given by: Vp = −Rep � V F u · (us · ∇us + us · ∇u∞ + u∞ · ∇us) dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S1) where Vp is scaled by Vmaxa/H or Vmaxλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The actual problem in the reciprocal theorem framework corresponds to the one of interest, that is, a neutrally buoyant sphere freely moving in wall-bounded plane Poiseuille flow for Rep and λ small but finite, with Rec = λ−2Rep being arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' For purposes of calculating the inertial lift to O(Rep), the disturbance velocity field in the actual problem may be replaced by its Stokesian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Thus, us, in the inertial acceleration terms in (S1), is the Stokesian disturbance field due to a force-free and torque-free sphere translating with Up in an ambient plane Poiseuille flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In a reference frame moving with the sphere center, the latter flow is given by u∞ = (α + βr2 + λγr2 2)11 − Up where, with the sphere at a distance λ−1s (in units of a) from the lower wall, one has α = 4λ−1s(1 − s), β = 4(1 − 2s) and γ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Here, α11 −Up and β are the ambient slip and shear rate at the sphere center, the latter varying linearly across the channel and equalling zero at the centerline (s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' γ denotes the constant curvature of the Poiseuille profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Up is determined using the force-free constraint in Faxen’s law for translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This leads to Up = (α+λγ/3)11 and, as a result, the ambient flow in the said reference frame takes the form u∞ = (βr2 + λγr2 2 − λγ/3)11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The test problem in the reciprocal theorem framework corresponds to the Stokesian translation of a sphere in an otherwise quiescent fluid confined between parallel walls (ones that bound the Poiseuille flow in the actual problem), under a constant force acting along the gradient direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The test disturbance field u multiplies the inertial acceleration terms involving us in (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' We now examine the length scales that contribute dominantly to the integral in (S1), beginning with the limit Rec ≪ 1, when the inertial screening length (HRe−1/2 c ) is much larger than the channel width, and therefore, irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The dominant contributions to the volume integral in this limit may arise from either scales of O(a) (the inner region) or those of O(H) (the outer region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In order to assess their relative magnitudes, we consider the intermediate asymptotic interval 1 ≪ r ≪ λ−1 (r is measured in units of a) where both the finite size of the sphere and wall-induced image contributions may be neglected at leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' For r in this interval, u ∼ 1/r corresponding to the farfield Stokeslet, and us ∼ β/r2 + γλ/r3 corresponding to the farfield stresslet and force-quadrupole contributions associated with the linear and quadratic ambient flow components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Using these forms along with u∞ ∼ βr + γλr2, one obtains the following estimates for parts of the integrand involving the linear and nonlinear components of the inertial acceleration: u · (us · ∇u∞ + u∞ · ∇us) ∼ β2 r3 + λβγ r2 + λγβ r4 + γ2 r3 (linear), u · (us · ∇us) ∼ β2 r6 + λβγ r7 + λ2γ2 r8 (nonlinear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Using dV ∼ O(r2dr), one obtains the following estimates for contributions to the lift velocity integral: V linear p ∼ Rep � r dr′ � β2 r′ + λβγ + λγβ r′2 + λ2γ2 r′ � , ∼ Rep � β2 ln r + λβγr + λγβ r + λ2γ2 ln r � , (S2a) V non-linear p ∼ Rep � r dr′ � β2 r′4 + λβγ r′5 + λ2γ2 r′6 � , ∼ Rep � β2 r3 + λβγ r4 + λ2γ2 r5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S2b) The algebraically growing terms in (S2a) will be dominated by scales of O(H), and the resulting contributions to the lift velocity are obtained by cutting off the divergence (for r → ∞) at r ∼ O(λ−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' this cut-off recognizes the wall- induced screening of the unbounded-domain behavior that eventually leads to a more rapid decay for r ≫ O(λ−1), and thence, convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The algebraically decaying terms in both (S2a) and (S2b) will be dominated by scales of 2 O(a), with the lift velocity contributions now obtained by cutting off the divergence (for r → 0) at r ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The ln r terms in (S2a) imply the dominance of the intermediate (matching) interval 1 ≪ r ≪ λ−1, leading, in principle, to contributions of O(β2 ln λ−1) and O(λ2γ2 ln λ−1) to the lift velocity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' logarithmically smaller contributions of O(β2) and O(λ2γ2) must arise from both scales of O(a) (r ∼ O(1)) and O(H) (r ∼ O(λ−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' However, contributions from the inner and matching regions turn out to be zero by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Owing to the absence of walls at leading order, the O(β2) and O(β2 ln λ−1) inner and matching-region contributions must correspond to the lift on a neutrally buoyant sphere (or the equivalent point-particle singularity) in an unbounded simple shear flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' likewise, the O(λ2γ2) and O(λ2γ2 ln λ−1) inner and matching-region contributions must correspond to the lift on a sphere at the origin of an unbounded quadratic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' In both these scenarios, however, the two lateral directions are equivalent, and there can be no lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' That these contributions are zero may also be seen from the fact that one cannot construct a true vector, the inertial lift velocity, from any quadratic combination of the velocity gradient tensor associated with an ambient linear flow (simple shear for the present case of an ambient Poiseuille flow), or from any quadratic combination of the third order tensor that would characterize a generic quadratic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Crucially, the O(β2) and O(λ2γ2) outer-region contributions are not subject to the above symmetry-argument-based limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This is due to the importance of walls at leading order, and the implied availability of an additional vector (the unit normal characterizing the wall orientations) to construct the lift velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Based on the above arguments, one is led to the following lift velocity contributions from the linear and nonlinear inertial terms in the integrand: V linear p ∼ Rep � β2(outer) + βγ(outer) + λγβ(inner) + λ2γ2(outer) � , (S3a) V non-linear p ∼ Rep λβγ(inner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S3b) From (S3a) and (S3b), the leading order contribution to the lift velocity is seen to come from the linearized inertial terms, and from scales of O(H), with there being two such contributions: one proportional to β2 that characterizes wall-induced repulsion in an ambient linear flow, and the other proportional to βγ that denotes the contribution due to the ambient profile curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The dominance of scales of O(H) implies that, for purposes of evaluating the above contributions, the sphere in both the actual and test problems can be replaced by the corresponding point singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Thus, at leading order, one only need consider the terms in (S1) that are linear in us, and further, us and u may be approximated as the disturbance fields due to a stresslet (ustr) and Stokeslet (uSt), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Note that uSt and ustr are a combination of the unbounded domain components and additional wall-image contributions, both of which are of comparable importance on scales of O(H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' detailed expressions are given in [S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Thus, the inertial lift velocity at leading order in Rep and λ reduces to: Vp = −Rep � V F +V P uSt · (ustr · ∇U ∞ + U ∞ · ∇ustr) dV, (S4) where the O(λγ) term in u∞ has been neglected, so U ∞ = (βr2 + λγr2 2)11 in (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Further, on account of the subdominance of scales of O(a), the domain of integration has been extended to include the particle volume (V P ), with an accompanying change in the integration variable that is now the position vector scaled by H (rather than a as in (S1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S4) is the point-particle approximation for the lift velocity for Rec ≪ 1, and corresponds to a dimensional lift velocity of O(V 2 maxλ2a/ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The use of a rescaled (with λ) integration variable, as mentioned above, leads to the integral in (S4) being only a function of s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' the detailed evaluation of this integral, via a partial Fourier transform, is discussed in [S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' For Rec ≳ O(1), the scaling arguments used above to establish the dominance of the outer region contributions still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The outer region now corresponds to scales of order the inertial screening length or larger, so that the farfield Stokesian estimate for us, used above to establish outer-region dominance, remains valid only for 1 ≪ r ≪ λ−1Re − 1 2 c , with the disturbance velocity field in the actual problem decaying more rapidly for larger r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The outer- region dominance for Rec ≳ O(1) implies that this disturbance field may still be approximated as being driven by a stresslet forcing, although the forcing appears in the linearized Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' While there still exist physically distinct contributions arising from profile curvature and wall-induced repulsion, the lift velocity can no longer be written as an additive superposition of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Furthermore, despite the relevance of the inertial screening length, asymptotically larger scales of O(H) continue to be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The ratio of these two outer scales involves Rec which appears in the governing linearized equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Thus, the version of the reciprocal theorem integral in (S4) for Rec ≳ O(1), with us replaced by its finite-Rec analog, will be a function of both s and Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Although the actual calculation is more easily accomplished via a direct solution of the partially Fourier transformed ODE’s using a shooting method [S2, S3], one may nevertheless write the point-particle lift velocity contribution formally in the form (V 2 maxλ2a/ν)F1(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Rec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 3 The modification of the leading order lift velocity, for λ small but finite, arises from contributions in (S3) that are of a smaller order in λ than the outer-region contributions included in (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The largest such contributions are proportional to λ(βγ), and pertain to the inner region, arising from both the linear and nonlinear inertial terms in (S3a) and (S3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The βγ-dependence implies that these contributions arise solely due to the coupling of the shear and curvature of the ambient profile, consistent with earlier symmetry arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Further, scales of O(H) are irrelevant, implying that the λ(βγ) terms will lead to contributions independent of Rec, with the dimensional lift velocity being of the form (V 2 maxλ3a/ν)F2(s), as in equation (2) of the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' A second implication of the O(λβγ) contributions being from the inner region is that they arise independently of the outer-region point-particle contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Said differently, the (V 2 maxλ2a/ν)F1(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Rec) and (V 2 maxλ3a/ν)F2(s) contributions to the inertial lift correspond to the leading order terms of the underlying asymptotic expansions of the integrand in the outer and inner regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This implies that the O(V 2 maxλ3a/ν)F2(s) contribution is not a correction to the leading order point-particle result, and thereby, not constrained to be small in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This feature is especially significant since the emergence of multiple equilibria in the lift profiles (Region 2 in Fig 4 of the main manuscript) is only made possible by allowing the finite-size contribution to be comparable to the leading point-particle one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Note that there are other corrections for finite λ: for instance, the O(λ2γ2) outer-region contribution in (S3), the correction to us arising from λ-dependent corrections to the stresslet coefficient, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' While both of these may be shown to be of a smaller order in λ for Rec ≪ 1, they will be far smaller for large Rec owing to an overall weakening of wall-induced corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This weakening arises from the more rapid decay of the disturbance velocity field for distances larger than O(HRe − 1 2 c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' This Rec-dependence is important since, as explained in the main manuscript, the finite-size contributions become significant only at large Rec owing to the reduction in the magnitude of the point- particle contribution with increasing Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Clearly, the only finite-size contributions of relevance correspond to the O(λβγ) terms in (S3a) and (S3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' To calculate these, we return to the original reciprocal theorem result, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S1), and subtract and add back the leading point-particle contribution given by (S4): Vp = − Rep � V F +V P uSt · � ustr · ∇U ∞ + U ∞ · ∇ustr � dV − Rep �� V F u · � us · ∇us + us · ∇u∞ + u∞ · ∇us � dV − � V F +V P uSt · � ustr · ∇U ∞ + U ∞ · ∇ustr � dV � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S5) Splitting the point-particle contribution within brackets into separate integrals over the fluid (V F ) and particle (V P ) domains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' separating the nonlinear and linear inertial terms in the original reciprocal theorem integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' and then combining the integrals involving the linear inertial terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' one obtains: Vp = − Rep � V F +V P uSt · � ustr · ∇U ∞ + U ∞ · ∇ustr � dV − Rep �� V F u · � us · ∇us � dV + � V F � u · � us · ∇u∞ + u∞ · ∇us � − uSt · � ustr · ∇U ∞ + U ∞ · ∇ustr �� dV − � V P uSt · � ustr · ∇U ∞ + U ∞ · ∇ustr � dV � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S6) The last integral within brackets, over V P , evidently involves scales of O(a), and we therefore focus on the remaining two integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The first integral within brackets contains the nonlinear inertial term, and as already seen in (S2b), the integrand exhibits an algebraic decay for r ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The second integral involves the difference between the linearized inertial terms, and their approximations based on point-particle representations of the corresponding velocity fields, as a result of which growing terms cancel out, and only the algebraically decaying one in (S3a) survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The algebraic decay implies the dominance of scales of O(a), and therefore, that wall-induced image contributions do not contribute at leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Thus, the leading approximations of all of the integrals within brackets, in (S6), may be obtained by 4 replacing the original fluid domain (V F ) by a completely unbounded one (V ∞), and (S6) may be written in the form: Vp = − Rep � V F +V P uSt · � ustr · ∇U ∞ + U ∞ · ∇ustr � dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' + λRep � − � V ∞ ui · � us,i · ∇us,i � dV − � V ∞ � ui · � us,i · ∇u∞ + u∞ · ∇us,i � − uSt,i · � ustr,i · ∇U ∞ + U ∞ · ∇u(1) str,i �� dV + � V P uSt,i · � ustr,i · ∇U ∞ + U ∞ · ∇ustr,i � dV � , (S7) where both the actual velocity fields and their point-particle approximations, that appear in the three bracketed integrals, are now approximated by simpler expressions pertaining to an unbounded fluid domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' these are indicated by the additional subscript ‘i’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' and are given below:: ui = 12 8π · �I r + rr r3 � + 12 8π · � I 3r3 − rr r5 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S8a) us,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='i = −5 2 (E : rr)r r5 − �E · r r5 − 5(E : rr)r 2r7 � + λγ � − 1 24r9 � 3r6 + r4(−23r2 1 + r2 2 − 8r2 3 − 3) + 15r2� r2 1(7r2 2 + 1) + r2 2 � − 105r2 1r2 2 � 11 + 5 8r9 (r2 − 1)r1r2(3r2 − 7r2 2)12 + 5 8r9 (r2 − 1)r1r3(r2 − 7r2 2)13 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S8b) uSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='i = 12 8π · �I r + rr r3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S8c) ustr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='i = −5 2 (E : rr)r r5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S8d) (S8a) is the disturbance velocity field induced by a translating sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S8b) is the disturbance field due to a force-free sphere in an ambient flow with both linear and quadratic components: the part involving E (= β 2 (1112 + 1211), the rate of strain tensor of the ambient Poiseuille flow), constitutes the disturbance in an ambient linear flow, while that proportional to γ constitutes the response to the quadratic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' As originally shown (but not used) by [S1], the latter may be obtained using an expansion in spherical harmonics [S4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S8c) and (S8d) are the usual velocity fields for a Stokeslet and the stresslet in an unbounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Finally, note that the integrals over V P and V ∞ extend down to the origin (the sphere center) with both integrands including an O(β2) contribution arising from the coupling of the linear ambient flow component with the stresslet field, that is O(1/r3) for r → 0 (see initial integrand estimates);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' while this leads to a conditionally convergent behavior, doing the angular integration first leads to a trivial answer, as must be the case based on symmetry arguments above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' For the integral over V ∞, a similar conditionally convergent behavior arises at infinity from the leading O(γ2) contribution, that is again resolved by appropriate choice of the order of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Even the leading point-particle integral is conditionally convergent at the origin, an issue (implicitly) addressed by a particular order of integration in the calculation procedure[S2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' The volume integrals, within brackets in (S8), are readily calculated analytically: � V ∞ ui · � us,i · ∇us,i � dV =6143 βγ 120960 , (S9) � V ∞ � ui· � us,i·∇u∞+u∞·∇us,i � −uSt,i· � ustr,i·∇U ∞+U ∞·∇ustr,i �� dV =37 βγ 1260 , (S10) � V P uSt,i · � ustr,i · ∇U ∞ + U ∞ · ∇ustr,i � dV =− βγ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (S11) Substituting (S9-S11) in (S6), with β = 4(1 − 2s) and γ = −4, gives: Vp = Rep � F1(s, Rec) + λF2(s) � , (S12) with F2(s) = 1141(1−2s) 216 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' 5 ∗ sganesh@jncasr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content='in [S1] Ho, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Leal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of fluid mechanics, 65(2), 365-400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [S2] Anand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Subramanian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' To be submitted to JFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [S3] Schonberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Hinch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Journal of Fluid Mechanics, 203, 517-524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' [S4] Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=', & Karrila, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Microhydrodynamics: principles and selected applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} +page_content=' Courier Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQf4frj/content/2301.00789v1.pdf'} diff --git a/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf b/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..044d73bf70c078ccbc2460f215da69bea1cf91b9 --- /dev/null +++ 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State University, San Diego, California, 92182, USA +Xiaofeng Liu‡ +San Diego State University, San Diego, California, 92182, USA +Many numerical algorithms have been established to reconstruct pressure fields from mea- +sured kinematic data with noise by Particle Image Velocimetry (PIV), such as the Pressure Pois- +son solver and the Omni-Directional Integration (ODI) method. This study adopts Gaussian +Process Regression (GPR), a probabilistic framework with an intrinsic de-noising mechanism +to tackle drawbacks of traditional Pressure Poisson solver and compares the performance with +ODI. To evaluate the accuracy of the algorithm, GPR and ODI are tested in detail in a canon- +ical setup of forced homogeneous isotropic turbulence from the Johns Hopkins Turbulence +Database. According to the result, GPR has the same level of accuracy as ODI with optimized +hyper-parameters for the isotropic turbulence flow. However, GPR has the tendency to flatten +impulsive signals. Therefore, without further modifications, it is not suitable to detect flow +structures with impulsive true signals. The error propagation of the proposed framework is +also analyzed and discussed in both physical and spectral spaces. +Nomenclature +𝑝(𝒙) += +pressure field +�𝑝(𝒙) += +true pressure field +𝒙 += +spatial coordinates +𝒖 += +velocity field +𝜌 += +density +𝑡 += +time +𝜇 += +dynamic viscosity +𝑘 += +realization number +¯𝑝(𝒙) += +mean of the pressure field +GP += +Gaussian Process +N += +Gaussian distribution +C += +covariance matrix of the Gaussian process +𝑙 += +correlation length in the radial basis function kernel +𝑙𝑝 += +correlation length scale from the true pressure field +𝜎(𝒙) += +standard deviation of Gaussian Process at a spatial location 𝒙 +𝜎𝑝 += +standard deviation of the pressure field +𝜎𝜖 += +standard deviation of assumed noise level in pressure gradient +𝜎∇𝑝 += +standard deviation of embedded noise in pressure gradient +𝑶 += +vector formed by observations of material derivatives +𝑿∗, 𝑿 += +vector formed by observation locations and general spatial locations +Σ𝑖 𝑗 += +Covariance matrices +𝑅𝜆 += +Reynolds number based on Taylor micro scale +𝑝𝐺𝑃𝑅, 𝑝𝑂𝐷𝐼 += +pressure field reconstructed by GPR or ODI +∗Doctoral student, Department of Aerospace Engineering, San Diego State University, Student Member AIAA. +†Assistant Professor, Department of Aerospace Engineering, San Diego State University, Member AIAA. E-mail: qwang4@sdsu.edu +‡Associate Professor, Department of Aerospace Engineering, San Diego State University, Associate Fellow AIAA. +1 +arXiv:2301.13282v1 [physics.flu-dyn] 30 Jan 2023 + +I. Introduction +A. Pressure field estimation from PIV +U +nderstanding the pressure field is essential in various turbulence and hydrodynamic researches. The generation +of acoustic noise [1, 2] and the onset of boundary separation [3, 4], for example, are instances where an +accurate estimation of the pressure is of paramount importance. However, non-intrusive measurements of the detailed +instantaneous pressure field is a leading challenge in experimental studies. By applying Particle Image Velocimetry +(PIV), the gradient information of pressure can be obtained from the balance of the Navier-Stokes equation, in which the +material acceleration is the dominant term, while the viscous term is negligible at regions away from the wall for high +Reynolds number flow [5], +∇𝑝 = −𝜌 𝐷𝒖 +𝐷𝑡 + 𝜇∇2𝒖 ≈ −𝜌 𝐷𝒖 +𝐷𝑡 . +(1) +Numerical tools have been established to further reconstruct the instantaneous pressure field based on the measured +pressure gradient. While the traditional approach solves the pressure-Poisson equation, which often utilizes an inaccurate +boundary condition, the state-of-art tool, named Omni-directional Integration (ODI), integrates the pressure gradients +in a collection of directions and averages the resulting field [5–8]. This has led to a robust de-noising framework +that greatly mitigates the effect of measurement error [9]. A recent example shows that the time-averaged pressure +reconstructed from Reynolds Averaged Navier-Stokes (RANS) equation based on stereo PIV measurements even reaches +spatial resolution beyond that of PIV [10]. The goal of the current study is to extend the idea of denoising and establish +a probabilistic framework using Gaussian Process Regression to reconstruct pressure from its gradient information with +uncertainties. +B. Gaussian process regression +Gaussian process regression has been used to solve regression problem when the underlying function is unknown +and hard to evaluate analytically. For example, apply Gaussian Process upper confidence bound sampling (GP-UCB) +to provide movie recommendation[11]; reconstruct missing temporal and spatial sensor data of a dynamic nonlinear +response [12, 13]; analyze motion trajectory of moving targets from sparse observations [14]. This non-parametric +Bayesian approach has been proven to be very powerful in exploration and exploitation scenarios. Furthermore, GPR +can capture a wide variety of relations between inputs and outputs by the kernel-encoded prior assumption. In this study, +we reconstruct the pressure from pressure gradient by deliberately encoding the prior assumption with different kernel +functions while previous works mainly focus on obtaining function from observations of the function itself. +The rest of the paper is structured as follows: in §II we introduce the mathematical formulation for Gaussian +Process Regression when applied to pressure field reconstruction. In §III we explain how we choose the optimal +hyper-parameters for GPR and how we evaluate the performance of GPR and ODI through the forced homogeneous +isotropic turbulence database from Johns Hopkins Turbulence Database (JHTDB). In §IV we show some comparison +results and analyses of reconstructed pressure field by GPR and ODI. In §V, we conclude current results and propose +some prospective future works. +II. Mathematical formulation +In the current study, we propose a new approach, adopting the idea of Gaussian Process Regression (GPR) [see 11, +for a brief tutorial] This probability framework takes into account measurement noise and could help to perform field +inversion from gradient information and mitigate the effect of measurement noise. +In Gaussian process regression, we assume the observation of pressure gradient at any spatial location 𝒙 can be +expressed by the true pressure gradient with an additional noise: +∇𝑝(𝒙) = ∇�𝑝(𝒙) + 𝜖 +(2) +where the noise term 𝜖 follows normal distribution +𝜖 ∼ N (0, 𝜎2 +𝜖 ). +(3) +Furthermore, GPR regards the pressure field, 𝑝(𝒙) as a Gaussian process in infinite dimensional space, e.g. +𝑝(𝒙) ∼ GP( ¯𝑝(𝒙), C(𝒙, 𝒙′)), +(4) +2 + +Fig. 1 +(a): An example of one-dimensional GPR when observations of the values of a smooth function are +available. Red dashed lines mark the location of observations while the black dashed lines are samples drawn +from the posterior distribution. Two red solid lines mark the region within one standard deviation in the +posterior distribution. (b): When observations of function derivatives are available, a similar approach can also +be adopted to infer the values of the function. +in which ¯𝑝(𝒙) represents the mean, or the expected value of the prior distribution for the pressure, ¯𝑝(𝒙) = E[𝑝(𝒙)], +and 𝒙, 𝒙′ represent two different spatial locations. The kernel of the Gaussian process, C(𝒙, 𝒙′) represents the +covariance of the uncertain pressure fields at two different spatial locations, 𝑝(𝒙) and 𝑝(𝒙′). +Or equivalently, +C(𝒙, 𝒙′) = E [(𝑝(𝒙) − ¯𝑝(𝒙))(𝑝(𝒙′) − ¯𝑝(𝒙′))]. +For a continuous and stationary random process, radial basis function kernel is popularly introduced. The Gaussian +kernel, for example, reads +C(𝒙, 𝒙′) = 𝜎(𝒙)𝜎(𝒙′) exp(−1 +2 +||𝒙 − 𝒙′||2 +𝑙2 +), +(5) +in which 𝜎(𝒙) is the standard deviation and 𝑙 is the correlation length. The standard deviation 𝜎(𝒙) will be updated +during the inference and can naturally lead to uncertainty quantification (UQ) of the reconstruction. The correlation +length 𝑙 is a hyper-parameter representing the smoothness of the pressure field, which can be tuned to achieve the best +performance for different scenarios. +The corresponding correlation function of the pressure field for the kernel function C(𝒙, 𝒙′) is defined as +K(𝒙, 𝒙′) = +C(𝒙, 𝒙′) +𝜎(𝒙)𝜎(𝒙′) +(6) +where 𝜎𝑝 is the root mean square of entire pressure field. For radial basis function kernel, the correlation function +K(𝒙, 𝒙′) is only function of distance 𝑟 between two point, +K(𝒙, 𝒙′) = K(𝑟) +(7) +in which 𝑟 = ||𝒙 −𝒙′||. In order to better comprehend the physical meaning of pressure field, we introduce the correlation +length scale 𝑙𝑝, which could be obtained by fitting the correlation function K(𝑟) of true pressure field into the correlation +function of Gaussian kernel. +While most applications of GPR deal with observing the values of the function directly [13], as shown in Figure +1(𝑎), the formulation of GPR is general and can be applied to observations of the gradient information. An example of +such reconstruction for a one-dimensional case is shown in Figure 1(𝑏), where the dashed lines marks the observation +location for the gradient of the function 𝑝(𝑥). Given 𝑿∗ a vector collection of 𝑛 spatial locations 𝒙𝒏 where data are +observed, the samples of the pressure gradient observations would follow a multivariate Gaussian distribution, +𝑶 = ∇𝑝 (𝑿∗) ∼ N +� +∇ ¯𝑝 +�� +𝑿∗, ∇𝒙∇𝒙′C (𝑿∗, 𝑿∗) + 𝜎2 +𝜖 I∗ +� +. +(8) +Here 𝜎𝜖 is the assumed noise level of synthetic noise introduced as a mimic of experimental data. I∗ is the identity +matrix. Moreover, once the observations are drawn from the above distribution, the observations of pressure gradient +at 𝑿∗ and unknown values of pressure field at 𝑿, a vector collection of spatial locations where reconstruction are +3 + +α) +2 +1 +1 +p +I +1 +-1 +1 +1 +- +-2 +2 +0.2 +0.2 +0.4 +0 +0.4 +0.6 +0.8 +0 +0.6 +0.8 +1 +1Fig. 2 +(a): True pressure field from an isotropic turbulence DNS database. (b)-(c): Pressure gradient obtained +from the DNS pressure field by central finite difference method. (d)-(e): Sample realization of 1000 error +embedded pressure gradient. +conducted, would follow the joint Gaussian distribution, +������� +𝑶 +𝑝(𝑿) +������� +∼ N +������ +� +��������� +∇𝑝 (𝑿∗) +¯𝑝(𝑿) +��������� +, +���������� +∇𝒙∇𝒙′C (𝑿∗, 𝑿∗) + 𝜎2 +𝜖 I∗ +�������������������������������������������������������� +Σ11 +, ∇𝒙′C (𝑿, 𝑿∗) +�������������������������� +Σ12 +∇𝒙C (𝑿∗, 𝑿) +������������������������ +Σ21 +, C (𝑿, 𝑿) +�������������� +Σ22 +���������� +������ +� +(9) +The last piece of this joint Gaussian distribution is the reference pressure. Since the value of reference pressure does not +influence the dynamic structure of reconstructed pressure fields, we include one additional observation of pressure +𝑝(𝒙0) = 0 at location 𝒙0 in the formulation. Once noisy measurements for the pressure gradient become available, the +pressure field can be recovered using Bayes’ theorem. And the posterior conditional distribution is given by, +𝑝 (𝑿) ∼ N +� +¯𝑝 − Σ21Σ−1 +11 +� +𝑶 − ∇ ¯𝑝 +�� +𝑿∗ +� +, Σ22 − Σ21Σ−1 +11Σ12 +� +. +(10) +The updated mean of the posterior distribution is regarded as the reconstructed pressure field from GPR, +𝑝𝐺𝑃𝑅(𝑿) = ¯𝑝 − Σ21Σ−1 +11 +� +𝑶 − ∇ ¯𝑝 +�� +𝑿∗ +� +(11) +Notice that the matrix inversion in the above expression requires O(𝑁3) operations, with 𝑁 being the number of +observations. The inversion would therefore be computationally intractable for large 𝑁. Nevertheless, for large-scale +computations, we could transform the matrix inversion into an iterative algorithm using the Conjugate Gradient method +[15], which is part of future efforts. +Moreover, the covariance matrix of the posterior probability distribution can be computed from, +𝐶𝐺𝑃𝑅(𝑿, 𝑿) = Σ22 − Σ21Σ−1 +11Σ12. +(12) +Square roots of the diagonal elements of the above matrix are the standard deviation 𝜎𝑝(𝑿), representing the uncertainty +of the reconstructed pressure fields at different spatial locations. +III. Problem Setup +A. Data acquisition +As the first step, we test our algorithm in a homogeneous isotropic turbulence flow field with Reynolds number around +𝑅𝜆 ∼ 433 based on Taylor microscale. Instead of using data from experiments, we extract data in the Johns Hopkins +4 + +a +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +ab) +c) +0.8 +30 +0.6 +20 +0.4 +0.2 +10 +0.0 +0 +0.8 +-10 +0.6 +0.4 +-20 +0.2 +30 +0.0 +0.00 +0.25 +0.50 +0.75 +0.00 +0.25 +0.50 +0.75Turbulent database [16–18] from a direct numerical simulation (DNS) as a surrogate. The simulated data provides +access to the fully resolved velocity and pressure fields, enabling us to test our algorithm thoroughly. Although the +database is three-dimensional, the algorithm of both ODI and GPR is able to reconstruct pressure on a two-dimensional +plane given observations of the in-plane components of the gradient vectors, as shown in Figure 2. +The observations are the projection of material derivatives onto the 𝑥 − 𝑦 plane at certain observation locations +extracted from the database, at four times the DNS grid spacing in 𝑥 and 𝑦 directions. A total number of 150 × 150 +observations of pressure gradient are obtained on a two-dimensional plane in the turbulent field on the 𝑥 − 𝑦 plane with +𝐿𝑥 = 𝐿𝑦 = 0.9. +In order to compare the performance of GPR and ODI, we adopted 1000 realizations of error-embedded pressure +gradient generated by Liu and Moreto [9] and reconstruct the pressure field by ODI as well as GPR. The error- +embedded pressure gradient is generated by adding random noise of uniform distribution with the magnitude as 40% +of (|∇𝑝|DNS)max, the maximum magnitude of the true pressure gradient, at each point. The true pressure field, true +pressure gradient as well as one sample realization of error embedded pressure gradient are shown in Figure 2. +B. Evaluation of pressure reconstruction +To quantify the accuracy of pressure reconstruction when subject to noise in the measurements, as often observed in +PIV, we evaluate the cumulative error over 1000 realizations of noisy measurements. First, we calculate the error of the +reconstructed pressure field by GPR and ODI by subtracting the true pressure field �𝑝(𝒙) at each point, i.e., +𝜖𝑖 𝑗 = 𝑝𝑖 𝑗 − �𝑝𝑖 𝑗, +(13) +where 𝑖, 𝑗 refer to Cartesian indices. Then, the standard deviation of error 𝜖std is defined as +𝜖std = +� +� +� +� +1 +𝑁𝑥 × 𝑁𝑦 − 1 +𝑁𝑦 +∑︁ +𝑗=1 +𝑁𝑥 +∑︁ +𝑖=1 +(𝜖𝑖 𝑗 − 𝜖𝑖 𝑗)2. +(14) +Finally, we calculate the averaged standard deviation over 1000 realizations, the cumulative error 𝜀std, as +𝜀std = 1 +𝑘 +𝑘 +∑︁ +𝑛=1 +� 𝜖std +𝑝std +� +𝑛 +. +(15) +C. Hyper-parameter Optimization +The performance of both ODI and GPR methods depends on setting up proper parameters, or hyper-parameters. +For Omni-directional Integration, we adopt the state-of-art formulation with Rotating Parallel Rays to create different +integration paths [see 7, for details]. The parallel rays are rotated with an increment of 0.2 degrees, which creates a +total number of 1800 different orientations of parallel integration paths over the entire domain of calculation. The +optimal separation between adjacent parallel rays at 0.4 times of the grid size as suggested by [9] is adopted in the ODI +calculation. For GPR, the prior distribution is set with mean ¯𝑝(𝒙) = 0 and variance 𝜎(𝒙) = 1. However, correlation +length 𝑙 and assumed noise level 𝜎𝜖 still need to be optimized, which is going to be elaborated in the following sections. +In order to optimize the performance of GPR, we calculate the averaged error of 10 realizations with different +hyper-parameters sets and find the optimal hyper-parameters with the lowest error as the correlation length 𝑙 = 0.0625 +and the assumed noise level 𝜎𝜖 = 6. The error is slightly smaller with assumed noise level 𝜎𝜖 less than 6. However, +because the influence of equivalent noise level is not very significant once it is smaller than 6, we choose 𝜎𝜖 = 6 as +our optimal hyper-parameter. The correlation length scale 𝑙𝑝 could be obtained by fitting the Gaussian kernel into the +correlation function of the true pressure field, which is equal to 0.0574 for the isotropic turbulence. So the correlation +length scale 𝑙𝑝 = 0.0574. Embedded noise is a uniform distribution from -12 to 12, the standard deviation of which +is 6.928. Thus the magnitude of embedded noise is 𝜎∇𝑝 = 6.918. We could find out that optimal hyper-parameters +correlation length 𝑙 approximates to correlation length scale 𝑙𝑝 while assumed noise level 𝜎𝜖 approximates to the +magnitude of embedded noise 𝜎∇𝑝. This phenomenon proves that the kernel of GPR depends on the correlation function +of the true pressure field as well as the standard deviation of noise in the observations. +5 + +Fig. 3 +(a): Correlation function 𝐾(𝑟) of the true pressure field and the optimal Gaussian kernel function from +curve-fitting. (b): The cumulative error of 10 realizations with different correlation length 𝑙 and assumed noise +level 𝜎𝜖 as well as the location of optimal hyper-parameters. +IV. Results +A. Error Analysis in Physical Space +The cumulative error of GPR and ODI on 150 by 150 grids as well as the cumulative error of ODI on 254 by +254 grids are shown in Figure 4. From the result, we could observe that error of GPR converges to 0.153 over 1000 +realizations. Furthermore, GPR has a similar level of accuracy with ODI method on 150 by 150 grids domain. However, +ODI on 254 by 254 grids can reach to lower cumulative error 0.148, the same as the result of previous work by Liu and +Moreto [9]. But GPR requires more memory to conduct the matrix inversion in the regression. Therefore, for large-scale +problems, it is necessary to convert matrix inversion into an iterative algorithm, such as the conjugate gradient iteration +for better efficiency, which is part of future work. +To further investigate the performance of GPR and ODI, Figure 5 shows the exact pressure field �𝑝, the pressure +field reconstructed by GPR for the instance of the worst case scenario among the 1000 realizations, pressure field +reconstructed by ODI for the instance of the worst case scenario among the 1000 realizations, the standard deviation of +reconstructed pressure field by GPR, error of GPR result as well as error of ODI result of the worst performance case +among 1000 realizations. In Figure 5, we could observe that the reconstructed pressure field by GPR is significantly +smoother than ODI result and both reconstructed pressure fields are fairly accurate compared to the exact pressure +field. Although the reconstructed pressure field by GPR has a smaller global error, it also has a larger local error +compared to the reconstructed pressure field by ODI, indicating the tendency of GPR in flattening impulsive local +pressure changes. Furthermore, the reconstructed pressure field by ODI preserves more fine structures (combined with +noise and high-frequency pressure signal) while GPR seems to have a stronger denoising effect according to the smooth +distribution. +While the mean of the posterior distribution can be used to represent the reconstructed pressure field, we could also +evaluate the accuracy of reconstruction based on the standard deviation of the posterior distribution. Notice that adding +or subtracting a constant field on the pressure does not alter the agreement with the observation of the pressure gradient. +This property, sometimes coined as “gauge invariance", has to be eliminated in order to obtain meaningful results for +uncertainty analysis. Therefore, we assume that the reconstructed pressure at a given reference point, e.g. 𝒙0 at the +center of the domain is solely due to such freedom of adding an arbitrary constant field. The variances of the pressure +field at other locations further take into account the effect of 𝜎𝑝(𝒙0), which should be subtracted to obtain a reasonable +estimation of the reconstruction uncertainty without the influence of such gauge invariance. For this reason, we here +plot 𝜎𝑝(𝑿) − 𝜎𝑝(𝒙0) in Figure 5(d). +In order to visualize how the error is propagated throughout the entire pressure field by different methods, we add a +6 + +a) +b) +×10-1 +1.0 +correlation function of p +★ +optimal parameters +4 +1.6 +fitted correlation function +0.8 +1.4 +0.6 +1.2 +2 +0.4 +三 1.0 + +p$3 +6 +0.2 +0.8 +1 +0.0 +0.6 +-0.2 +0.4 - +0.0 +0.1 +0.2 +0.3 +0.4 +100 +5 × 100 +1/lpFig. 4 +Cumulative error 𝜀std and standard deviation (𝜖std)𝑘 of 1000 realizations with correlation length 𝑙 = +0.0625 and assumed noise level 𝜎𝜖 = 6. The bottom histogram shows the standard deviation of error (𝜖std)𝑘 in +the 𝑘-th realization. The blue bar represents the standard deviation of error by GPR and red line represents +the standard deviation of error by ODI in 150 × 150 grids. Three converged lines on the top show the cumulative +error of GPR and ODI in 150 × 150 and 254 × 254 grids. +unit impulse in 𝜕𝑝 +𝜕𝑥 at the middle of the computational domain and use GPR and ODI to reconstruct the pressure field, +respectively. Differences between the reconstructed pressure field with and without the unit impulse are then visualized +in Figure 6 to quantify the domain of influence for such point-wise perturbation. Such approaches have been adopted in +previous research to study the domain of influence or the domain of dependence to fully understand the forward and +backward propagation of perturbations in a dynamical system or data assimilation algorithm [19, 20]. +The results of this impulse response have profound implications. First of all, the ODI method indicates a clear +singularity point at the location of perturbation, manifested by very large positive and negative values. The implication +here is that although ODI averages the error across the whole computational domain, the reconstructed pressure still +relies heavily on the local pressure gradient information near the point of interest. In other words, most of the local error +remains local, and correspondingly, the error diffusion is relatively not strong in comparison with that of GPR. On the +other hand, the GPR method exhibits nearly zero influence at the point of perturbation and has a larger influence slightly +farther away. This difference could explain the stronger de-noising effect in GPR than in ODI. +B. Error Analysis in Wave Number Space +The different behavior of GPR and ODI in Figure 5 leads to a comparison of power spectrum density of the +reconstructed pressure field by GPR and ODI, as shown in Figure 7. From Figure 7(a), we could see that both methods +perform well on low wave number space: the power spectrum density of GPR and ODI coincides with the power +spectrum density of true pressure field perfectly. However, in high wave number space, the power spectrum density +of different pressure field seem to diverge: GPR has a lower power spectrum density while ODI has a larger power +spectrum density compared to the power spectrum density of true pressure field. This phenomenon validates previous +observations in the pressure field shown in Figure 5, as well as the impulse response in Figure 6. GPR has a stronger +denoising effect but also smooths out some pressure information in high wave number space while ODI preserves +more fine structures. Some of them are dynamic behavior of the pressure field, others are noise in the observation. +Furthermore, Figure 7(b) clearly indicates that if the pressure gradient information is accurate, ODI can faithfully +replicate the dynamic behavior of the pressure field over the entire spectral range. However, in contrast, because +7 + +0.50 +0.20 +GPR (150×150) +0.45 +ODI (254×254) +ODI (150×150) +0.18 +0.40 +0.35 +0.16 +Estd +p4s +0.30 +0.14 +0.25 +0.20 +0.12 +0.15 +0.10 +0.10 +0 +200 +400 +600 +800 +1000 +Realization kFig. 5 +(a): True pressure field from isotropic turbulence DNS database. (b): Instant of realization of recon- +structed pressure field by GPR with the largest standard deviation of error 𝜖std. (c): Instant of realization of +reconstructed pressure field by ODI with the largest standard deviation of error 𝜖std. (d): Standard deviation +of reconstructed pressure field by GPR, 𝜎(𝑿) subtracted by the standard deviation of reference point 𝜎(𝒙0). +(e): Error distribution of reconstructed pressure field by GPR, obtained by subtracting (a) from (b). (f): Error +distribution of reconstructed pressure field by ODI, obtained by subtracting (a) from (c). +Fig. 6 +Error Propagation of GPR and ODI, represented by the change of reconstructed pressure field when +perturbation in the form of a unit impulse of 𝜕𝑝𝜕𝑥 is added at the center of the computational domain. +8 + +a) +b) +c) +1.0 +0.8 +0.5 +0.6 +Pressure +0.0 +0.4- +-0.5 +0.2 +-1.0 +0.0 - +a +d) +e) +X10-4 +2 +0.8 +4 +1 +2 +0.6 +E +Error +0 +0 +0.4 +-2 +0.2 +-1 +-4 +0.0 - +-2 +0.25 +0.50 +0.75 +0.25 +0.50 +0.75 +0.00 +0.00 +0.00 +0.25 +0.50 +0.75ODI +GPR +×10-4 +0.8 +2 +1 +0.6 +0 +0.4 +-1 +0.2 +-2 +0.0 - +0.00 +0.25 +0.50 +0.75 +0.00 +0.25 +0.50 +0.75 +a +aFig. 7 +(a): Power spectrum density of true pressure field as well as reconstructed pressure field by GPR and +ODI from error embedded pressure gradients of 150 by 150 grids. (b): Power spectrum density of true pressure +field as well as reconstructed pressure field by GPR and ODI from true pressure gradients of 150 by 150 grids +the hyperparameters used in this GPR computation were optimized with the error-embedded data (therefore are flow +dependent), GPR produces a reconstructed pressure spectrum with an overall lower fluctuation amplitude over the +entire spectral domain. This indicates that the optimization of GPR needs to be adjusted according to the actual flow +properties. This problem might be solved by switching a proper kernel function rather than a Gaussian kernel in this +case since its denoising effect is too strong to preserve necessary information, which requires future work. +V. Conclusion and future work +We adopt the framework of Gaussian Process Regression (GPR) to the problem of determining the pressure fields +from measured pressure gradients, with the potential of reconstructing pressure from sparsely measured data. The +formulation naturally avoids the burden of solving Poisson equation with inaccurate boundary conditions and takes into +account the effect of measurement noise. Furthermore, this framework provides more possibilities to improvement. +From the comparison between the reconstructed pressure field by GPR and ODI, we are able to conclude that the +reconstructed pressure field by GPR achieves accuracy comparable to the state-of-art Omni-directional integration +method (ODI) and has a stronger denoising effect compared to ODI. However, pressure reconstruction by GPR might +also smooth out some pressure information in high wave number space by mistake, especially dealing with accurate +pressure gradient data. This problem might be able to be solved by switching the Gaussian kernel, which requires +further investigation. +In future directions, the following will be studied in detail: (a) improvement of computational efficiency for +large-scale problems. (b) different kernel functions for the Gaussian Process. (c) effect of the sparseness of the +observation in terms of reconstruction quality. +VI. Acknowledgments +The support from San Diego State University is gratefully acknowledged. Thanks to Jose Moreto for his assistance in +the usage of the parallel-ray Omni-directional integration code and for providing the error-embedded pressure gradients +data in previous study by Liu and Moreto [9]. +References +[1] Buchta, D. A., and Freund, J. B., “The near-field pressure radiated by planar high-speed free-shear-flow turbulence,” Journal of +Fluid Mechanics, Vol. 832, 2017, pp. 383–408. +9 + +a) +b) +true pressure +real pressure +TI +GPR +GPR +T +ODI +ODI +10-1 +10-1, +88 +@ +更 +x +10-3, +10-3, +10-4 +10-4 ↓ +10-2 +10-1 +100 +10-2 +10-1 +100 +rn +kn[2] Wu, S. 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A., “Spatial reconstruction of steady scalar sources from remote measurements in +turbulent flow,” Journal of Fluid Mechanics, Vol. 870, 2019, pp. 316–352. +10 + diff --git a/HNAzT4oBgHgl3EQfHfvA/vector_store/index.pkl b/HNAzT4oBgHgl3EQfHfvA/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..913ef50183378c11c65ca979aedc428bf99d582e --- /dev/null +++ b/HNAzT4oBgHgl3EQfHfvA/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f4fec1f8d9aaed8cecac2daefc0878ea51e19d5592125c467aeb7e14bc1b48fc +size 102107 diff --git a/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf b/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fe03416cb2cb6c9a6b79ae0fdc1be04933456525 --- /dev/null +++ b/HNE1T4oBgHgl3EQfrQW4/content/2301.03353v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:071efb764273ef030c7e07bdb620bf8a5ef976f031f3cf355478a11c5d0d417f +size 1078900 diff --git a/IdE3T4oBgHgl3EQfuguG/content/tmp_files/2301.04685v1.pdf.txt b/IdE3T4oBgHgl3EQfuguG/content/tmp_files/2301.04685v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fde5543b5478003e28e884d90b09ce6a2a34aaa3 --- /dev/null +++ b/IdE3T4oBgHgl3EQfuguG/content/tmp_files/2301.04685v1.pdf.txt @@ -0,0 +1,2048 @@ +SHUNIT: Style Harmonization for Unpaired Image-to-Image Translation +Seokbeom Song1, Suhyeon Lee1, Hongje Seong1, Kyoungwon Min2, and Euntai Kim1* +1Yonsei University, Seoul, Korea +2Korea Electronics Technology Institute, Seongnam, Korea +{lgs5751, hyeon93, hjseong, etkim}@yonsei.ac.kr, minkw@keti.re.kr +light +red light +red light +red light +light +red light +red light +red light +light +light +light +light +input image +translated +image +I2I translation +annotation +: source domain +: target domain +: source-to-target +style mapping +Red, blue, and purple: +styles from different +categories +zoom-in +zoom-in +zoom-in +target domain images +source domain images +(a) Global I2I +(b) Class-level I2I +(c) Style harmonization I2I +Figure 1: Illustration of the concepts in unpaired I2I translation. The results are obtained on Cityscapes → ACDC (night) setting. +In the image, many head and tail lamps should be bright at night. (a) Global I2I (Huang et al. 2018b) converts all classes to +bright because it translates the image with a single source-to-target style mapping function. (b) Class-level I2I (Jeong et al. +2021) leverages additional annotations to address the problem of (a) and performs per-class source-to-target style mapping. It +can effectively deal with multiple classes in an image, but loses the original style: All white lights and red lights become white +lights. (c) Style harmonization I2I also performs class-wise style mapping, while adaptively preserving the original styles. +Abstract +We propose a novel solution for unpaired image-to-image +(I2I) translation. To translate complex images with a wide +range of objects to a different domain, recent approaches of- +ten use the object annotations to perform per-class source-to- +target style mapping. However, there remains a point for us to +exploit in the I2I. An object in each class consists of multiple +components, and all the sub-object components have differ- +ent characteristics. For example, a car in CAR class consists +of a car body, tires, windows and head and tail lamps, etc., +and they should be handled separately for realistic I2I trans- +lation. The simplest solution to the problem will be to use +more detailed annotations with sub-object component annota- +tions than the simple object annotations, but it is not possible. +The key idea of this paper is to bypass the sub-object com- +*Corresponding authors. +ponent annotations by leveraging the original style of the in- +put image because the original style will include the informa- +tion about the characteristics of the sub-object components. +Specifically, for each pixel, we use not only the per-class +style gap between the source and target domains but also the +pixel’s original style to determine the target style of a pixel. +To this end, we present Style Harmonization for unpaired I2I +translation (SHUNIT). Our SHUNIT generates a new style +by harmonizing the target domain style retrieved from a class +memory and an original source image style. Instead of direct +source-to-target style mapping, we aim for source and target +styles harmonization. We validate our method with extensive +experiments and achieve state-of-the-art performance on the +latest benchmark sets. The source code is available online: +https://github.com/bluejangbaljang/SHUNIT. +arXiv:2301.04685v1 [cs.CV] 11 Jan 2023 + +Introduction +Unpaired image-to-image (I2I) translation aims to learn +source-to-target style mapping, where source and target im- +ages are unpaired. It can be applied to data augmenta- +tion (Antoniou, Storkey, and Edwards 2017; Mariani et al. +2018; Huang et al. 2018a; Xie et al. 2020), domain adapta- +tion (Hoffman et al. 2018; Murez et al. 2018) and various im- +age editing applications, such as style transfer (Gatys, Ecker, +and Bethge 2016; Huang and Belongie 2017; Ulyanov, +Vedaldi, and Lempitsky 2017), colorization (Zhang, Isola, +and Efros 2016; Zhang et al. 2017), and image inpaint- +ing (Iizuka, Simo-Serra, and Ishikawa 2017; Pathak et al. +2016). +In the I2I translation, the biggest problem is how to deal +with the style variations among objects or classes. In other +words, when a global style gap is applied to an entire im- +age as in Fig. 1a, the I2I translation often results in unreal- +istic images because each class has different the style gaps +between the source domain and target domain. Recent ad- +vanced methods (Shen et al. 2019; Bhattacharjee et al. 2020; +Jeong et al. 2021; Kim et al. 2022) addressed the problem +by leveraging additional object annotations. They simplify +the task into class-level I2I translation and then perform per- +class source-to-target style mapping. This enables the net- +works to explicitly estimate class-wise target styles, but it +has a critical limitation: An object in each class consists +of multiple components, and all the sub-object components +might also have different characteristics. Let us consider the +example given in Fig. 1b. +Understandably, when a road image taken on a sunny day +is translated into a night image, head and tail lamps in a car +should be brighter while the rest of the components, such as +car body, tires, and windows, should be darker than before. +Therefore, each component in a car should be handled sep- +arately for realistic I2I translation. However, if the previous +approaches are applied to perform per-class source-to-target +style mapping, they will translate all head and tail lamps into +white lights, making unrealistic images, as shown in Fig. 1b. +Here, one might think that this issue can be addressed by an- +notating more detailed sub-object components than the sim- +ple object categories, but it is actually impossible. A brief +example is as follows. A car consists of body, window, and +tires. A tire consists of wheel and gum. In this way, sub- +object components can be divided endlessly. +To solve the above limitation of the previous class-level +I2I methods, we present Style Harmonization for unpaired +I2I translation (SHUNIT). The key idea of SHUNIT is to by- +pass the sub-object component annotations by leveraging the +original style of the input image because the original style +will include the information about the characteristics of the +sub-object components. Thus, instead of mapping source- +to-target style directly, SHUNIT harmonizes the source and +target styles to realize realistic and practical I2I translation. +As illustrated in Fig. 1c, SHUNIT uses not only the per- +class style gap between the source and target domains but +also the pixel’s original style to determine the target style +of a pixel. To achieve this, we disentangle the target style +into class-aware memory style and image-specific compo- +nent style. The class-aware memory style is stored in a style +memory, and image-specific component style is taken from +the original input image. +The goal of the style memory is to obtain class-wise +source-to-target style gaps and it is motivated by (Jeong et al. +2021). Compared to the memory in (Jeong et al. 2021), our +style memory differs in two aspects. First, the output from +the style memory was used alone as a target style in (Jeong +et al. 2021), but the output from the memory is adaptively +aggregated (=harmonized) in this paper with the style of +the original input image to make a target style. Second, the +memory was simply updated in (Jeong et al. 2021), whereas +our style memory is jointly trained and optimized with the +other parts of SHUNIT. Specifically, the class-aware mem- +ory in (Jeong et al. 2021) was not trained but simply was +updated using the input features during the training, memo- +rizing the style features from the target domain. The gradi- +ent was not propagated to the style memory. Thus, the style +memory in (Jeong et al. 2021) cannot update their param- +eters based on the error of memory. In SHUNIT, however, +we overcome this problem by enabling the memory to learn +through backpropagation. To this end, we train the style +memory from randomly initialized parameters and introduce +style contrastive loss to constrain the memory to learn class- +wise style representations. The backpropagation forces the +style memory to reduce the final loss jointly and effectively +along with the other parts of SHUNIT. To demonstrate the +superiority of our SHUNIT, we conduct extensive experi- +ments on the latest benchmark sets and achieve state-of-the- +art performance. +Overall, the contributions of our work are summarized as +follows: +• We present a novel challenge in I2I translation: an object +might have various styles. +• We propose a new I2I method, style harmonization, that +leverages two distinct styles: class-aware memory style +and image-specific component style. To the best of our +knowledge, the style harmonization is the first method +to estimate the target style in multiple perspectives for +unpaired I2I translation. +• We achieve new state-of-the-art performance on latest +benchmarks and provide extensive experimental results +with analysis. +Related Work +Image-to-image translation. +The goal of I2I is to +learn source-to-target style mapping. For I2I translation, +pix2pix (Isola et al. 2017) proposes a general solution us- +ing conditional generative adversarial networks (Mirza and +Osindero 2014). However, it has a significant limitation: +paired training data should be used for training networks. +CycleGAN (Zhu et al. 2017) successfully addresses this +problem with a cycle consistency loss. The loss allows us to +train the networks with unpaired training data by supervis- +ing the reconstructed original image only. Based on Cycle- +GAN, many approaches (Kim et al. 2017; Choi et al. 2018) +have been proposed to tackle I2I translation take in an un- +paired manner. UNIT (Liu, Breuel, and Kautz 2017) pro- +poses another unpaired I2I translation solution by mapping + +𝑳𝒙 +𝐸�� +𝐸�� +𝑰𝒙 +𝒄𝒙 +𝒔𝒙 +Read +𝐺� +𝑰�𝒚 +Learnable Style Memory +𝑘 +𝑣 +class 1 +class 2 +class N +𝑴𝒚 +𝒔�𝒚 +SH ResBlk +SH ResBlk +SH ResBlk +ReLU +SHL +Conv +𝒔�𝒚 +𝒔𝒙 +ReLU +SHL +Conv +𝒔�𝒚 +𝒔𝒙 +(a) Overall architecture +(b) SH ResBlk +Figure 2: An overview of SHUNIT. (a) From a pair of image Ix and label Lx in source domain X, the two encoders (i.e., +Ex +c and Ex +s ) extract the content cx and component style sx, respectively. The content retrieves the memory style ˆsy. The three +features cx, sx, and ˆsy are fed to the target generator Gy, which consists of several Style Harmonization Residual Blocks (SH +ResBlk). The generator Gy outputs the translated image ˆIy. (b) The input of the SH ResBlk is the content cx for the first layer, +and the output of the previous block is used as input for the remainders. Each SH ResBlk includes two Style Harmonization +Layers (SHL) that transfer target styles, i.e., sx and ˆsy. +two images in different domains into the same latent code +in a shared-latent space. MUNIT (Huang et al. 2018b) and +DRIT (Lee et al. 2018) introduce a disentangled representa- +tion to achieve diverse and multi-modal I2I translation from +unpaired data. Basically, they perform global I2I translation +which focuses on mapping a global style on all pixels in an +image. Although they work well on object-centric images, +they bring severe artifacts for complex images, such as mul- +tiple objects being presented or large domain gap scenarios, +as illustrated in Fig. 1a. To complement the problem, recent +approaches leverage additional object annotations and per- +form class-level I2I translation. +Class-level image-to-image translation. +Recent several +approaches (Mo, Cho, and Shin 2019; Shen et al. 2019; +Bhattacharjee et al. 2020; Jeong et al. 2021; Kim et al. +2022) propose class-level image-to-image translation so- +lutions with object annotations. Specifically, INIT (Shen +et al. 2019) generates instance-wise target domain images. +DUNIT (Bhattacharjee et al. 2020) additionally employs +an object detection network and jointly trains it with the +I2I translation network. MGUIT (Jeong et al. 2021) pro- +poses an approach to store and read class-wise style repre- +sentations with key-value memory networks (Miller et al. +2016). This approach, however, cannot directly supervise +the memory with objective functions for I2I translation. In- +staformer (Kim et al. 2022) proposes a transformer-based +(Vaswani et al. 2017; Dosovitskiy et al. 2021) architec- +ture that mixes instance-aware content and style represen- +tations. The existing methods that leverage object annota- +tions learns the direct class-wise source-to-target style map- +ping, as shown in Fig. 1b. This effectively simplifies the I2I +translation problem into per-class I2I translation, but they +overlook an important point that not all pixels in the same +class should be translated with the same style. Our approach, +style harmonization, addresses this problem by introducing +the component style that facilitates preserving the original +style of the source image, as illustrated in Fig. 1c. +Proposed Method +Definition and Overview +Let X and Y be the visual source and target domains, respec- +tively. Given an image and the corresponding label (=bound- +ing box or segmentation mask) in X domain, our frame- +work generates a new image in Y domain while remain- +ing the semantic information in the given image. We assume +that each domain consists of images and labels denoted by +(Ix, Lx) ∈ X and (Iy, Ly) ∈ Y, and both domains have the +same set of N classes. Our framework contains the source +encoder Ex = {Ex +c , Ex +s }, target generator Gy, and target +style memory M y for source-to-target mapping, and the tar- +get encoder Ey = {Ey +c , Ey +s }, source generator Gx, source +style memory M x for target-to-source mapping. For conve- +nience, we will only describe the source-to-target direction, +and the overview of our framework is depicted in Fig. 2. +Following the previous studies (Huang et al. 2018b; Lee +et al. 2018), we assume that an image can be disentangled +into domain-invariant content and domain-specific style. For +this, we basically follow the MUNIT (Huang et al. 2018b) +architecture. The content encoder Ex +c consists of several +strided convolutional layers and residual blocks (He et al. +2016), and all the convolutional layers are followed by +Instance Normalization (Ulyanov, Vedaldi, and Lempitsky +2016). The content encoder extracts the domain-invariant +content feature cx from the image Ix and label Lx. The +style encoder Ex +s also consists of several strided convolu- +tional layers and residual blocks, and it extracts the compo- + +𝒄𝒙 +𝒔�𝒚 +class-wise separation +𝑣� +𝑘� +𝑑 +softmax +class 1 +𝑣� +𝑘� +class 2 +𝑣� +𝑘� +class N +••• +𝑐� +� +𝑐� +� +𝑐� +� +𝑠̂� +� +𝑠̂� +� +𝑠̂� +� +Figure 3: Read operation in style memory. The content +feature cx is separated by the label Lx. For each class, the +memory read is independently performed. After class-wise +memory read, the memory style ˆsy is obtained by gathering +the retrieved class-wise values into the original locations. � +denotes matrix multiplication. +nent style feature sx from the image Ix. The memory style +ˆsy is read by retrieving from the learnable style memory M y +to the content feature cx. The generator Gy consists of sev- +eral style harmonization layers and residual blocks, and it +produces the translated image ˆIy from cx, sx, and ˆsy. +Style Harmonization for Unpaired Image-to-Image +Translation (SHUNIT) +The important point of SHUNIT is that two styles are em- +ployed to determine the target style: One is image-specific +component style, and the other one is class-aware memory +style. We focus on extracting two distinct styles accurately +and then harmonizing them. In what follows, we describe +the detail of each step. +Component style. +The style encoder Ex +s takes the im- +age Ix as input and extracts the style feature sx of size +H × W × C, where H, W, and C are the height, width, and +number of channels of the feature, respectively. Here, sx ex- +plicitly represents the style of the input image, thus we use +it as image-specific component style. The component style +is used together with the memory style in the style harmo- +nization layer to reduce the artifacts of the generated target +image. +Memory style. +The component style is not sufficient to +handle complex scenes with multiple objects. Therefore, we +exploit the class-specific style that leverages an object anno- +tation. To this end, we construct the class-wise style mem- +ory retrieved by the content feature. The target style memory +M y consists of N class memories to store class-wise style +representations of the target domain Y. Each class memory +M y +n has U key-value pairs (ky, vy) (Jeong et al. 2021), con- +sidering that various styles exist in one class (e.g., different +styles of headlamp and tires in CAR class). The key ky is +used for matching with the content feature and the value vy +has the class-aware style representations. The key and value +are learnable vectors, each one of size 1 × 1 × C. +Fig. 3 shows a detailed implementation of the process of +reading the corresponding memory style ˆsy from the mem- +𝒔𝒙 +𝜷𝒙 +𝜸𝒙 +𝒔�𝒚 +𝜷𝒚 +𝜸𝒚 +𝛼� +1 − 𝛼� +Conv +Conv +Conv +Conv +𝜷 +𝜸 +input +output +••• +class-wise 𝛼 +broadcasting +norm +Figure 4: Style harmonization layer. From the component +style sx and memory style ˆsy, scale (γx and γy) and shift (βx +and βy) factors are extracted via four convolutional layers. +The alpha mask ˆα is obtained by broadcasting the class-wise +alpha α with the semantic label Lx. We weighted sum the +image and memory styles with the alpha mask, and transfer +it to the input feature. ⊙ denotes Hadamard product. +ory M y. With the semantic label, we separate the content +feature cx into {cx +1, cx +2, · · · , cx +N}, where cx +n denotes source +content feature for the n-th class. Let cx +n,i be the i-th pixel +of the n-th class source content feature and (ky +n,j, vy +n,j) be +the j-th key-value pair of the n-th class target style memory +M y +n. In this work, we aim to read the target memory style +ˆsy +n,i corresponding to the source content cx +n,i using the simi- +larity between the source content and key of target memory. +To this end, we calculate the similarity wx +n,i,j between cx +n,i +and ky +n,j as: +wx +n,i,j = +exp +� +d(cx +n,i, ky +n,j) +� +�U +u=1 exp +� +d(cx +n,i, ky +n,u) +� +(1) +where d(·, ·) is the cosine similarity. We then read the +memory style ˆsy +n,i corresponding to cx +n,i by calculating the +weighted sum of values in n-th class style memory: +ˆsy +n,i = +U +� +j=1 +wx +n,i,jvy +n,j. +(2) +The same process is applied for content features of other +classes, finally extracting spatially varying target memory +style features ˆsy of size H × W × C. +Different from the previous key-value memory net- +works (Jeong et al. 2021) that learn the memory via updating +mechanism, we learn the memory through backpropagation. +The updating mechanism is used to directly store the ex- +ternal input features. However, it has a critical drawback: +The memory cannot be trained with the network jointly with +the same objective function because the gradient should be +stopped at the updated memory. To solve the problem, we +discard the update mechanism and learn the memory with +the loss functions presented in Eq. (7). The effectiveness of +our memory learning strategy is validated in the experiments +section. +Style harmonization layer. +Our goal is harmonizing the +source and target styles instead of mapping source-to-target + +style directly. To this end, we propose the style harmoniza- +tion layer to adaptively aggregate the component style and +memory style. The style harmonization layer consists of sev- +eral convolution layers and class-wise alpha parameters, and +it is illustrated in Fig. 4. Here we use three conditional in- +puts: memory style, component style, and label. Convolu- +tional layers are used to compute pixel-wise scale γ and shift +β factors from the two styles. Following (Jiang et al. 2020; +Park et al. 2019; Zhu et al. 2020; Ling et al. 2021), we trans- +fer the harmonized target style by scaling and shifting the +normalized input feature with the computed factors (i.e., γ +and β). In the layer, we additionally set class-wise alpha pa- +rameters (α1, α2, · · · , αN). It is used to decide which style +has more influence for each class in the generated images. +If the alpha value is large, the component style sx has more +influence, and vice versa. +Let f, of size H × W × C, be the input feature of the +current style harmonization layer in the generator Gy. With +the style harmonizing scale γ and shift β factors, the feature +is denormalized by +γc,h,w +(fc,h,w − µc) +σc ++ βc,h,w +(3) +where µc and σc are the mean and standard deviation of the +input feature f at the channel c, respectively. The modula- +tion parameters γc,h,w and βc,h,w are obtained from γx +c,h,w, +γy +c,h,w, βx +c,h,w, and βy +c,h,w, which are the scale and shift fac- +tors of the component style sx and memory style, sy respec- +tively, and they are computed by +γc,h,w = ˆαh,wγx +c,h,w + (1 − ˆαh,w)γy +c,h,w, +βc,h,w = ˆαh,wβx +c,h,w + (1 − ˆαh,w)βy +c,h,w, +(4) +where ˆα denotes the alpha mask. It is obtained by broadcast- +ing the class-wise alpha parameters to their corresponding +semantic regions of the label Lx. We experimentally demon- +strate that our style harmonization layer adaptively controls +the style of each object well, and the results are given in the +experiments section. +Loss Functions +We leverage standard loss functions used in MUNIT (Huang +et al. 2018b) to generate proper target domain images. It in- +cludes self-reconstruction Lself (Zhu et al. 2017), cycle con- +sistency Lcycle (Zhu et al. 2017), perceptual Lperc (Johnson, +Alahi, and Fei-Fei 2016) and adversarial loss Ladv (Good- +fellow et al. 2014). The detailed explanations of those loss +functions are given in the supplementary material. +In this paper, we propose two advanced loss functions to +facilitate style harmonization: content contrastive loss and +style contrastive loss. It is used with the aforementioned +standard loss functions jointly. In what follows, we intro- +duce the proposed two loss functions. +Content contrastive loss. +To extract domain invariant +content features from the content encoder, MUNIT (Huang +et al. 2018b) simply reduces the L1 distances between cx and +ˆcy. We replace this with contrastive representation learning +to improve discrimination within a class. For a content fea- +ture ˆcy +i at pixel i, which is the content feature extracted from +translated target image, we set the positive sample to cx +i and +we set the remaining features at the other pixels as negative +samples. The content contrastive loss is defined with the a +form of InfoNCE (Oord, Li, and Vinyals 2018) as: +Lcontent = − +HW +� +i=1 +log +� +exp((cx +i · ˆcy +i )/τ) +�HW +j=1 exp((cx +j · ˆcy +i )/τ) +� +(5) +where τ is a temperature parameter. In this equation, the +features in the same class at the pixel i can be considered +as negative samples. This encourages the content encoder +to extract more diverse style representations from the style +memory within the same class. It is also applied to the target- +to-source pipeline with ˆcx and cy +Style constrative loss. +We propose the style contrastive +loss to allow the style memory to learn class-wise style rep- +resentations. Similar to the content contrastive loss, for a +style feature ˆsx +i at pixel i, which is the memory style of +target-to-source mapping, we set the positive sample to the +source component style sx +i and we set the remaining features +at the other pixels as negative samples. The style constrative +loss is defined as follows: +Lstyle = − +HW +� +i=1 +log +� +exp((sx +i · ˆsx +i )/τ) +�HW +j=1 exp((sx +j · ˆsx +i )/τ) +� +(6) +This loss directly supervises the memory style ˆsx for the +translated target image. This improves the stability of cycle +consistency learning for reconstructing Ix from ˆIy. It is also +applied to the target styles, i.e., ˆsy and sy +Finally, all loss functions are summarized as follows: +min +(Ex,Ey,Gx,Gy) max +(Dx,Dy)L(Ex, Ey, Gx, Gy, Dx, Dy) = +λselfLself + λcycleLcycle + λpercLperc ++λadvLadv + λcontentLcontent + λstyleLstyle +(7) +where Dx and Dy denote the multi-scale discrimina- +tors (Wang et al. 2018) for each visual domain, X and Y. +The details of Lself, Lcycle, Lperc, and Ladv are described +in the supplementary material. +Experiments +In this section, we present extensive experimental results and +analysis. To demonstrate the superiority of our method, we +compare our SHUNIT with state-of-the-art I2I translation +methods. The implementation details of our method are pro- +vided in the supplementary material. +Datasets +We evaluate our SHUNIT on three I2I translation scenar- +ios: Cityscapes (Cordts et al. 2016) → ACDC (Sakaridis, +Dai, and Van Gool 2021) and INIT (Shen et al. 2019), +and KITTI (Geiger et al. 2013) → Cityscapes (Cordts +et al. 2016). In all scenarios, INIT (Shen et al. 2019), +DUNIT (Bhattacharjee et al. 2020), MGUIT (Jeong et al. +2021), InstaFormer (Kim et al. 2022), and our method use +semantic labels provided in each dataset. + +Input +CycleGAN +MUNIT +MGUIT +SHUNIT (ours) +Target domain +Figure 5: Qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image (first +column), we generate four adverse condition images using (Zhu et al. 2017; Huang et al. 2018b; Jeong et al. 2021) and SHUNIT. +In the last column, we show a sample of the real image for each adverse condition. +clear → snow +clear → rain +clear → fog +clear → night +cFID ↓ +mIoU ↑ +cFID ↓ +mIoU ↑ +cFID ↓ +mIoU ↑ +cFID ↓ +mIoU ↑ +CycleGAN (Zhu et al. 2017) +21.88 +23.68 +20.16 +35.96 +31.72 +13.73 +15.26 +31.33 +UNIT (Liu, Breuel, and Kautz 2017) +13.89 +31.24 +16.25 +39.39 +29.36 +28.70 +12.28 +35.29 +MUNIT (Huang et al. 2018b) +13.79 +33.83 +12.62 +44.20 +29.34 +27.44 +12.56 +37.43 +TSIT (Jiang et al. 2020) +10.47 +38.08 +14.16 +46.40 +25.16 +36.68 +11.62 +35.92 +MGUIT (Jeong et al. 2021) +8.75 +33.33 +10.76 +42.60 +24.36 +10.22 +15.83 +31.36 +SHUNIT (ours) +6.62 +45.15 +8.47 +48.84 +6.53 +38.96 +14.08 +33.66 +Table 1: Quantitative comparison on Cityscapes → ACDC. We measure class-wise FID (lower is better) and mIoU (higher +is better). For brevity, class-wise FID is written as cFID. +clear → snow +clear → rain +clear → fog +clear → night +AdaptSegNet (Tsai et al. 2018) +35.3 +49.0 +31.8 +29.7 +ADVENT (Vu et al. 2019) +32.1 +44.3 +32.9 +31.7 +BDL (Li, Yuan, and Vasconcelos 2019) +36.4 +49.7 +37.7 +33.8 +CLAN (Luo et al. 2019) +37.7 +44.0 +39.0 +31.6 +FDA (Yang and Soatto 2020) +46.9 +53.3 +39.5 +37.1 +SIM (Wang et al. 2020) +33.3 +44.5 +36.6 +28.0 +MRNet (Zheng and Yang 2021) +38.7 +45.4 +38.8 +27.9 +SHUNIT (ours) +45.2 +48.8 +39.0 +33.7 +Table 2: Quantitative Comparison on domain adaptation for semantic segmentation. We report mIoU for Cityscapes → +ACDC. +Cityscapes → ACDC +Cityscapes (Cordts et al. 2016) +is one of the most popular urban scene dataset. ACDC +(Sakaridis, Dai, and Van Gool 2021) is the latest dataset +with multiple adverse condition images and consists of +four conditions of street scenes: snow, rain, fog, and night. +ACDC dataset provides images with corresponding dense +pixel-level semantic annotations, and it has 19 classes the +same as Cityscapes dataset for all adverse conditions. Fol- +lowing (Sakaridis, Dai, and Van Gool 2021), we leverage +Cityscapes dataset as a clear condition and translate it to the +adverse conditions (i.e., snow, rain, fog, and night) in ACDC +dataset. Therefore, this scenario is challenging because not +only the weather conditions, but also layouts, such as cam- +era model, view, and angle, are different. To train the net- +works, 2975, 400, 400, 400, and 400 images are used for +clear, snow, rain, fog, and night conditions, respectively. For +a fair comparison on this benchmark, we reproduce existing +state-of-the-art methods (Zhu et al. 2017; Liu, Breuel, and +Kautz 2017; Huang et al. 2018b; Jiang et al. 2020; Jeong +et al. 2021) in our system. For fair comparison, we set the +number of key-value pairs for style memory to be the same +as our setting and use segmentation mask for reproducing + +国Input +CycleGAN +UNIT +MUNIT +DRIT +MGUIT +InstaFormer +SHUNIT (ours) +Figure 6: Qualitative comparison on INIT dataset. (Top to bottom) sunny→night, night→sunny, cloudy→sunny results. Our +method preserves object details and looks more realistic. +sunny → night +night → sunny +sunny → rainy +sunny → cloudy +cloudy → sunny +Average +CIS ↑ +IS ↑ +CIS ↑ +IS ↑ +CIS ↑ +IS ↑ +CIS ↑ +IS ↑ +CIS ↑ +IS ↑ +CIS ↑ +IS ↑ +CycleGAN (Zhu et al. 2017) +0.014 +1.026 +0.012 +1.023 +0.011 +1.073 +0.014 +1.097 +0.090 +1.033 +0.025 +1.057 +UNIT (Liu, Breuel, and Kautz 2017) +0.082 +1.030 +0.027 +1.024 +0.097 +1.075 +0.081 +1.134 +0.219 +1.046 +0.087 +1.055 +MUNIT (Huang et al. 2018b) +1.159 +1.278 +1.036 +1.051 +1.012 +1.146 +1.008 +1.095 +1.026 +1.321 +1.032 +1.166 +DRIT (Lee et al. 2018) +1.058 +1.224 +1.024 +1.099 +1.007 +1.207 +1.025 +1.104 +1.046 +1.321 +1.031 +1.164 +INIT (Shen et al. 2019) +1.060 +1.118 +1.045 +1.080 +1.036 +1.152 +1.040 +1.142 +1.016 +1.460 +1.043 +1.179 +DUNIT (Bhattacharjee et al. 2020) +1.166 +1.259 +1.083 +1.108 +1.029 +1.225 +1.033 +1.149 +1.077 +1.472 +1.079 +1.223 +MGUIT (Jeong et al. 2021) +1.176 +1.271 +1.115 +1.130 +1.092 +1.213 +1.052 +1.218 +1.136 +1.489 +1.112 +1.254 +Instaformer (Kim et al. 2022) +1.200 +1.404 +1.115 +1.127 +1.158 +1.394 +1.130 +1.257 +1.141 +1.585 +1.149 +1.353 +SHUNIT (ours) +1.205 +1.503 +1.308 +1.585 +1.136 +1.609 +1.111 +1.405 +1.085 +1.315 +1.169 +1.483 +Table 3: Quantitative Comparison on INIT dataset. We measure CIS and IS (higher is better). +(Jeong et al. 2021). +INIT +INIT (Shen et al. 2019) is a public benchmark set for +I2I translation. It contains street scenes images including 4 +weather categories (i.e., sunny, night, rainy, and cloudy) with +the corresponding bounding box labels. Following (Shen +et al. 2019), we split the 155K images into 85% for train- +ing and 15% for testing. We conduct five translation exper- +iments: sunny ↔ night, sunny ↔ cloudy, sunny → rainy. +In this dataset, we directly copied the results of the existing +methods from (Shen et al. 2019; Bhattacharjee et al. 2020; +Jeong et al. 2021; Kim et al. 2022). Similarly, for fair com- +parison with MGUIT, the number of key-value pairs in style +memory is set equally. +KITTI → Cityscapes +KITTI is a public benchmark set +for object detection. It contains 7481 images with bounding +boxes annotations for training and 7518 images for testing. +Following the previous I2I translation methods (Bhattachar- +jee et al. 2020; Jeong et al. 2021; Kim et al. 2022), we select +the common 4 object classes (person, car, truck, bicycle) for +evaluatation. +Qualitative Comparison +Fig. 5 shows qualitative results on Cityscapes → ACDC. +Since our I2I translation setting, Cityscapes → ACDC, is +very challenging as discussed in the datasets section, ex- +isting methods cannot generate realistic images in several +scenarios. Specifically, CycleGAN (Zhu et al. 2017) often +destroys the semantic layout. MUNIT (Huang et al. 2018b) +translates images with a global style, thus it also often gen- +erates artifacts, as shown in the snow, rain, and fog images. +MGUIT (Jeong et al. 2021) also includes artifacts in the car +even though leveraging memory style. It shows the limita- +tion of the updating mechanism for training memory style, +and the limitation is clearly depicted in the challenging sce- +nario. In contrast to them, our SHUNIT accurately generates +images in the target domains without losing the original style +in the input image. In the supplementary material, we further +provide the results of UNIT (Liu, Breuel, and Kautz 2017) +and TSIT (Jiang et al. 2020). +As shown in Fig. 6, which depicts qualitative results on +INIT dataset, our method generates high-quality images in +various scenarios. In the night → sunny scenario (second +row), InstaFormer (Kim et al. 2022) translates the color of +the road lane to yellow. On the other hand, our method keeps +the color of the lane as white and generates a sunny scene by +harmonizing the target domain style retrieved from a style +memory and an image style. +Quantitative Comparison +The quantitative results on Cityscapes → ACDC are pre- +sented in Table 1. To quantify the per-class image-to-image +translation quality, we measure class-wise FID (Shim et al. +2022). We further measure mIoU on ACDC test set. The + +Pers. +Car +Truc. +Bic. +mAP +DT (Inoue et al. 2018) +28.5 +40.7 +25.9 +29.7 +31.2 +DAF (Huang et al. 2018b) +39.2 +40.2 +25.7 +48.9 +38.5 +DARL (Kim et al. 2019) +46.4 +58.7 +27.0 +49.1 +45.3 +DAOD (Rodriguez and Mikolajczyk 2019) +47.3 +59.1 +28.3 +49.6 +46.1 +DUNIT (Bhattacharjee et al. 2020) +60.7 +65.1 +32.7 +57.7 +54.1 +MGUIT (Jeong et al. 2021) +58.3 +68.2 +33.4 +58.4 +54.6 +InstaFormer (Kim et al. 2022) +61.8 +69.5 +35.3 +55.3 +55.5 +SHUNIT (ours) +56.3 +74.4 +51.9 +53.2 +59.0 +Table 4: Quantitative Comparison on domain adaptation for object detection. We report per-class AP for KITTI → +Cityscapes. +clear → snow +clear → rain +Mem. Comp. +α +cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ +✓ +12.93 +26.33 +7.83 +39.06 +✓ +✓ +8.72 +37.20 +8.47 +38.97 +✓ +✓ +✓ +6.62 +44.35 +8.47 +42.77 +(a) Style ablation study on SHL. +clear → snow +clear → rain +Lcontent +Lstyle +cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ +✓ +11.06 +32.43 +13.23 +38.39 +✓ +12.38 +30.89 +9.33 +39.47 +✓ +✓ +6.62 +44.35 +8.47 +42.77 +(b) Ablation study on loss functions. +clear → snow +clear → rain +cFID ↓ +mIoU ↑ +cFID ↓ +mIoU ↑ +Updating +10.55 +36.70 +12.05 +38.21 +Backprop. +6.62 +44.35 +8.47 +42.77 +(c) Style memory training strategies. +clear → snow +clear → rain +cFID ↓ +mIoU ↑ +cFID ↓ +mIoU ↑ +L1 +12.21 +38.98 +7.72 +39.78 +Contrastive +6.62 +44.35 +8.47 +42.77 +(d) L1 vs. Contrastive loss for style and content +losses. +clear → snow +clear → rain +cFID ↓ mIoU ↑ +cFID ↓ mIoU ↑ +w/o +6.73 +44.11 +9.03 +38.72 +w/ +6.62 +44.35 +8.47 +42.77 +(e) Experimental results of label input for +content encoder. +Table 5: Ablation Study. We report class-wise FID and mIoU in two scenarios: clear → {snow, rain}. +mIoU metric is used to validate the results on the practical +problem, semantic segmentation. We generate a training set +by Cityscapes → ACDC and then train DeepLabV2 (Chen +et al. 2017) on it. The mIoU score is obtained with the +trained DeepLabV2 by evaluating on ACDC test set. As +shown in Table 1, we surpass the state-of-the-art I2I trans- +lation methods by a significant margin in most scenarios, +demonstrating the superiority of our style harmonization for +unpaired I2I translation. Table 2 shows the quantitative re- +sults of domain adaptation for semantic segmentation with +the state-of-the-art methods. Despite we trained DeepLabV2 +with only a simple cross-entropy loss, our SHUNIT achieves +comparable performance with the domain adaptation meth- +ods that were trained DeepLabV2 with additional loss func- +tions and several techniques for boosting the performance of +mIoU score on the target domain. +Table 3 shows another quantitative results on the testing +split of INIT (Shen et al. 2019). To directly compare our +method with the public results, we evaluate our method with +Inception Score (IS) (Salimans et al. 2016) and Conditional +Inception Score (CIS) (Huang et al. 2018b). As shown in +Table 3, we achieve the best performance in most scenarios. +We further evaluate our method on domain adaptation +benchrmark following DUNIT (Bhattacharjee et al. 2020). +We use Faster-RCNN (Ren et al. 2015) trained on the source +domain as a detector. As shown in Table 4, we achieve the +state-of-the-art performance. +Ablation Study +In this section, we study the effectiveness of each component +in our method. We validate on Cityscapes (clear) → ACDC +(snow/rain) scenarios and use ACDC validation set for both +class-wise FID and mIoU1. +Ablation study on style harmonization layer. +We ablate +the memory style, component style, and class-wise α in the +style harmonization layer, and they are denoted as “Mem.”, +“Comp.”, and “α” in Table 5a, respectively. As shown in the +table, the memory style-only is far behind the full model. +With component style, we can achieve performance im- +provement on clear → snow while decreasing on clear → +rain. We obtain significant improvement on most scenarios +with class-wise α. The results demonstrate that the exist- +ing approach, which only leverages the memory style, is not +sufficient for I2I translation, and we successfully address the +problem by adaptively harmonizing two styles. +Ablation study on content and style losses. +We study the +effectiveness of the proposed two loss functions, Lstyle and +Lcontent, by ablating them step-by-step, and the results are +1mIoU on test set should be evaluated on the online server +(Sakaridis, Dai, and Van Gool 2021) and it has a limit on the num- +ber of submissions. Therefore, we use validation set for ablation +study. + +given in Table 5b. As shown in the table, our model is effec- +tive when two losses are used jointly. +Style memory training strategy. +As described in the pro- +posed method section, we opt for backpropagation to train +the style memory rather than updating the mechanism used +in (Jeong et al. 2021). The results are shown in Table 5c. We +surpass the existing updating method by a large margin. +L1 vs. Contrastive loss. +Table 5d shows the efficacy of our +contrastive-based approach by replacing Lcontent and Lstyle +with L1 losses as used in MUNIT (Huang et al. 2018b). The +L1 losses are designed to reduce L1 distances within posi- +tive pairs without consideration of negative pairs. As we dis- +cussed in loss functions section, our method effectively en- +courages extracting more diverse style representations, lead- +ing to performance improvement. +Label input for content encoder. +Table 5e shows that la- +bel input is not significant but always leads to performance +improvements. Therefore, we have no reason to omit the la- +bel input. +Limitations +Since our framework leverages the component style of the +source image, the generated image’s quality relies on the +source image’s quality. If the source image has a very bright +colors, SHUNIT often generates a relatively bright night im- +age. In Fig. 5, SHUNIT struggles to generate geometrically +distinct lights from the source image. Due to the above prob- +lems, SHUNIT cannot achieve the best performance on clear +→ night scenario in Table. 1. We believe that these problems +can be alleviated by leveraging geometric information such +as depth or camera pose. Additionally, our approach can give +limited benefit to some I2I translation scenarios, such as dog +→ cat, because these tasks need to change the content; how- +ever, we tackle the unpaired I2I translation task under the +condition that the content will not be changed, and only the +style will be changed. +Conclusion +We present a new perspective of the target style: It can +be disentangled into class-aware and image-specific styles. +Furthermore, our SHUNIT effectively harmonizes the two +styles, and its superiority is demonstrated through extensive +experiments. We believe that our proposal has the potential +to break new ground in style-based image editing applica- +tions such as style transfer, colorization, and image inpaint- +ing. +Acknowledgement +This work was supported by Institute of Information & +communications Technology Planning & Evaluation (IITP) +grant funded by the Korea government (MSIT) (No.2021- +0-00800, Development of Driving Environment Data Trans- +formation and Data Verification Technology for the Mutual +Utilization of Self-driving Learning Data for Different Vehi- +cles). +References +Antoniou, A.; Storkey, A.; and Edwards, H. 2017. Data aug- +mentation generative adversarial networks. arXiv preprint +arXiv:1711.04340. +Bhattacharjee, D.; Kim, S.; Vizier, G.; and Salzmann, M. +2020. Dunit: Detection-based unsupervised image-to-image +translation. In CVPR, 4787–4796. +Chen, L.-C.; Papandreou, G.; Kokkinos, I.; Murphy, K.; and +Yuille, A. L. 2017. 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Our +models are trained with a single NVIDIA RTX A6000 GPU. +For the experiments on Cityscapes (Cordts et al. 2016) → +ACDC (Sakaridis, Dai, and Van Gool 2021), we use RGB +images with a size of 512 × 256 during both training and +inference. We train the model for 50K iterations. Since the +ACDC dataset has been published recently, no official re- +sults are available on this dataset for the existing meth- +ods. Therefore, we reproduced CycleGAN (Zhu et al. 2017), +UNIT (Liu, Breuel, and Kautz 2017), MUNIT (Huang et al. +2018b), TSIT (Jiang et al. 2020), and MGUIT (Jeong et al. +2021) with their official implementation code1, 2, 3, 4. +For the experiments on INIT (Shen et al. 2019), we use +RGB images with a size of 360 × 360 during training while +we use RGB images with a size of 572 × 360 during infer- +ence. We train the model for 250K iterations. For the ex- +periment on KITTI (Geiger et al. 2013) → Cityscapes, we +use RGB images with a size of 540 × 360 during training +while we use RGB images with a size of 1242 × 375 during +inference. We train the model for 200K iterations. +For fair comparisons with INIT (Shen et al. 2019), +DUNIT (Bhattacharjee et al. 2020), MGUIT (Jeong et al. +2021), and InstaFormer (Kim et al. 2022), which used +bounding-box labels as an additional object annotation, +we did not use segmentation labels but used bounding- +box labels for all experiments on both INIT and KITTI +→ Cityscapes scenarios. In addition, we reproduced +MGUIT (Jeong et al. 2021) on Cityscapes → ACDC exper- +iments with segmentation labels. +Details of Network Architecture +In the content encoder Ex +c , we employ two convolutional +networks, one network takes RGB input, and the other takes +one-hot encoded semantic label input. Each network con- +sists of Conv(input channel, 64, 7, 1, 3, IN, relu) - Conv(64, +128, 4, 2, 1, IN, relu) - Conv(128, 256, 4, 2, 1, IN, relu) - +Resblk ×4 - Conv(256, 128, 1, 1, IN, relu), where IN de- +notes the instance normalization layer (Ulyanov, Vedaldi, +and Lempitsky 2016); Resblk denotes the residual block (He +et al. 2016); and the components in Conv(·) denotes (input +channel, output channel, kernel size, stride, padding, nor- +malization, activation). The encoded RGB and semantic la- +bel features are then concatenated along the channel dimen- +sion, and it is the content feature cx. The style encoder Ex +s +1The code of CycleGAN was taken from +https://github.com/junyanz/pytorch-CycleGAN-and-pix2pix +2The code of UNIT and MUNIT were taken from +https://github.com/NVlabs/MUNIT +3The code of TSIT was taken from +https://github.com/EndlessSora/TSIT +4The code of MGUIT was taken from +https://github.com/somijeong/MGUIT +has the same network architecture as a convolutional net- +work in the content encoder, except for the normalization +layer. The style encoder does not use normalization layers +to keep the original style of the input image. The generator +Gy consists of SH Resblk ×4 - Conv(256, 128, 5, 1, 2, IN, +relu) - Conv(128, 64, 5, 1, 2, IN, relu) - Conv(64, 3, 7, 1, 3, -, +tanh). For the experiments on Cityscapes → ACDC, we use +20 key-value pairs for each class in the style memory M y +n. +In the rest of the experiments, we follow key-value pairs of +MGUIT (Jeong et al. 2021). +Details of Loss Functions +Following MUNIT (Huang et al. 2018b), we leverage self- +reconstruction Lself (Zhu et al. 2017), cycle consistency +Lcycle (Zhu et al. 2017), perceptual Lperc (Johnson, Alahi, +and Fei-Fei 2016), and adversarial Ladv (Goodfellow et al. +2014) losses as follows: +Lself = ∥Ix − Gx(cx, sx)∥1 + ∥Iy − Gy(cy, sy)∥1 , (8) +Lcycle = ∥Ix − Gx(ˆcy, ˆsx)∥1 + ∥Iy − Gy(ˆcx, ˆsy)∥1 , (9) +Lperc = +���F(Ix) − F( ˆIy) +��� +1 + +���F(Iy) − F( ˆIx) +��� +1 , (10) +Ladv = +� +log (1 − Dx(ˆIx)) + log Dx(Ix) +� ++ +� +log (1 − Dy(ˆIy)) + log Dy(Iy) +� +, +(11) +where F(·) is a feature map extracted from relu5 3 +layer of the ImageNet pretrained VGG-16 network (Si- +monyan and Zisserman 2014); Dx(·) and Dy(·) are +features +extracted +from +the +multi-scale +discrimina- +tors (Wang et al. 2018) for each visual domain, X +and Y, respectively. We empirically set the hyperpa- +rameters +{λself, λcycle, λperc, λadv, λcontent, λstyle} +to +{10, 10, 1, 1, 10, 10}. The temperature τ in Lcontent and +Lstyle is set to 0.7. +Analysis of class-wise FID +Recent work (Shim et al. 2022) introduced class-wise FID +(cFID) to quantify the per-class image quality in the image +generation task. We realized that the I2I translation task also +has advantages if we evaluate the results with cFID. In this +section, we provide a detailed analysis of the necessity for +cFID on our benchmark set and the implementation detail of +cFID for I2I translation. +Necessity of CFID. +By the nature of unsupervised I2I, as +shown in Fig. 7, the class statistics between the source do- +main (Cityscapes (Cordts et al. 2016)) and the target do- +main (ACDC (Sakaridis, Dai, and Van Gool 2021)) are un- +matched. However, FID ignores the unmatched class statis- +tics because FID only uses the global embeddings to com- +pute the distance. Therefore, the different class statistics pre- +vent FID improvement, regardless of the generated image +quality. + +Figure 7: The class statistics in Cityscapes (Cordts et al. 2016) validation set vs. ACDC (Sakaridis, Dai, and Van Gool +2021) validation set. Even though the two datasets share the class categories, the class statistics are naturally different. +To demonstrate the problem of FID intuitively, we il- +lustrate an example in Fig. 8. In the figure, we translate +a clear image sampled from Cityscapes dataset to snow +with three methods: (a) CycleGAN (Zhu et al. 2017), (b) +MGUIT (Jeong et al. 2021), and (c) SHUNIT. As can be +compared qualitatively, SHUNIT generates a more realis- +tic snow image than CycleGAN. For a quantitatively com- +parison, we take a similar strategy used in FID: We extract +global embeddings from a real snow image sampled from +ACDC dataset and the generated image using Inception- +V3 (Szegedy et al. 2016). Then, we compute the squared Eu- +clidean distance between the two global embeddings. How- +ever, as depicted in Fig. 8, CycleGAN (267.47) achieves bet- +ter performance than SHUNIT (295.45) in global distance. +The reason is that CycleGAN destroys the layout in the input +image and follows the layout of the real snow image, result- +ing in severe artifacts in the generated image while better +performance in the global distance. If we extract class-wise +embeddings and compute class-wise distance then average +them, we can obtain a reasonable rank: SHUNIT (6.64) +achieves the best, MGUIT (10.61) achieves the second-best, +and CycleGAN (31.07) achieves the third. As can be ob- +served in this example, comparing class-wise embeddings is +a more reasonable method to evaluate the generated qual- +ity than comparing global embeddings in unsupervised I2I +translation. +To further demonstrate the necessity of cFID, we plot the +pixel-wise features of Inception-V3 (Szegedy et al. 2016) in +Fig. 9. In Figs. 9(a), 9(b), and 9(c), Cityscapes validation +set is used as clear images for I2I translation. In Fig. 9(d), +ACDC validation set is used as real snow images. In the +figure, we use the same color for the same category of the +pixel-wise feature. As shown in the zoom-in regions, pixel- +wise features extracted from (d) real snow images are well +separated by class categories because the pretrained network +is used. However, (a) CycleGAN includes various categories +in the zoom-in region, which clearly indicates that the qual- +ity of the generated images is low. In contrast, (c) features in +SHUNIT are well separated by class categories as in (d) real +snow images. This class information should be considered +to evaluate the quality of the generated images. +However, as shown in Table 6, CycleGAN achieves the +best performance of FID in clear → snow scenario. There- +fore, FID score is not reliable in unsupervised I2I transla- +tion with multiple classes. In this paper, we simply and ef- +fectively address the problem by computing class-wise dis- +tances separately. +Implementation detail of cFID for I2I translation. +We +measure cFID based on the official implementation of +FID (Seitzer 2020). cFID is calculated with the Inception- +V3 (Szegedy et al. 2016) features. We extract the features +from the first block of the network to obtain fine-scale fea- +tures, which effectively include features of small objects. To +extract class-wise embeddings, we upsample the features to +their original input size by bilinear interpolation, and then +semantic region pooling is applied to the features. Here, we +use the ground truth of the semantic segmentation for the +semantic region pooling. With the class-wise embeddings, +we compute FID for each class separately, and then cFID is +obtained by averaging them. +Stability of SHUNIT +To show the stability of our model, we train SHUNIT five +times with different random seeds on Cityscapes (clear) → +ACDC (snow, rain), and the results are given in Table 7. +As shown in the table, the performance gap between the +best and worst results is reasonably small. In addition, the +worst result also achieves the state-of-the-art performance +over previous works. This result demonstrates that our re- +sults for quantitative comparison are not cherry-picked and +actually lead to performance improvements. Furthermore, +the result promises that our strong performance can be easily +reproduced. + +Semantics statistics +Cityscapes val +ACDC val +0.30 +0.25 +semantic ratio +0.20 +0.15 +0.10 +0.05 +0.00 +0 +15 +16 +17 +1819I2I +Global distance: 267.47 +Class-wise distance: 31.07 +Global distance: 357.54 +Class-wise distance: 10.61 +Global distance: 295.45 +Class-wise distance: 6.64 +(a) CycleGAN (Zhu et al. 2017) +Input +(b) MGUIT (Jeong et al. 2021) +(d) Target domain +(c) SHUNIT (ours) +Figure 8: Illustration of comparison of global distance and class-wise distance. From a clear image, we generate snow +images using (a) CycleGAN (Zhu et al. 2017), (b) MGUIT (Jeong et al. 2021), and (c) SHUNIT. To quantitatively evaluate the +results, we compute distances between (a,b,c) the generated images and (d) real snow image in two methods: One is computed +with global embeddings of Inception-V3 and the other one is computed with class-wise embeddings. +clear → snow +clear → rain +clear → fog +clear → night +FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑ +CycleGAN (Zhu et al. 2017) +134.21 21.88 +23.68 160.91 20.16 +35.96 154.23 31.72 +13.73 105.40 15.26 +31.33 +UNIT (Liu, Breuel, and Kautz 2017) 136.63 13.89 +31.24 145.79 16.25 +39.39 190.47 29.36 +28.70 116.39 12.28 +35.29 +MUNIT (Huang et al. 2018b) +136.91 13.79 +33.83 151.48 12.62 +44.20 195.28 29.34 +27.44 114.94 12.56 +37.43 +TSIT (Jiang et al. 2020) +162.19 10.47 +38.08 158.12 14.16 +46.40 197.62 25.16 +36.68 127.09 11.62 +35.92 +MGUIT (Jeong et al. 2021) +166.76 +8.75 +33.33 178.59 10.76 +42.60 190.86 24.36 +10.22 119.11 15.83 +31.36 +SHUNIT (ours) +143.21 +6.62 +45.15 158.96 +8.47 +48.84 153.19 +6.53 +38.96 125.80 14.08 +33.66 +Table 6: Quantitative comparison on Cityscapes → ACDC. We measure FID, CFID (lower is better) and mIoU (higher is +better). +More Qualitative Results +Cityscapes → ACDC +We show more qualitative results on Cityscapes (clear) → +ACDC (snow/rain/fog/night) in Figs 10 to 13. In the results, +we additionally show the results of UNIT (Liu, Breuel, and +Kautz 2017) and TSIT (Jiang et al. 2020), which are omitted +in the main paper. The qualitative results demonstrate that +our method generate more realistic images than the previ- +ous works (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; +Huang et al. 2018b; Jiang et al. 2020; Jeong et al. 2021) in +most cases. We further provide a comparison video in the +supplementary material. +KITTI → Cityscapes +Fig. 14 shows the visual comparison of InstaFormer (Kim +et al. 2022) and our method for domain adaptive detection on +KITTI → Cityscapes. As shown in the figure, InstaFormer +generates artifacts in the sky and struggles to maintain the +sub-object components of a small object, such a car located +in the center. In contrast, our method translates the sky to +the Cityscapes style without artifacts, and preserves the sub- +objects components of a small object. The more realistic +image translation is demonstrated by a detection result: the +Faster-RCNN (Ren et al. 2015) detects the small car located +in the center in our image (Fig. 14c) while cannot detect it +in InstaFormer image (Fig. 14b) +More Quantitative Results +Tables 8 to 11 provide the per-class IoU performance on +ACDC test set. As shown in the tables, we achieve the best +performance not only in mIoU but also in per-class IoU of +the most classes on snow, rain, and fog. + +zoom-in +zoom-in +zoom-in +zoom-in +zoom-in +zoom-in +zoom-in +zoom-in +(a) CycleGAN (Zhu et al. 2017) +(b) MGUIT (Jeong et al. 2021) +(c) SHUNIT (ours) +(d) Target domain +Figure 9: t-SNE (Van der Maaten and Hinton 2008) visualization. We plot pixel-wise features extracted from Inception- +V3 (Szegedy et al. 2016). In (a), (b), and (c), the generated snow images, which are translated from clear images, are used for +extracting features, while in (d), the real snow images in the ACDC validation set are used. In the figure, the same color of dots +are in the same class category. +clear → snow +clear → rain +cFID ↓ +mIoU ↑ +cFID ↓ +mIoU ↑ +CycleGAN (Zhu et al. 2017) +21.88 +23.68 +20.16 +35.96 +UNIT (Liu, Breuel, and Kautz 2017) +13.89 +31.24 +16.25 +39.39 +MUNIT (Huang et al. 2018b) +13.79 +33.83 +12.62 +44.20 +TSIT (Jiang et al. 2020) +10.47 +38.08 +14.16 +46.40 +MGUIT (Jeong et al. 2021) +8.75 +33.33 +10.76 +42.60 +SHUNIT (trial 1, reported in the main paper) +6.62 +45.15 +8.47 +48.84 +SHUNIT (trial 2) +7.18 +44.71 +9.53 +49.14 +SHUNIT (trial 3) +6.41 +45.94 +8.44 +48.01 +SHUNIT (trial 4) +7.56 +38.41 +8.34 +48.67 +SHUNIT (trial 5) +6.86 +45.41 +8.56 +48.70 +Table 7: Results of five trials on Cityscapes → ACDC. We +measure class-wise FID (lower is better) and mIoU (higher +is better). In our result, the best results are bold-faced while +the worst results are red-colored. For brevity, class-wise FID +is written as cFID. + +oo +1 +. +:Methods +road +sidewalk +building +wall +fence +pole +traffic light +traffic sign +vegetation +terrain +sky +person +rider +car +truck +bus +train +motorcycle +bicycle +mIoU +CycleGAN (Zhu et al. 2017) +54.55 12.74 41.24 8.42 9.71 11.78 23.77 28.34 38.86 1.68 2.07 17.85 17.20 62.00 6.05 34.29 50.93 8.94 19.57 23.68 +UNIT (Liu, Breuel, and Kautz 2017) 58.88 16.27 43.10 9.74 14.35 17.47 38.47 39.71 47.97 6.98 4.17 41.25 30.11 73.35 20.49 36.20 60.74 6.75 27.48 31.24 +MUNIT (Huang et al. 2018b) +65.09 21.06 44.42 13.53 18.20 19.04 41.24 40.68 51.58 7.94 5.85 41.21 27.10 77.44 24.08 31.01 60.19 18.93 34.14 33.83 +TSIT (Jiang et al. 2020) +79.32 42.91 47.03 13.01 18.48 21.82 45.05 42.71 55.98 10.05 7.55 47.79 39.11 78.97 19.21 49.75 58.25 19.55 27.07 38.08 +MGUIT (Jeong et al. 2021) +69.14 31.58 49.37 7.13 6.84 18.31 28.06 32.98 64.92 9.20 51.08 32.05 37.06 74.26 4.57 30.31 60.42 6.81 19.15 33.33 +SHUNIT(ours) +81.62 46.12 52.33 16.03 27.34 25.07 49.22 43.19 80.08 9.41 54.61 52.95 36.23 77.66 20.22 48.18 69.04 26.47 43.12 45.15 +Table 8: Per-class IoU results on Cityscapes (clear) → ACDC (snow). +Methods +road +sidewalk +building +wall +fence +pole +traffic light +traffic sign +vegetation +terrain +sky +person +rider +car +truck +bus +train +motorcycle +bicycle +mIoU +CycleGAN (Zhu et al. 2017) +73.51 31.73 51.75 20.75 18.66 20.99 24.80 26.39 71.58 24.46 28.91 34.06 11.75 74.66 37.32 23.02 52.42 16.18 40.19 35.96 +UNIT (Liu, Breuel, and Kautz 2017) 74.20 37.16 58.67 24.76 17.98 23.51 38.65 44.07 58.10 20.11 18.63 40.51 15.17 78.89 41.53 40.16 57.62 18.57 40.20 39.39 +MUNIT (Huang et al. 2018b) +77.68 40.79 71.64 20.81 27.55 26.40 44.48 43.77 67.15 24.84 55.84 42.86 15.69 79.60 41.18 38.71 57.21 20.00 43.53 44.20 +TSIT (Jiang et al. 2020) +78.08 46.43 71.28 29.44 25.12 29.23 44.95 46.64 78.14 24.68 67.75 44.24 16.37 80.94 46.28 31.09 57.90 25.87 41.74 46.40 +MGUIT (Jeong et al. 2021) +73.69 31.94 80.06 19.54 14.44 24.74 37.17 34.91 79.99 21.61 93.74 34.46 13.60 76.69 27.54 38.64 60.43 16.32 31.04 42.60 +SHUNIT(ours) +76.41 28.42 82.03 23.44 28.12 28.81 46.15 46.78 86.54 36.26 94.36 41.39 12.79 80.74 46.90 25.71 58.57 29.93 45.09 48.84 +Table 9: Per-class IoU results on Cityscapes (clear) → ACDC (rain). +Methods +road +sidewalk +building +wall +fence +pole +traffic light +traffic sign +vegetation +terrain +sky +person +rider +car +truck +bus +train +motorcycle +bicycle +mIoU +CycleGAN (Zhu et al. 2017) +46.89 35.49 33.79 27.63 27.07 17.65 14.74 10.70 7.82 7.14 6.96 5.39 5.28 3.81 3.08 2.57 2.28 2.25 0.32 13.73 +UNIT (Liu, Breuel, and Kautz 2017) 64.21 26.65 30.18 16.54 11.36 10.77 23.89 30.93 31.44 25.16 1.68 14.22 4.55 58.45 34.28 45.42 41.64 27.20 5.79 28.70 +MUNIT (Huang et al. 2018b) +54.73 26.48 28.43 14.60 8.54 7.36 30.30 29.62 33.86 24.07 3.95 8.33 32.37 56.23 30.05 58.80 51.89 12.87 8.93 27.44 +TSIT (Jiang et al. 2020) +80.10 46.44 19.39 14.77 15.83 16.11 27.27 39.82 72.30 44.35 2.44 25.78 41.78 65.67 41.98 49.48 42.54 31.12 19.71 36.68 +MGUIT (Jeong et al. 2021) +54.26 11.66 21.85 5.81 0.89 8.73 3.86 11.28 22.51 7.30 1.37 1.97 3.08 34.31 1.08 0.05 3.86 0.00 0.34 10.22 +SHUNIT(ours) +72.35 43.36 41.68 21.42 18.13 16.08 39.11 35.90 71.56 35.10 67.93 27.53 50.45 52.55 37.31 37.06 28.32 27.64 16.57 38.96 +Table 10: Per-class IoU results on Cityscapes (clear) → ACDC (fog). +Methods +road +sidewalk +building +wall +fence +pole +traffic light +traffic sign +vegetation +terrain +sky +person +rider +car +truck +bus +train +motorcycle +bicycle +mIoU +CycleGAN (Zhu et al. 2017) +86.03 46.64 60.57 24.69 15.16 26.56 11.75 31.92 42.44 27.05 4.47 36.84 10.44 61.32 0.04 15.97 51.54 16.27 25.51 31.33 +UNIT (Liu, Breuel, and Kautz 2017) 86.03 46.74 63.33 26.69 14.80 28.41 19.25 30.92 43.94 35.14 15.34 40.08 19.55 64.43 2.51 30.81 44.15 29.44 29.00 35.29 +MUNIT (Huang et al. 2018b) +87.73 68.58 66.92 54.84 51.74 44.49 42.45 35.94 33.97 32.46 30.67 29.12 27.61 26.24 20.50 19.91 18.52 16.51 3.01 37.43 +TSIT (Jiang et al. 2020) +88.69 56.81 63.49 21.67 16.59 27.28 21.60 33.81 43.53 38.21 3.50 42.42 19.94 67.13 4.08 30.27 45.36 25.55 32.62 35.92 +MGUIT (Jeong et al. 2021) +86.67 44.75 66.36 24.50 16.13 28.26 8.87 32.15 42.24 8.32 8.66 37.88 10.85 61.65 2.38 19.17 46.22 20.52 30.16 31.36 +SHUNIT(ours) +85.65 44.84 67.98 24.53 18.98 26.66 20.57 30.32 45.02 26.21 16.61 39.13 14.40 62.68 0.56 21.13 42.92 23.59 27.76 33.66 +Table 11: Per-class IoU results on Cityscapes (clear) → ACDC (night). + +Input +CycleGAN +UNIT +MUNIT +TSIT +MGUIT +SHUNIT (ours) +Target domain +Figure 10: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image +(first row), we generate four adverse condition images using (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; Huang et al. 2018b; +Jiang et al. 2020; Jeong et al. 2021) and SHUNIT (top row to bottom row order). In the last row, we show a sample of the real +image for each adverse condition. + +Input +CycleGAN +UNIT +MUNIT +TSIT +MGUIT +SHUNIT (ours) +Target domain +Figure 11: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image +(first row), we generate four adverse condition images using (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; Huang et al. 2018b; +Jiang et al. 2020; Jeong et al. 2021) and SHUNIT (top row to bottom row order). In the last row, we show a sample of the real +image for each adverse condition. + +Input +CycleGAN +UNIT +MUNIT +TSIT +MGUIT +SHUNIT (ours) +Target domain +Figure 12: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image +(first row), we generate four adverse condition images using (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; Huang et al. 2018b; +Jiang et al. 2020; Jeong et al. 2021) and SHUNIT (top row to bottom row order). In the last row, we show a sample of the real +image for each adverse condition. + +HOTEL-RESTAURANT +HAGNAUERHOFInput +CycleGAN +UNIT +MUNIT +TSIT +MGUIT +SHUNIT (ours) +Target domain +Figure 13: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night). From the given clear image +(first row), we generate four adverse condition images using (Zhu et al. 2017; Liu, Breuel, and Kautz 2017; Huang et al. 2018b; +Jiang et al. 2020; Jeong et al. 2021) and SHUNIT (top row to bottom row order). In the last row, we show a sample of the real +image for each adverse condition. + +(b) InstaFormer +(a) Input image (KITTI) +(c) SHUNIT (ours) +Figure 14: Qualitative comparison on domain adaptive detection for KITTI → Cityscapes. Given (a) input image (KITTI), +(b) and (c) show the translated image and the object detection result generated by InstaFormer and our method, respectively. +Note that the results are taken from InstaFormer paper. + diff --git a/IdE3T4oBgHgl3EQfuguG/content/tmp_files/load_file.txt b/IdE3T4oBgHgl3EQfuguG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a93c7ae8d909a8133f6a1bdf29a0d610dd191a3 --- /dev/null +++ b/IdE3T4oBgHgl3EQfuguG/content/tmp_files/load_file.txt @@ -0,0 +1,2345 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf,len=2344 +page_content='SHUNIT: Style Harmonization for Unpaired Image-to-Image Translation Seokbeom Song1, Suhyeon Lee1, Hongje Seong1, Kyoungwon Min2, and Euntai Kim1* 1Yonsei University, Seoul, Korea 2Korea Electronics Technology Institute, Seongnam, Korea {lgs5751, hyeon93, hjseong, etkim}@yonsei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='kr, minkw@keti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='kr light red light red light red light light red light red light red light light light light light input image translated image I2I translation annotation : source domain : target domain : source-to-target style mapping Red,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' blue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' and purple: styles from different categories zoom-in zoom-in zoom-in target domain images source domain images (a) Global I2I (b) Class-level I2I (c) Style harmonization I2I Figure 1: Illustration of the concepts in unpaired I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The results are obtained on Cityscapes → ACDC (night) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the image, many head and tail lamps should be bright at night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (a) Global I2I (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) converts all classes to bright because it translates the image with a single source-to-target style mapping function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (b) Class-level I2I (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) leverages additional annotations to address the problem of (a) and performs per-class source-to-target style mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It can effectively deal with multiple classes in an image, but loses the original style: All white lights and red lights become white lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (c) Style harmonization I2I also performs class-wise style mapping, while adaptively preserving the original styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Abstract We propose a novel solution for unpaired image-to-image (I2I) translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To translate complex images with a wide range of objects to a different domain, recent approaches of- ten use the object annotations to perform per-class source-to- target style mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' However, there remains a point for us to exploit in the I2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' An object in each class consists of multiple components, and all the sub-object components have differ- ent characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For example, a car in CAR class consists of a car body, tires, windows and head and tail lamps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=', and they should be handled separately for realistic I2I trans- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The simplest solution to the problem will be to use more detailed annotations with sub-object component annota- tions than the simple object annotations, but it is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The key idea of this paper is to bypass the sub-object com- Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' ponent annotations by leveraging the original style of the in- put image because the original style will include the informa- tion about the characteristics of the sub-object components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Specifically, for each pixel, we use not only the per-class style gap between the source and target domains but also the pixel’s original style to determine the target style of a pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To this end, we present Style Harmonization for unpaired I2I translation (SHUNIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Our SHUNIT generates a new style by harmonizing the target domain style retrieved from a class memory and an original source image style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Instead of direct source-to-target style mapping, we aim for source and target styles harmonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We validate our method with extensive experiments and achieve state-of-the-art performance on the latest benchmark sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The source code is available online: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='com/bluejangbaljang/SHUNIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='04685v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='CV] 11 Jan 2023 Introduction Unpaired image-to-image (I2I) translation aims to learn source-to-target style mapping, where source and target im- ages are unpaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It can be applied to data augmenta- tion (Antoniou, Storkey, and Edwards 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Mariani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020), domain adapta- tion (Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Murez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018) and various im- age editing applications, such as style transfer (Gatys, Ecker, and Bethge 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang and Belongie 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Ulyanov, Vedaldi, and Lempitsky 2017), colorization (Zhang, Isola, and Efros 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017), and image inpaint- ing (Iizuka, Simo-Serra, and Ishikawa 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the I2I translation, the biggest problem is how to deal with the style variations among objects or classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In other words, when a global style gap is applied to an entire im- age as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 1a, the I2I translation often results in unreal- istic images because each class has different the style gaps between the source domain and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Recent ad- vanced methods (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Bhattacharjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022) addressed the problem by leveraging additional object annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' They simplify the task into class-level I2I translation and then perform per- class source-to-target style mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' This enables the net- works to explicitly estimate class-wise target styles, but it has a critical limitation: An object in each class consists of multiple components, and all the sub-object components might also have different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Let us consider the example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Understandably, when a road image taken on a sunny day is translated into a night image, head and tail lamps in a car should be brighter while the rest of the components, such as car body, tires, and windows, should be darker than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Therefore, each component in a car should be handled sep- arately for realistic I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' However, if the previous approaches are applied to perform per-class source-to-target style mapping, they will translate all head and tail lamps into white lights, making unrealistic images, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Here, one might think that this issue can be addressed by an- notating more detailed sub-object components than the sim- ple object categories, but it is actually impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' A brief example is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' A car consists of body, window, and tires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' A tire consists of wheel and gum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In this way, sub- object components can be divided endlessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To solve the above limitation of the previous class-level I2I methods, we present Style Harmonization for unpaired I2I translation (SHUNIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The key idea of SHUNIT is to by- pass the sub-object component annotations by leveraging the original style of the input image because the original style will include the information about the characteristics of the sub-object components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Thus, instead of mapping source- to-target style directly, SHUNIT harmonizes the source and target styles to realize realistic and practical I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 1c, SHUNIT uses not only the per- class style gap between the source and target domains but also the pixel’s original style to determine the target style of a pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To achieve this, we disentangle the target style into class-aware memory style and image-specific compo- nent style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The class-aware memory style is stored in a style memory, and image-specific component style is taken from the original input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The goal of the style memory is to obtain class-wise source-to-target style gaps and it is motivated by (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Compared to the memory in (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), our style memory differs in two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' First, the output from the style memory was used alone as a target style in (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), but the output from the memory is adaptively aggregated (=harmonized) in this paper with the style of the original input image to make a target style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Second, the memory was simply updated in (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), whereas our style memory is jointly trained and optimized with the other parts of SHUNIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Specifically, the class-aware mem- ory in (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) was not trained but simply was updated using the input features during the training, memo- rizing the style features from the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The gradi- ent was not propagated to the style memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Thus, the style memory in (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) cannot update their param- eters based on the error of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In SHUNIT, however, we overcome this problem by enabling the memory to learn through backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To this end, we train the style memory from randomly initialized parameters and introduce style contrastive loss to constrain the memory to learn class- wise style representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The backpropagation forces the style memory to reduce the final loss jointly and effectively along with the other parts of SHUNIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To demonstrate the superiority of our SHUNIT, we conduct extensive experi- ments on the latest benchmark sets and achieve state-of-the- art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Overall, the contributions of our work are summarized as follows: We present a novel challenge in I2I translation: an object might have various styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We propose a new I2I method, style harmonization, that leverages two distinct styles: class-aware memory style and image-specific component style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To the best of our knowledge, the style harmonization is the first method to estimate the target style in multiple perspectives for unpaired I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We achieve new state-of-the-art performance on latest benchmarks and provide extensive experimental results with analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Related Work Image-to-image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The goal of I2I is to learn source-to-target style mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For I2I translation, pix2pix (Isola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) proposes a general solution us- ing conditional generative adversarial networks (Mirza and Osindero 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' However, it has a significant limitation: paired training data should be used for training networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) successfully addresses this problem with a cycle consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The loss allows us to train the networks with unpaired training data by supervis- ing the reconstructed original image only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Based on Cycle- GAN, many approaches (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018) have been proposed to tackle I2I translation take in an un- paired manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' UNIT (Liu, Breuel, and Kautz 2017) pro- poses another unpaired I2I translation solution by mapping 𝑳𝒙 𝐸�� 𝐸�� 𝑰𝒙 𝒄𝒙 𝒔𝒙 Read 𝐺� 𝑰�𝒚 Learnable Style Memory 𝑘 𝑣 class 1 class 2 class N 𝑴𝒚 𝒔�𝒚 SH ResBlk SH ResBlk SH ResBlk ReLU SHL Conv 𝒔�𝒚 𝒔𝒙 ReLU SHL Conv 𝒔�𝒚 𝒔𝒙 (a) Overall architecture (b) SH ResBlk Figure 2: An overview of SHUNIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (a) From a pair of image Ix and label Lx in source domain X, the two encoders (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=', Ex c and Ex s ) extract the content cx and component style sx, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The content retrieves the memory style ˆsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The three features cx, sx, and ˆsy are fed to the target generator Gy, which consists of several Style Harmonization Residual Blocks (SH ResBlk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The generator Gy outputs the translated image ˆIy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (b) The input of the SH ResBlk is the content cx for the first layer, and the output of the previous block is used as input for the remainders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Each SH ResBlk includes two Style Harmonization Layers (SHL) that transfer target styles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=', sx and ˆsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' two images in different domains into the same latent code in a shared-latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) and DRIT (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018) introduce a disentangled representa- tion to achieve diverse and multi-modal I2I translation from unpaired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Basically, they perform global I2I translation which focuses on mapping a global style on all pixels in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Although they work well on object-centric images, they bring severe artifacts for complex images, such as mul- tiple objects being presented or large domain gap scenarios, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To complement the problem, recent approaches leverage additional object annotations and per- form class-level I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Class-level image-to-image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Recent several approaches (Mo, Cho, and Shin 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Bhattacharjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022) propose class-level image-to-image translation so- lutions with object annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Specifically, INIT (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019) generates instance-wise target domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' DUNIT (Bhattacharjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020) additionally employs an object detection network and jointly trains it with the I2I translation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) pro- poses an approach to store and read class-wise style repre- sentations with key-value memory networks (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' This approach, however, cannot directly supervise the memory with objective functions for I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In- staformer (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022) proposes a transformer-based (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) architec- ture that mixes instance-aware content and style represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The existing methods that leverage object annota- tions learns the direct class-wise source-to-target style map- ping, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' This effectively simplifies the I2I translation problem into per-class I2I translation, but they overlook an important point that not all pixels in the same class should be translated with the same style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Our approach, style harmonization, addresses this problem by introducing the component style that facilitates preserving the original style of the source image, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Proposed Method Definition and Overview Let X and Y be the visual source and target domains, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Given an image and the corresponding label (=bound- ing box or segmentation mask) in X domain, our frame- work generates a new image in Y domain while remain- ing the semantic information in the given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We assume that each domain consists of images and labels denoted by (Ix, Lx) ∈ X and (Iy, Ly) ∈ Y, and both domains have the same set of N classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Our framework contains the source encoder Ex = {Ex c , Ex s }, target generator Gy, and target style memory M y for source-to-target mapping, and the tar- get encoder Ey = {Ey c , Ey s }, source generator Gx, source style memory M x for target-to-source mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For conve- nience, we will only describe the source-to-target direction, and the overview of our framework is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Following the previous studies (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018), we assume that an image can be disentangled into domain-invariant content and domain-specific style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For this, we basically follow the MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The content encoder Ex c consists of several strided convolutional layers and residual blocks (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016), and all the convolutional layers are followed by Instance Normalization (Ulyanov, Vedaldi, and Lempitsky 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The content encoder extracts the domain-invariant content feature cx from the image Ix and label Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The style encoder Ex s also consists of several strided convolu- tional layers and residual blocks, and it extracts the compo- 𝒄𝒙 𝒔�𝒚 class-wise separation 𝑣� 𝑘� 𝑑 softmax class 1 𝑣� 𝑘� class 2 𝑣� 𝑘� class N ••• 𝑐� � 𝑐� � 𝑐� � 𝑠̂� � 𝑠̂� � 𝑠̂� � Figure 3: Read operation in style memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The content feature cx is separated by the label Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For each class, the memory read is independently performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' After class-wise memory read, the memory style ˆsy is obtained by gathering the retrieved class-wise values into the original locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' � denotes matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' nent style feature sx from the image Ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The memory style ˆsy is read by retrieving from the learnable style memory M y to the content feature cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The generator Gy consists of sev- eral style harmonization layers and residual blocks, and it produces the translated image ˆIy from cx, sx, and ˆsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Style Harmonization for Unpaired Image-to-Image Translation (SHUNIT) The important point of SHUNIT is that two styles are em- ployed to determine the target style: One is image-specific component style, and the other one is class-aware memory style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We focus on extracting two distinct styles accurately and then harmonizing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In what follows, we describe the detail of each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Component style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The style encoder Ex s takes the im- age Ix as input and extracts the style feature sx of size H × W × C, where H, W, and C are the height, width, and number of channels of the feature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Here, sx ex- plicitly represents the style of the input image, thus we use it as image-specific component style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The component style is used together with the memory style in the style harmo- nization layer to reduce the artifacts of the generated target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Memory style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The component style is not sufficient to handle complex scenes with multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Therefore, we exploit the class-specific style that leverages an object anno- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To this end, we construct the class-wise style mem- ory retrieved by the content feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The target style memory M y consists of N class memories to store class-wise style representations of the target domain Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Each class memory M y n has U key-value pairs (ky, vy) (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), con- sidering that various styles exist in one class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=', different styles of headlamp and tires in CAR class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The key ky is used for matching with the content feature and the value vy has the class-aware style representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The key and value are learnable vectors, each one of size 1 × 1 × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 3 shows a detailed implementation of the process of reading the corresponding memory style ˆsy from the mem- 𝒔𝒙 𝜷𝒙 𝜸𝒙 𝒔�𝒚 𝜷𝒚 𝜸𝒚 𝛼� 1 − 𝛼� Conv Conv Conv Conv 𝜷 𝜸 input output ••• class-wise 𝛼 broadcasting norm Figure 4: Style harmonization layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' From the component style sx and memory style ˆsy, scale (γx and γy) and shift (βx and βy) factors are extracted via four convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The alpha mask ˆα is obtained by broadcasting the class-wise alpha α with the semantic label Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We weighted sum the image and memory styles with the alpha mask, and transfer it to the input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' ⊙ denotes Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' ory M y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' With the semantic label, we separate the content feature cx into {cx 1, cx 2, · · · , cx N}, where cx n denotes source content feature for the n-th class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Let cx n,i be the i-th pixel of the n-th class source content feature and (ky n,j, vy n,j) be the j-th key-value pair of the n-th class target style memory M y n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In this work, we aim to read the target memory style ˆsy n,i corresponding to the source content cx n,i using the simi- larity between the source content and key of target memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To this end, we calculate the similarity wx n,i,j between cx n,i and ky n,j as: wx n,i,j = exp � d(cx n,i, ky n,j) � �U u=1 exp � d(cx n,i, ky n,u) � (1) where d(·, ·) is the cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We then read the memory style ˆsy n,i corresponding to cx n,i by calculating the weighted sum of values in n-th class style memory: ˆsy n,i = U � j=1 wx n,i,jvy n,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (2) The same process is applied for content features of other classes, finally extracting spatially varying target memory style features ˆsy of size H × W × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Different from the previous key-value memory net- works (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) that learn the memory via updating mechanism, we learn the memory through backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The updating mechanism is used to directly store the ex- ternal input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' However, it has a critical drawback: The memory cannot be trained with the network jointly with the same objective function because the gradient should be stopped at the updated memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To solve the problem, we discard the update mechanism and learn the memory with the loss functions presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The effectiveness of our memory learning strategy is validated in the experiments section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Style harmonization layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Our goal is harmonizing the source and target styles instead of mapping source-to-target style directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To this end, we propose the style harmoniza- tion layer to adaptively aggregate the component style and memory style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The style harmonization layer consists of sev- eral convolution layers and class-wise alpha parameters, and it is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Here we use three conditional in- puts: memory style, component style, and label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Convolu- tional layers are used to compute pixel-wise scale γ and shift β factors from the two styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Following (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), we trans- fer the harmonized target style by scaling and shifting the normalized input feature with the computed factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=', γ and β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the layer, we additionally set class-wise alpha pa- rameters (α1, α2, · · · , αN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It is used to decide which style has more influence for each class in the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' If the alpha value is large, the component style sx has more influence, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Let f, of size H × W × C, be the input feature of the current style harmonization layer in the generator Gy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' With the style harmonizing scale γ and shift β factors, the feature is denormalized by γc,h,w (fc,h,w − µc) σc + βc,h,w (3) where µc and σc are the mean and standard deviation of the input feature f at the channel c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The modula- tion parameters γc,h,w and βc,h,w are obtained from γx c,h,w, γy c,h,w, βx c,h,w, and βy c,h,w, which are the scale and shift fac- tors of the component style sx and memory style, sy respec- tively, and they are computed by γc,h,w = ˆαh,wγx c,h,w + (1 − ˆαh,w)γy c,h,w, βc,h,w = ˆαh,wβx c,h,w + (1 − ˆαh,w)βy c,h,w, (4) where ˆα denotes the alpha mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It is obtained by broadcast- ing the class-wise alpha parameters to their corresponding semantic regions of the label Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We experimentally demon- strate that our style harmonization layer adaptively controls the style of each object well, and the results are given in the experiments section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Loss Functions We leverage standard loss functions used in MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) to generate proper target domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It in- cludes self-reconstruction Lself (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017), cycle con- sistency Lcycle (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017), perceptual Lperc (Johnson, Alahi, and Fei-Fei 2016) and adversarial loss Ladv (Good- fellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The detailed explanations of those loss functions are given in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In this paper, we propose two advanced loss functions to facilitate style harmonization: content contrastive loss and style contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It is used with the aforementioned standard loss functions jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In what follows, we intro- duce the proposed two loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Content contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To extract domain invariant content features from the content encoder, MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) simply reduces the L1 distances between cx and ˆcy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We replace this with contrastive representation learning to improve discrimination within a class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For a content fea- ture ˆcy i at pixel i, which is the content feature extracted from translated target image, we set the positive sample to cx i and we set the remaining features at the other pixels as negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The content contrastive loss is defined with the a form of InfoNCE (Oord, Li, and Vinyals 2018) as: Lcontent = − HW � i=1 log � exp((cx i · ˆcy i )/τ) �HW j=1 exp((cx j · ˆcy i )/τ) � (5) where τ is a temperature parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In this equation, the features in the same class at the pixel i can be considered as negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' This encourages the content encoder to extract more diverse style representations from the style memory within the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It is also applied to the target- to-source pipeline with ˆcx and cy Style constrative loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We propose the style contrastive loss to allow the style memory to learn class-wise style rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Similar to the content contrastive loss, for a style feature ˆsx i at pixel i, which is the memory style of target-to-source mapping, we set the positive sample to the source component style sx i and we set the remaining features at the other pixels as negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The style constrative loss is defined as follows: Lstyle = − HW � i=1 log � exp((sx i · ˆsx i )/τ) �HW j=1 exp((sx j · ˆsx i )/τ) � (6) This loss directly supervises the memory style ˆsx for the translated target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' This improves the stability of cycle consistency learning for reconstructing Ix from ˆIy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It is also applied to the target styles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=', ˆsy and sy Finally, all loss functions are summarized as follows: min (Ex,Ey,Gx,Gy) max (Dx,Dy)L(Ex, Ey, Gx, Gy, Dx, Dy) = λselfLself + λcycleLcycle + λpercLperc +λadvLadv + λcontentLcontent + λstyleLstyle (7) where Dx and Dy denote the multi-scale discrimina- tors (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018) for each visual domain, X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The details of Lself, Lcycle, Lperc, and Ladv are described in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Experiments In this section, we present extensive experimental results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To demonstrate the superiority of our method, we compare our SHUNIT with state-of-the-art I2I translation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The implementation details of our method are pro- vided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Datasets We evaluate our SHUNIT on three I2I translation scenar- ios: Cityscapes (Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016) → ACDC (Sakaridis, Dai, and Van Gool 2021) and INIT (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019), and KITTI (Geiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2013) → Cityscapes (Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In all scenarios, INIT (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019), DUNIT (Bhattacharjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020), MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), InstaFormer (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022), and our method use semantic labels provided in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Input CycleGAN MUNIT MGUIT SHUNIT (ours) Target domain Figure 5: Qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' From the given clear image (first column), we generate four adverse condition images using (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) and SHUNIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the last column, we show a sample of the real image for each adverse condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain clear → fog clear → night cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='88 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='68 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='16 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='96 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='72 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='73 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='26 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='33 UNIT (Liu, Breuel, and Kautz 2017) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='89 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='24 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='39 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='36 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='70 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='28 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='29 MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='79 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='83 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='20 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='34 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='44 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='56 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='43 TSIT (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='08 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='16 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='40 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='16 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='68 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='92 MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='75 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='33 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='60 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='36 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='22 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='83 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='36 SHUNIT (ours) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='84 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='53 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='96 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='08 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='66 Table 1: Quantitative comparison on Cityscapes → ACDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We measure class-wise FID (lower is better) and mIoU (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For brevity, class-wise FID is written as cFID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain clear → fog clear → night AdaptSegNet (Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 ADVENT (Vu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='1 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='9 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 BDL (Li, Yuan, and Vasconcelos 2019) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='8 CLAN (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='6 FDA (Yang and Soatto 2020) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='1 SIM (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='0 MRNet (Zheng and Yang 2021) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='9 SHUNIT (ours) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='2 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 Table 2: Quantitative Comparison on domain adaptation for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We report mIoU for Cityscapes → ACDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Cityscapes → ACDC Cityscapes (Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016) is one of the most popular urban scene dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' ACDC (Sakaridis, Dai, and Van Gool 2021) is the latest dataset with multiple adverse condition images and consists of four conditions of street scenes: snow, rain, fog, and night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' ACDC dataset provides images with corresponding dense pixel-level semantic annotations, and it has 19 classes the same as Cityscapes dataset for all adverse conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Fol- lowing (Sakaridis, Dai, and Van Gool 2021), we leverage Cityscapes dataset as a clear condition and translate it to the adverse conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=', snow, rain, fog, and night) in ACDC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Therefore, this scenario is challenging because not only the weather conditions, but also layouts, such as cam- era model, view, and angle, are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To train the net- works, 2975, 400, 400, 400, and 400 images are used for clear, snow, rain, fog, and night conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For a fair comparison on this benchmark, we reproduce existing state-of-the-art methods (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Liu, Breuel, and Kautz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For fair comparison, we set the number of key-value pairs for style memory to be the same as our setting and use segmentation mask for reproducing 国Input CycleGAN UNIT MUNIT DRIT MGUIT InstaFormer SHUNIT (ours) Figure 6: Qualitative comparison on INIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (Top to bottom) sunny→night, night→sunny, cloudy→sunny results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Our method preserves object details and looks more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' sunny → night night → sunny sunny → rainy sunny → cloudy cloudy → sunny Average CIS ↑ IS ↑ CIS ↑ IS ↑ CIS ↑ IS ↑ CIS ↑ IS ↑ CIS ↑ IS ↑ CIS ↑ IS ↑ CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='026 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='057 UNIT (Liu, Breuel, and Kautz 2017) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='082 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='027 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='097 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='081 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='219 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='087 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='146 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='008 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='095 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='026 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='321 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='032 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='166 DRIT (Lee et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='046 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='321 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='031 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='164 INIT (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019) 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='489 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='112 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='254 Instaformer (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='404 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='115 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='127 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='158 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='394 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='130 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='257 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='141 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='585 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='149 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='353 SHUNIT (ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='205 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='503 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='308 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='585 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='136 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='609 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='111 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='405 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='085 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='315 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='169 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='483 Table 3: Quantitative Comparison on INIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We measure CIS and IS (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' INIT INIT (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019) is a public benchmark set for I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It contains street scenes images including 4 weather categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=', sunny, night, rainy, and cloudy) with the corresponding bounding box labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Following (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019), we split the 155K images into 85% for train- ing and 15% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We conduct five translation exper- iments: sunny ↔ night, sunny ↔ cloudy, sunny → rainy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In this dataset, we directly copied the results of the existing methods from (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Bhattacharjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Similarly, for fair com- parison with MGUIT, the number of key-value pairs in style memory is set equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' KITTI → Cityscapes KITTI is a public benchmark set for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It contains 7481 images with bounding boxes annotations for training and 7518 images for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Following the previous I2I translation methods (Bhattachar- jee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022), we select the common 4 object classes (person, car, truck, bicycle) for evaluatation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Qualitative Comparison Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 5 shows qualitative results on Cityscapes → ACDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Since our I2I translation setting, Cityscapes → ACDC, is very challenging as discussed in the datasets section, ex- isting methods cannot generate realistic images in several scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Specifically, CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) often destroys the semantic layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) translates images with a global style, thus it also often gen- erates artifacts, as shown in the snow, rain, and fog images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) also includes artifacts in the car even though leveraging memory style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' It shows the limita- tion of the updating mechanism for training memory style, and the limitation is clearly depicted in the challenging sce- nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In contrast to them, our SHUNIT accurately generates images in the target domains without losing the original style in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the supplementary material, we further provide the results of UNIT (Liu, Breuel, and Kautz 2017) and TSIT (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 6, which depicts qualitative results on INIT dataset, our method generates high-quality images in various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the night → sunny scenario (second row), InstaFormer (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022) translates the color of the road lane to yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' On the other hand, our method keeps the color of the lane as white and generates a sunny scene by harmonizing the target domain style retrieved from a style memory and an image style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Quantitative Comparison The quantitative results on Cityscapes → ACDC are pre- sented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To quantify the per-class image-to-image translation quality, we measure class-wise FID (Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We further measure mIoU on ACDC test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The Pers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Car Truc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Bic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' mAP DT (Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='2 DAF (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='9 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='5 DARL (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 DAOD (Rodriguez and Mikolajczyk 2019) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='1 DUNIT (Bhattacharjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='1 MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='4 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='6 InstaFormer (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='5 SHUNIT (ours) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='0 Table 4: Quantitative Comparison on domain adaptation for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We report per-class AP for KITTI → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' α cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ ✓ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='93 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='83 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='06 ✓ ✓ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='72 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='20 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='97 ✓ ✓ ✓ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='77 (a) Style ablation study on SHL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain Lcontent Lstyle cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ ✓ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='06 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='43 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='23 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='39 ✓ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='38 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='89 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='33 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 ✓ ✓ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='77 (b) Ablation study on loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ Updating 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='55 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='70 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='05 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='21 Backprop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='77 (c) Style memory training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ L1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='21 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='72 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='78 Contrastive 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='77 (d) L1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Contrastive loss for style and content losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ w/o 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='73 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='03 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='72 w/ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='77 (e) Experimental results of label input for content encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Table 5: Ablation Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We report class-wise FID and mIoU in two scenarios: clear → {snow, rain}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' mIoU metric is used to validate the results on the practical problem, semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We generate a training set by Cityscapes → ACDC and then train DeepLabV2 (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The mIoU score is obtained with the trained DeepLabV2 by evaluating on ACDC test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in Table 1, we surpass the state-of-the-art I2I trans- lation methods by a significant margin in most scenarios, demonstrating the superiority of our style harmonization for unpaired I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Table 2 shows the quantitative re- sults of domain adaptation for semantic segmentation with the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Despite we trained DeepLabV2 with only a simple cross-entropy loss, our SHUNIT achieves comparable performance with the domain adaptation meth- ods that were trained DeepLabV2 with additional loss func- tions and several techniques for boosting the performance of mIoU score on the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Table 3 shows another quantitative results on the testing split of INIT (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To directly compare our method with the public results, we evaluate our method with Inception Score (IS) (Salimans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016) and Conditional Inception Score (CIS) (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in Table 3, we achieve the best performance in most scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We further evaluate our method on domain adaptation benchrmark following DUNIT (Bhattacharjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We use Faster-RCNN (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2015) trained on the source domain as a detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in Table 4, we achieve the state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Ablation Study In this section, we study the effectiveness of each component in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We validate on Cityscapes (clear) → ACDC (snow/rain) scenarios and use ACDC validation set for both class-wise FID and mIoU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Ablation study on style harmonization layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We ablate the memory style, component style, and class-wise α in the style harmonization layer, and they are denoted as “Mem.”, “Comp.”, and “α” in Table 5a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in the table, the memory style-only is far behind the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' With component style, we can achieve performance im- provement on clear → snow while decreasing on clear → rain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We obtain significant improvement on most scenarios with class-wise α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The results demonstrate that the exist- ing approach, which only leverages the memory style, is not sufficient for I2I translation, and we successfully address the problem by adaptively harmonizing two styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Ablation study on content and style losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We study the effectiveness of the proposed two loss functions, Lstyle and Lcontent, by ablating them step-by-step, and the results are 1mIoU on test set should be evaluated on the online server (Sakaridis, Dai, and Van Gool 2021) and it has a limit on the num- ber of submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Therefore, we use validation set for ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' given in Table 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in the table, our model is effec- tive when two losses are used jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Style memory training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As described in the pro- posed method section, we opt for backpropagation to train the style memory rather than updating the mechanism used in (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The results are shown in Table 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We surpass the existing updating method by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' L1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Table 5d shows the efficacy of our contrastive-based approach by replacing Lcontent and Lstyle with L1 losses as used in MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The L1 losses are designed to reduce L1 distances within posi- tive pairs without consideration of negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As we dis- cussed in loss functions section, our method effectively en- courages extracting more diverse style representations, lead- ing to performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Label input for content encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Table 5e shows that la- bel input is not significant but always leads to performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Therefore, we have no reason to omit the la- bel input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Limitations Since our framework leverages the component style of the source image, the generated image’s quality relies on the source image’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' If the source image has a very bright colors, SHUNIT often generates a relatively bright night im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 5, SHUNIT struggles to generate geometrically distinct lights from the source image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Due to the above prob- lems, SHUNIT cannot achieve the best performance on clear → night scenario in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We believe that these problems can be alleviated by leveraging geometric information such as depth or camera pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Additionally, our approach can give limited benefit to some I2I translation scenarios, such as dog → cat, because these tasks need to change the content;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' how- ever, we tackle the unpaired I2I translation task under the condition that the content will not be changed, and only the style will be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Conclusion We present a new perspective of the target style: It can be disentangled into class-aware and image-specific styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Furthermore, our SHUNIT effectively harmonizes the two styles, and its superiority is demonstrated through extensive experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We believe that our proposal has the potential to break new ground in style-based image editing applica- tions such as style transfer, colorization, and image inpaint- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Acknowledgement This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='2021- 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Sean: Image synthesis with semantic region-adaptive normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In CVPR, 5104–5113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Implementation Details Experiments Settings We implement our model with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='1 version of Py- Torch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To train SHUNIT, we use a fixed learning rate of 10−4 and use the Adam op- timizer (Kingma and Ba 2015) with β1 and β2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The weight decay is set to 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Our models are trained with a single NVIDIA RTX A6000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For the experiments on Cityscapes (Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016) → ACDC (Sakaridis, Dai, and Van Gool 2021), we use RGB images with a size of 512 × 256 during both training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We train the model for 50K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Since the ACDC dataset has been published recently, no official re- sults are available on this dataset for the existing meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Therefore, we reproduced CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017), UNIT (Liu, Breuel, and Kautz 2017), MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b), TSIT (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020), and MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) with their official implementation code1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For the experiments on INIT (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019), we use RGB images with a size of 360 × 360 during training while we use RGB images with a size of 572 × 360 during infer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We train the model for 250K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For the ex- periment on KITTI (Geiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2013) → Cityscapes, we use RGB images with a size of 540 × 360 during training while we use RGB images with a size of 1242 × 375 during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We train the model for 200K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For fair comparisons with INIT (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2019), DUNIT (Bhattacharjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020), MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), and InstaFormer (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022), which used bounding-box labels as an additional object annotation, we did not use segmentation labels but used bounding- box labels for all experiments on both INIT and KITTI → Cityscapes scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In addition, we reproduced MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) on Cityscapes → ACDC exper- iments with segmentation labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Details of Network Architecture In the content encoder Ex c , we employ two convolutional networks, one network takes RGB input, and the other takes one-hot encoded semantic label input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Each network con- sists of Conv(input channel, 64, 7, 1, 3, IN, relu) - Conv(64, 128, 4, 2, 1, IN, relu) - Conv(128, 256, 4, 2, 1, IN, relu) - Resblk ×4 - Conv(256, 128, 1, 1, IN, relu), where IN de- notes the instance normalization layer (Ulyanov, Vedaldi, and Lempitsky 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Resblk denotes the residual block (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' and the components in Conv(·) denotes (input channel, output channel, kernel size, stride, padding, nor- malization, activation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The encoded RGB and semantic la- bel features are then concatenated along the channel dimen- sion, and it is the content feature cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The style encoder Ex s 1The code of CycleGAN was taken from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='com/junyanz/pytorch-CycleGAN-and-pix2pix 2The code of UNIT and MUNIT were taken from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='com/NVlabs/MUNIT 3The code of TSIT was taken from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='com/EndlessSora/TSIT 4The code of MGUIT was taken from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='com/somijeong/MGUIT has the same network architecture as a convolutional net- work in the content encoder, except for the normalization layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The style encoder does not use normalization layers to keep the original style of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The generator Gy consists of SH Resblk ×4 - Conv(256, 128, 5, 1, 2, IN, relu) - Conv(128, 64, 5, 1, 2, IN, relu) - Conv(64, 3, 7, 1, 3, -, tanh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For the experiments on Cityscapes → ACDC, we use 20 key-value pairs for each class in the style memory M y n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the rest of the experiments, we follow key-value pairs of MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Details of Loss Functions Following MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b), we leverage self- reconstruction Lself (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017), cycle consistency Lcycle (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017), perceptual Lperc (Johnson, Alahi, and Fei-Fei 2016), and adversarial Ladv (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2014) losses as follows: Lself = ∥Ix − Gx(cx, sx)∥1 + ∥Iy − Gy(cy, sy)∥1 , (8) Lcycle = ∥Ix − Gx(ˆcy, ˆsx)∥1 + ∥Iy − Gy(ˆcx, ˆsy)∥1 , (9) Lperc = ���F(Ix) − F( ˆIy) ��� 1 + ���F(Iy) − F( ˆIx) ��� 1 , (10) Ladv = � log (1 − Dx(ˆIx)) + log Dx(Ix) � + � log (1 − Dy(ˆIy)) + log Dy(Iy) � , (11) where F(·) is a feature map extracted from relu5 3 layer of the ImageNet pretrained VGG-16 network (Si- monyan and Zisserman 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Dx(·) and Dy(·) are features extracted from the multi-scale discrimina- tors (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018) for each visual domain, X and Y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We empirically set the hyperpa- rameters {λself, λcycle, λperc, λadv, λcontent, λstyle} to {10, 10, 1, 1, 10, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The temperature τ in Lcontent and Lstyle is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Analysis of class-wise FID Recent work (Shim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022) introduced class-wise FID (cFID) to quantify the per-class image quality in the image generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We realized that the I2I translation task also has advantages if we evaluate the results with cFID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In this section, we provide a detailed analysis of the necessity for cFID on our benchmark set and the implementation detail of cFID for I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Necessity of CFID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' By the nature of unsupervised I2I, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 7, the class statistics between the source do- main (Cityscapes (Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016)) and the target do- main (ACDC (Sakaridis, Dai, and Van Gool 2021)) are un- matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' However, FID ignores the unmatched class statis- tics because FID only uses the global embeddings to com- pute the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Therefore, the different class statistics pre- vent FID improvement, regardless of the generated image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Figure 7: The class statistics in Cityscapes (Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016) validation set vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' ACDC (Sakaridis, Dai, and Van Gool 2021) validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Even though the two datasets share the class categories, the class statistics are naturally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To demonstrate the problem of FID intuitively, we il- lustrate an example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the figure, we translate a clear image sampled from Cityscapes dataset to snow with three methods: (a) CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017), (b) MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), and (c) SHUNIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As can be compared qualitatively, SHUNIT generates a more realis- tic snow image than CycleGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For a quantitatively com- parison, we take a similar strategy used in FID: We extract global embeddings from a real snow image sampled from ACDC dataset and the generated image using Inception- V3 (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Then, we compute the squared Eu- clidean distance between the two global embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' How- ever, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 8, CycleGAN (267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47) achieves bet- ter performance than SHUNIT (295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='45) in global distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The reason is that CycleGAN destroys the layout in the input image and follows the layout of the real snow image, result- ing in severe artifacts in the generated image while better performance in the global distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' If we extract class-wise embeddings and compute class-wise distance then average them, we can obtain a reasonable rank: SHUNIT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='64) achieves the best, MGUIT (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='61) achieves the second-best, and CycleGAN (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='07) achieves the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As can be ob- served in this example, comparing class-wise embeddings is a more reasonable method to evaluate the generated qual- ity than comparing global embeddings in unsupervised I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To further demonstrate the necessity of cFID, we plot the pixel-wise features of Inception-V3 (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 9(a), 9(b), and 9(c), Cityscapes validation set is used as clear images for I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 9(d), ACDC validation set is used as real snow images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the figure, we use the same color for the same category of the pixel-wise feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in the zoom-in regions, pixel- wise features extracted from (d) real snow images are well separated by class categories because the pretrained network is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' However, (a) CycleGAN includes various categories in the zoom-in region, which clearly indicates that the qual- ity of the generated images is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In contrast, (c) features in SHUNIT are well separated by class categories as in (d) real snow images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' This class information should be considered to evaluate the quality of the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' However, as shown in Table 6, CycleGAN achieves the best performance of FID in clear → snow scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' There- fore, FID score is not reliable in unsupervised I2I transla- tion with multiple classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In this paper, we simply and ef- fectively address the problem by computing class-wise dis- tances separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Implementation detail of cFID for I2I translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We measure cFID based on the official implementation of FID (Seitzer 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' cFID is calculated with the Inception- V3 (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016) features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We extract the features from the first block of the network to obtain fine-scale fea- tures, which effectively include features of small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To extract class-wise embeddings, we upsample the features to their original input size by bilinear interpolation, and then semantic region pooling is applied to the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Here, we use the ground truth of the semantic segmentation for the semantic region pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' With the class-wise embeddings, we compute FID for each class separately, and then cFID is obtained by averaging them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Stability of SHUNIT To show the stability of our model, we train SHUNIT five times with different random seeds on Cityscapes (clear) → ACDC (snow, rain), and the results are given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in the table, the performance gap between the best and worst results is reasonably small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In addition, the worst result also achieves the state-of-the-art performance over previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' This result demonstrates that our re- sults for quantitative comparison are not cherry-picked and actually lead to performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Furthermore, the result promises that our strong performance can be easily reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Semantics statistics Cityscapes val ACDC val 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='25 semantic ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='00 0 15 16 17 1819I2I Global distance: 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 Class-wise distance: 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='07 Global distance: 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='54 Class-wise distance: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='61 Global distance: 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='45 Class-wise distance: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='64 (a) CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) Input (b) MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) (d) Target domain (c) SHUNIT (ours) Figure 8: Illustration of comparison of global distance and class-wise distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' From a clear image, we generate snow images using (a) CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017), (b) MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021), and (c) SHUNIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' To quantitatively evaluate the results, we compute distances between (a,b,c) the generated images and (d) real snow image in two methods: One is computed with global embeddings of Inception-V3 and the other one is computed with class-wise embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain clear → fog clear → night FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑ FID ↓ CFID ↓ mIoU ↑ CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='21 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='88 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='68 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='91 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='16 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='96 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='23 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='72 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='73 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='40 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='26 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='33 UNIT (Liu, Breuel, and Kautz 2017) 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='63 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='89 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='24 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='79 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='39 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='36 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='70 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='39 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='28 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='29 MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='91 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='79 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='83 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='48 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='20 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='28 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='34 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='44 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='94 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='56 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='43 TSIT (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020) 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='19 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='08 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='12 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='16 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='40 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='16 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='68 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='09 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='92 MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='76 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='75 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='33 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='59 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='60 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='86 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='36 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='22 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='11 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='83 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='36 SHUNIT (ours) 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='15 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='96 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='84 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='53 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='96 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='80 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='08 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='66 Table 6: Quantitative comparison on Cityscapes → ACDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We measure FID, CFID (lower is better) and mIoU (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' More Qualitative Results Cityscapes → ACDC We show more qualitative results on Cityscapes (clear) → ACDC (snow/rain/fog/night) in Figs 10 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the results, we additionally show the results of UNIT (Liu, Breuel, and Kautz 2017) and TSIT (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020), which are omitted in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The qualitative results demonstrate that our method generate more realistic images than the previ- ous works (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Liu, Breuel, and Kautz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We further provide a comparison video in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' KITTI → Cityscapes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 14 shows the visual comparison of InstaFormer (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2022) and our method for domain adaptive detection on KITTI → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in the figure, InstaFormer generates artifacts in the sky and struggles to maintain the sub-object components of a small object, such a car located in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In contrast, our method translates the sky to the Cityscapes style without artifacts, and preserves the sub- objects components of a small object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' The more realistic image translation is demonstrated by a detection result: the Faster-RCNN (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2015) detects the small car located in the center in our image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 14c) while cannot detect it in InstaFormer image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 14b) More Quantitative Results Tables 8 to 11 provide the per-class IoU performance on ACDC test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' As shown in the tables, we achieve the best performance not only in mIoU but also in per-class IoU of the most classes on snow, rain, and fog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' zoom-in zoom-in zoom-in zoom-in zoom-in zoom-in zoom-in zoom-in (a) CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) (b) MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) (c) SHUNIT (ours) (d) Target domain Figure 9: t-SNE (Van der Maaten and Hinton 2008) visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We plot pixel-wise features extracted from Inception- V3 (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In (a), (b), and (c), the generated snow images, which are translated from clear images, are used for extracting features, while in (d), the real snow images in the ACDC validation set are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the figure, the same color of dots are in the same class category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' clear → snow clear → rain cFID ↓ mIoU ↑ cFID ↓ mIoU ↑ CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='88 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='68 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='16 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='96 UNIT (Liu, Breuel, and Kautz 2017) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='89 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='24 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='39 MUNIT (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='79 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='83 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='20 TSIT (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='08 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='16 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='40 MGUIT (Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='75 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='33 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='60 SHUNIT (trial 1, reported in the main paper) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='62 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='84 SHUNIT (trial 2) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='18 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='71 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='53 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='14 SHUNIT (trial 3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='41 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='94 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='44 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='01 SHUNIT (trial 4) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='56 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='41 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='34 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='67 SHUNIT (trial 5) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='86 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='41 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='56 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='70 Table 7: Results of five trials on Cityscapes → ACDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' We measure class-wise FID (lower is better) and mIoU (higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In our result, the best results are bold-faced while the worst results are red-colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' For brevity, class-wise FID is written as cFID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' oo 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' :Methods road sidewalk building wall fence pole traffic light traffic sign vegetation terrain sky person rider car truck bus train motorcycle bicycle mIoU CycleGAN (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='55 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='74 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='42 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='71 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='78 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='77 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='34 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='07 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='85 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='20 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='05 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='29 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='93 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='94 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='57 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='68 UNIT (Liu, Breuel, and Kautz 2017) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='88 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='27 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='74 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='35 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='47 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='71 47.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='51 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='73 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='75 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='75 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='66 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='99 24.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='56 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='13 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='92 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='59 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='76 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content='66 Table 11: Per-class IoU results on Cityscapes (clear) → ACDC (night).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Input CycleGAN UNIT MUNIT TSIT MGUIT SHUNIT (ours) Target domain Figure 10: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' From the given clear image (first row), we generate four adverse condition images using (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Liu, Breuel, and Kautz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) and SHUNIT (top row to bottom row order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the last row, we show a sample of the real image for each adverse condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Input CycleGAN UNIT MUNIT TSIT MGUIT SHUNIT (ours) Target domain Figure 11: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' From the given clear image (first row), we generate four adverse condition images using (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Liu, Breuel, and Kautz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) and SHUNIT (top row to bottom row order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the last row, we show a sample of the real image for each adverse condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Input CycleGAN UNIT MUNIT TSIT MGUIT SHUNIT (ours) Target domain Figure 12: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' From the given clear image (first row), we generate four adverse condition images using (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Liu, Breuel, and Kautz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) and SHUNIT (top row to bottom row order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the last row, we show a sample of the real image for each adverse condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' HOTEL-RESTAURANT HAGNAUERHOFInput CycleGAN UNIT MUNIT TSIT MGUIT SHUNIT (ours) Target domain Figure 13: More qualitative comparison on Cityscapes (clear) → ACDC (snow/rain/fog/night).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' From the given clear image (first row), we generate four adverse condition images using (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Liu, Breuel, and Kautz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' 2021) and SHUNIT (top row to bottom row order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' In the last row, we show a sample of the real image for each adverse condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' (b) InstaFormer (a) Input image (KITTI) (c) SHUNIT (ours) Figure 14: Qualitative comparison on domain adaptive detection for KITTI → Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Given (a) input image (KITTI), (b) and (c) show the translated image and the object detection result generated by InstaFormer and our method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} +page_content=' Note that the results are taken from InstaFormer paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE3T4oBgHgl3EQfuguG/content/2301.04685v1.pdf'} diff --git a/JdAyT4oBgHgl3EQf5_qu/content/tmp_files/2301.00815v1.pdf.txt b/JdAyT4oBgHgl3EQf5_qu/content/tmp_files/2301.00815v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b51c8e1cc670a6c2ad9a3b31ad52327f302db7e8 --- /dev/null +++ b/JdAyT4oBgHgl3EQf5_qu/content/tmp_files/2301.00815v1.pdf.txt @@ -0,0 +1,898 @@ +NeuroExplainer: Fine-Grained Attention +Decoding to Uncover Cortical Development +Patterns of Preterm Infants +Chenyu Xue1, Fan Wang2, Yuanzhuo Zhu2, Hui Li3, Deyu Meng1, Dinggang +Shen4, and Chunfeng Lian1 +1 School of Mathematics and Statistics,Xi’an Jiaotong University, Xi’an, China +chunfeng.lian@xjtu.edu.cn +2 Key Laboratory of Biomedical Information Engineering of Ministry of Education, +School of Life Science and Technology, Xi’an Jiaotong University, Xi’an, China +fan.wang@xjtu.edu.cn +3 Department of Neonatology, The First Affiliated Hospital of Xi’an Jiaotong +University, Xi’an, China +4 School of Biomedical Engineering, ShanghaiTech University, Shanghai, China +Abstract. Deploying reliable deep learning techniques in interdisciplinary +applications needs learned models to output accurate and (even more im- +portantly) explainable predictions. Existing approaches typically expli- +cate network outputs in a post-hoc fashion, under an implicit assumption +that faithful explanations come from accurate predictions/classifications. +We have an opposite claim that explanations boost (or even determine) +classification. That is, end-to-end learning of explanation factors to aug- +ment discriminative representation extraction could be a more intu- +itive strategy to inversely assure fine-grained explainability, e.g., in those +neuroimaging and neuroscience studies with high-dimensional data con- +taining noisy, redundant, and task-irrelevant information. In this pa- +per, we propose such an explainable geometric deep network dubbed +as NeuroExplainer, with applications to uncover altered infant cortical +development patterns associated with preterm birth. Given fundamen- +tal cortical attributes as network input, our NeuroExplainer adopts a +hierarchical attention-decoding framework to learn fine-grained atten- +tions and respective discriminative representations to accurately rec- +ognize preterm infants from term-born infants at term-equivalent age. +NeuroExplainer learns the hierarchical attention-decoding modules un- +der subject-level weak supervision coupled with targeted regularizers de- +duced from domain knowledge regarding brain development. These prior- +guided constraints implicitly maximizes the explainability metrics (i.e., +fidelity, sparsity, and stability) in network training, driving the learned +network to output detailed explanations and accurate classifications. Ex- +perimental results on the public dHCP benchmark suggest that NeuroExplainer +led to quantitatively reliable explanation results that are qualitatively +consistent with representative neuroimaging studies. +Keywords: Geometric Deep Learning · Explainability · Infant Brain +Cortical Development · AI for Neuroscience +arXiv:2301.00815v1 [cs.LG] 1 Jan 2023 + +2 +Authors Suppressed Due to Excessive Length +1 +Introduction +Due to the capacity of learning highly nonlinear representations in task-oriented +fashions, deep neural networks are showing promising applications in many in- +terdisciplinary communities, including biomedical image computing, neuroimag- +ing, and neuroscience studies [33]. To quantitatively analyze brain develop- +ment/degeneration or timely diagnose associated disorders, various deep learn- +ing methods have been proposed [45,28]. Most of these studies focused on the +designs of network architectures and learning strategies to produce accurate pre- +dictions/classifications. A critical challenge is that the learned models typically +lack computational interpretability and results’ explainability. From the aspect +of practical usage, neuroimaging and neuroscience studies desire AI tools that +can make accurate and (even more importantly) explainable predictions. For +example, a key value of a brain disease diagnosis network is to identify from +high-dimensional neuroimage data individualized subtle changes leading to ac- +curate classification, which can be clues for human experts to analyze disease +heterogeneity [45]. Also, the classification task to differentiate between preterm +and term-born (or male and female) infants may not be practically meaningful; +in contrast, fine-grained differences on brain cortical surfaces, identified by the +learned classification network, could be valuable factors for better understanding +featured cortical development patterns of infants from different groups. +Recently, explainable and interpretable deep learning is being actively stud- +ied in the machine learning community, with obviously more works on gridded +data (e.g., images) [11] than on non-Euclidean data (e.g., 3D meshes) [46]. Accu- +rate classification and faithful explanation are highly correlated and inseparable. +Existing methods typically adopt post-hoc techniques to explain a deep network +[46], which is first trained for a specific classification task, and then the underly- +ing (sparse) correlations between its input and output are analyzed offline, e.g., +by backpropagating prediction gradients to the shallow layers [34]. Notably, such +post-hoc approaches are established upon a common assumption that reliable ex- +planations are the results caused by accurate predictions. This assumption could +work in general applications that have large-scale training data, while cannot al- +ways hold for neuroimaging and neuroscience research, where available data are +typically small-sized and much more complex (e.g., high-resolution cortical sur- +faces containing noisy, highly redundant, and task-irrelevant information). +Distinct to existing post-hoc methods, we have an opposite claim that ex- +plainability boosts or even determines classification, especially in those chal- +lenging tasks related to brain cortical development analyses based on high- +dimensional neuroimaging data. The key is how to construct an end-to-end +framework, where fine-grained explanation factors can be identified in a fully +learnable fashion to enhance discriminative representation extraction and finally +output accurate classification. This paper presents such an explainable geometric +deep network, called NeuroExplainer, with applications to uncover altered infant +cortical development patterns associated with preterm birth. NeuroExplainer +adopts high-resolution cortical attributes as the input to develop a hierarchi- +cal attention-decoding architecture. In the framework of weakly supervised dis- + +NeuroExplainer +3 +criminative localization, our NeuroExplainer is trained by minimizing general +classification losses coupled with a set of constraints designed according to prior +knowledge regarding brain development. These targeted regularizers drive the +network to implicitly optimize the explainability metrics from multiple aspects +(i.e., fidelity, sparsity, and stability), thus capturing fine-grained explanation fac- +tors to explicitly improve classification accuracies. Experimental results on the +public dHCP benchmark suggest that our NeuroExplainer led to quantitatively +reliable explanation results that are qualitatively consistent with representative +neuroimaging studies, implying that it could be a practically useful AI tool for +other related cortical surface-based neuroimaging studies. +2 +Related Work +Deep Learning for Cortical Surface Analyses. The human cerebral cortex +is a highly folded and thin sheet of gray matter [15]. By leveraging structural +magnetic resonance imaging (MRI), such complex topology can be rendered as a +3D mesh, with each vertex/cell presenting fundamental cortical attributes, e.g., +cortical thickness, mean curvature, and average convexity. For brain develop- +ment/degeneration analyses, advanced geometric deep learning methods can be +potentially applied to learning from cortical surface data a powerful and efficient +classification/prediction model, either over the original 3D mesh or after mapping +it onto a spherical surface to ease the computation [47]. Although these existing +studies suggested promising accuracies of geometric deep learning in multiple +tasks (e.g., parcellation [47], registration [36], and longitudinal prediction [22]), +the learned models typically lack explainability and interpretability. +Explainability in Deep Neural Networks. The instance-level explanation +approaches aim to study why a deep model makes a specific prediction for +a given input, e.g., which part of the input contributes more to its output +classification score. Such input-dependent explanation methods can be roughly +categorized into four branches, including gradient/feature-based, perturbation- +based, decomposition-based, and surrogate-based methods [46]. As the most +straightforward strategy, gradient/feature-based approaches have been actively +studied in deep learning over gridded data [11], with some extensions to non- +Euclidean cases (e.g., graphs) [46,24]. Currently, research regarding explainable +geometric deep learning on 3D meshes (like brain cortical surfaces) is still lim- +ited [30]. More importantly, existing explanation methods (in both gridded and +non-Euclidean spaces) typically investigate network outputs in a post-hoc fash- +ion [11,46,23], assuming that faithful explanations are the results of accurate pre- +dictions/classifications. In this paper, we have an opposite claim that leveraging +domain knowledge to perform end-to-end learning of feature-based explanation +factors for discriminative representation extraction (and classification perfor- +mance enhancement) could be a more intuitive strategy to produce fine-grained +explainability in challenging tasks. + +4 +Authors Suppressed Due to Excessive Length +EB-1 +DB-1 +A +EB-4 +T A +�𝑠�, 𝑠�} +�𝑠�, 𝑠�} +DB-2 +A +�𝑠�, 𝑠�} +DB-3 +A +�𝑠�, 𝑠�} +C +EB-2 +EB-3 +Cortical attributes +Left +Right +Convexity +Mean curvature +Cortical thickness +Spherical +mapping +1D +Conv +1D +Conv +⋯ +⋯ +Input features +Share +weights +GAP +�𝑠�, 𝑠�} +⊙ +Attention-gated +features +Spherical attention mechanism +Left +Right +Left +Right +Left +Right +Classification +score +Attention maps +EB +: Hexagonal Conv + Pooling (except EB-4) +: Cross-hemisphere Self-Attention +T +A : Spherical Attention Mechanism +DB +: Hexagonal Transposed Conv ++ Skip connection ++ Hexagonal Conv +C : GAP + 1D Conv +: Concatenation +: Multiplication +⊙: Dot-product +Fig. 1: The schematic diagram of our NeuroExplainer that learns to capture fine- +grained explanation factors in an end-to-end attention-decoding architecture to +boost discriminative representation extraction from cortical-surface data. +3 +Method +As the schematic diagram shown in Fig. 1, our NeuroExplainer works on the +high-resolution spherical surfaces of both brain hemispheres (each with 10, 242 +vertices). The inputs are fundamental vertex-wise cortical attributes, i.e., thick- +ness, mean curvature, and convexity. The architecture has two main parts, in- +cluding an encoding branch to produce initial task-related attentions on down- +sampled hemispheric surfaces, and a set of attention decoding blocks to hier- +archically propagate such vertex-wise attentions onto higher-resolution spheres, +finally capturing fine-grained explanation factors on the input high-resolution +surfaces to boost the prediction task (i.e., the differentiation between preterm +and fullterm infants in our study). The whole network is trained end-to-end by +minimizing multi-resolution classification losses, under the constraints provided +by prior-induced regularizations to enhance explanation metrics. +3.1 +Spherical Attention Encoding +The starting components of the encoding branch are four spherical convolution +blocks (i.e., EB-1 to EB-4 in Fig. 1), with the learnable parameters shared across +two hemispheric surfaces. Each EB adopts 1-ring hexagonal convolution [47] +followed by batch normalization (BN) and ReLU activation to extract vertex- +wise representations, which are then downsampled by hexagonal max pooling +[47] (except in EB-4) to serve as the input of the subsequent layer. Based on +the outputs from EB, we propose a learnable spherical attention mechanism to +conduct weakly-supervised discriminative localization. +Specifically, let Fl and Fr ∈ R162×M0 be the vertex-wise representations +(produced by EB-4) for the left and right hemispheres, respectively. We first +concatenate them as a 324 × M0 matrix, on which a self-attention operation +[38] is applied to capturing cross-hemisphere long-range dependencies to refine +the vertex-wise representations from both hemispheric surfaces, resulting in a +unified feature matrix denoted as F0 = [ˆFl; ˆFr] ∈ R324×M0. As shown in Fig. + +10242lrvatureNeuroExplainer +5 +1, F0 is further global average pooled (GAP) across all vertices to be a holistic +feature vector f0 ∈ R1×M0 representing the whole cerebral cortex. Both F0 and +f0 are then mapped by a same vertex-wise 1D convolution (i.e., W0 ∈ RM0×2, +without bias) into the categorical space, denoted as A0 = [Al +0; Ar +0] ∈ R324×2 +and s0, respectively. Notably, so is supervised by the one-hot code of subject’s +categorical label, by which Al +0 and Ar +0 highlight discriminative vertices on the +(down-sampled) left and right surfaces, respectively, considering that +s0[i] ∝ +� +1T F0 +� +W0[:, i] = 1T � +[ˆFl; ˆFr]W0[:, i] +� += 1T Al +0[:, i] + 1T Ar +0[:, i], +(1) +where so[i] (i = 0 or 1) in our study denote the prediction scores of preterm +and fullterm, respectively, and 1 is an unit vector having the same row size +with the subsequent matrix. Finally, we define the hemispheric attentions as +¯Al +0 = �1 +i=0 s0[i]Al +0[:, i] and ¯Ar +0 = �1 +i=0 s0[i]Ar +0[:, i] ∈ R324×1, respectively, with +values spatially varying and depending on the relevance to subject’s category. +3.2 +Hierarchically Spherical Attention Decoding +The explanation factors captured by the encoding branch are relatively coarse, +as the receptive field of a cell on the downsampled surfaces (with 162 vertices +after three pooling operations) is no smaller than a hexagonal region of 343 +cells on the input surfaces (with 10, 242 vertices). To tackle this challenge, we +design a spherical attention decoding strategy to hierarchically propagate coarse +attentions (from lower-resolution spheres) onto higher-resolution spheres, based +on which fine-grained attentions are finally produced to improve classification. +Specifically, NeuroExplainer contains three consecutive decoding blocks (i.e., +DB-1 to DB-3 in Fig. 1). Each DB adopts both the attention-gated discriminative +representations from the preceding DB (except DB-1 that uses EB-4 outputs) +and the local-detailed representations from the symmetric EB (at the same reso- +lution) as the input. Let the attention-gated representations from the preceding +DB be Fl +G = +� ¯Al +in11×Min +� +⊙ ˆFl +in and Fr +G = +� ¯Ar +in11×Min +� +⊙ ˆFr +in, respectively, +where each row of ˆFin has Min channels, and ⊙ denotes element-wise dot prod- +uct. We first upsample Fl +G and Fr +G to the spatial resolution of the current DB, +by using hexagonal transposed convolutions [47] with learnable weights shared +across hemispheres. Then, the upsampled discriminative representations from +each hemisphere (say ˜Fl +G and ˜Fr +G) are channel-wisely concatenated with the lo- +cal representations from the corresponding EB (say Fl +E and Fr +E), followed by an +1-ring convolution to produce a unified feature matrix, such as +FD = [Cθ(˜Fl +G ⊕ Fl +E); Cθ(˜Fr +G ⊕ Fr +E)], +(2) +where Cθ(·) denotes 1-ring conv parameterized by θ, and ⊕ stands for channel +concatenation. In terms of FD, the attention mechanism described in (1) is +further applied to producing refined spherical attentions and classification scores. +Finally, as shown in Fig. 1, based on the fine-grained attentions over the input +surfaces (each with 10, 242 vertices), we use GAP to aggregate the attention- +gated representations and apply an 1D conv to output the classification score. + +6 +Authors Suppressed Due to Excessive Length +Original convexity +… +Random coarsening +-15 +10 +Atypical subject +⋯ +Feature matrix 𝐅� +� +Attention 𝐀� +� +⋯ +Feature matrix 𝐅� +� +Attention 𝐀� +� +Typical subject +Pull +Discriminative +representation space +(a) +(b) +Fig. 2: Brief illustrations of (a) the explanation fidelity-aware contrastive learning +strategy, and (b) explanation stability-aware data augmentation strategy. +3.3 +Domain Knowledge-Guided Explanation Enhancement +In this study, we design NeuroExplainer under a central idea that task-oriented +learning of explanation factors to boost discriminative representation extraction +can inversely assure fine-grained explainability in challenging cases like learn- +ing on complex cortical-surface data. To effectively train our network for such a +purpose, we design a set of targeted regularization strategies by considering fun- +damental domain knowledge regarding infant brain development. Specifically, it +is reasonable to assume that human brains in infancy have generally consistent +developments, while the structural/functional discrepancies between different +groups (e.g., preterm and term-born) are typically localized [37,10]. Accordingly, +we require the preterm-altered cortical development patterns captured by our +NeuroExplainer to be discriminative, spatially sparse, and robust, which sug- +gests the design of the following constraints that concurrently optimize fidelity, +sparsity, and stability metrics [46] in deploying an explainable deep network. +Explanation Fidelity-Aware Contrastive Learning. Given the spherical +attention block at a specific resolution, we have A+ +i +and A− +j ∈ RV ×1 as the +output attentions for a positive and negative subjects (i.e., preterm and fullterm +infants in our study), respectively, and F+ +i and F− +j ∈ RV ×M are the correspond- +ing representation matrices. Based on the prior knowledge regarding infant brain +development, it is reasonable to assume that A+ +i highlights atypically-developed +cortical regions caused by preterm birth. In contrast, the remaining part of the +cerebral cortex of a preterm infant (corresponding to 1 − A+ +i ) still growths nor- +mally, i.e., looking globally similar to the cortex of a term-born infant. +Accordingly, as the illustration shown in Fig. 2(a), we design a fidelity- +aware contrastive penalty to regularize the learning of the attention maps and +associated representations to improve their discriminative power. Let f + +i += +1T � +A+ +i 11×M ⊙ F+ +i +� +and ¯f + +i += 1T � +{1 − A+ +i }11×M ⊙ F+ +i +� +be the holistic fea- +ture vector and its inverse for the ith (positive) sample, respectively. Similarly, +f − +j = 1T � +A− +j 11×M ⊙ F− +j +� +denotes the holistic feature vector for the compared +jth (negative) sample. By pushing f + +i +away from both ¯f + +i +and f − +j , while pulling + +Original convexitNeuroExplainer +7 +¯f + +i +close to f − +j , we define the respective loss as +Lcontrast = +N +� +i̸=j +|| ¯f + +i − f − +j || + max(m − || ¯f + +i − f + +i ||, 0) + max(m − ||f − +j − f + +i ||, 0), +(3) +where i and j indicate any a pair of positive and negative cases from totally N +training samples, and m is a margin setting as 1 in our implementation. +Explanation Sparsity-Aware Regularization. According to the specified +prior knowledge regarding infant brain development, the attention maps pro- +duced by our NeuroExplainer should have two featured properties in terms of +sparsity. That is, the attention map for a preterm infant (e.g., A+ +i ) should be +sparse, considering that altered cortical developments are assumed to be local- +ized. In contrast, the attention map for a healthy term-born infant (e.g., A− +j ) +should not be spatially informative, as all brain regions growth typically without +abnormality. To this end, we design a straightforward entropy-based regulariza- +tion to enhance results’ explainability, such as +Lentropy = +N +� +i̸=j +1T � +A+ +i ⊙ log(A+ +i ) − A− +j ⊙ log(A− +j ) +� +, +(4) +where i and j indicate a positive and a negative cases from totally N training +samples, respectively, and 1 is an unit vector to sum up the values of all vertices. +Explanation Stability-Aware Regularization. We enhance the explanation +stability of our NeuroExplainer from two aspects. First, we require the spherical +attention mechanisms to robustly decode from complex cortical-surface data (po- +tentially containing perturbed and noisy information) fine-grained explanation +factors to produce accurate predictions. To this end, a customized data augmen- +tation strategy is designed to explicitly increase perturbations and variances in +preprocessing the data for training such a network. Specifically, the inputs of our +network are cortical surfaces with 10, 242 vertices that are downsampled from +the original data with 163, 842 vertices. We randomize this surface coarsening +step by quantifying a vertex’s cortical attributes (on the downsampled surface) +as the average of a random subset of the vertices from the respective hexagonal +region of the highest-resolution surface. As the examples summarized in Fig. +2(b), such a data augmentation strategy can generate partially different training +samples from one single subject. Considering that the network is trained to pro- +duce consistently accurate predictions for all these variants with perturbations, +it inversely enhances the stability of learned explanation factors. +Second, as described in Sec. 3.2, the fine-grained explanation factors over +high-resolution cortical surface are captured by the proposed hierarchical atten- +tion decoding strategy, where coarse results from the encoder part serve as the +foundation. To further enhance the explanation stability, we design a cross-scale +consistency regularization to refine the decoding branch. Specifically, let Al +i and + +8 +Authors Suppressed Due to Excessive Length +Ah +i be the spherical attentions from two different DB blocks. We simply minimize +Lconsistent = +N +� +i=1 +� +Al +i − Ah +i +�2, +(5) +which encourages spherical attention maps at different spatial resolutions to be +consistent in network training. +Implementation Details. In our implementation, the feature representations +produced by EB-1 to EB-4 in Fig. 1 have 32, 64, 128, and 256 channels, respec- +tively. Correspondingly, DB-1 to DB-3, and the final classification layer have 256, +128, 64, and 32 channels, respectively. The network was trained end-to-end by +minimizing the cross-entropy classification losses defined at three different spa- +tial resolutions (overall denoted as LCE), coupled with the regularization terms +introduced in Sec. 3.3, such as +L = LCE + λ1Lcontrast + λ2Lentropy + λ3Lconsistent, +(6) +where the tuning parameters were empirically set as λ1 = 0.2, λ3 = 0.5, and +λ3 = 0.1. The network parameters were updated by using Adam optimizer for +500 epochs, with the initial learning rate setting as 0.001 and bath size as 20. +4 +Experiments +Dataset & Experimental Setup. We conducted experiments on the dHCP +benchmark [26]. The structural MRIs of 700 infants scanned at term-equivalent +ages (35-44 weeks postmenstrual age) were studied, including 143 preterm and +557 term-born infants. These subjects were randomly split as a training set of 500 +infants (89 preterm and 411 fullterm), and a test set of the remaining 200 infants +(54 preterm and 146 fullterm). Using the data-augmentation strategy described +in Sec. 3.3, the training set was augmented to have roughly 1, 250 subjects from +each category for balanced network training. The input spherical surfaces contain +10, 242 vertices, and each of them has three morphological attributes, i.e., cortical +thickness, mean curvature, and convexity. +For classification, our NeuroExplainer was compared with three representa- +tive geometric deep networks, including a spherical network based on 1-ring con- +volution (SphericalCNN) [47], a MoNet reimplementation working on spherical +surfaces (SphericalMoNet) [36], and SubdivNet [16] working on original cor- +tical meshes. In addition, we conducted detailed ablation studies to verify the +efficacy of each prior-guided regularization introduced in Sec. 3.3. The classifi- +cation performance was quantified in terms of accuracy (ACC), area under the +ROC curve (AUC), sensitivity (SEN), and specificity (SPE). +On the other hand, the explanation performance of our NeuroExplainer was +compared with two representative feature-based explanation approaches, i.e., +CAM [49] and Grad-CAM [31], which were coupled with the geometric net- +works described above for post-hoc analysis. The explanation performance was + +NeuroExplainer +9 +Table 1: Classification results obtained by the competing geometric deep net- +works and different variants of our NeuroExplainer. +Competing Mehtods +ACC +AUC +SEN +SPE +SphericalCNN [47] +0.93 +0.92 +0.76 +0.98 +SphericalMoNet [36] +0.85 +0.93 +0.65 +0.92 +SubdivNet [16] +0.79 +0.67 +0.74 +0.80 +NeuroExplainer (ours) +0.95 +0.97 +0.94 +0.95 +w/o Lcontrast (3) +0.88 +0.89 +0.80 +0.91 +w/o Lentropy (4) +0.91 +0.96 +0.74 +0.97 +w/o Lconsistent (5) +0.89 +0.95 +0.89 +0.88 +quantitatively evaluated in terms of three metrics [46], i.e., Fidelity that mea- +sures the classification differences between the network captured explanation +factors and the remaining part, Sparsity of the captured explanation factors +compared to the whole surface, and Stability that measures the average clas- +sification accuracy on different perturbations of a single subject. Please refer +to [46] for more details regarding these explainability evaluation metrics. +Classification Results. The classification results obtained by different com- +peting methods are summarized in Table 1, from which we can have at least +three observations. 1) Compared with SubdivNet working on original meshes, +and the other two networks working on spherical surfaces (i.e., SphericalCNN +and SphericalMoNet), our NeuroExplainer consistently led to better classifica- +tion accuracies in terms of all metrics. 2) The improvements brought by our +method are especially significant in terms of SEN and AUC, suggesting that +it can reliably identify featured development patterns associated with preterm +birth to make accurate predictions in such an imbalanced learning task. These +results imply that our idea to capture fine-grained explanation factors in an end- +to-end fashion to boost discriminative representation extraction is beneficial for +deploying an accurate classification model in the task of learning on complex +surface data containing noisy, redundant, and task-irrelevant information. +3) To check the efficacy of the prior-induced regularization strategies, we +orderly removed them from the loss function (6) to quantify the respective in- +fluence on classification results. From Table 1, we can see that all the three reg- +ularizations (i.e., Lcontrast, Lentropy, and Lconsistent) demonstrated significant +but different improvements on classification. Specifically, according to the com- +parison between NeuroExplainer and its variant w/o Lentropy, we can see that +such an explanation sparsity-aware regularization boosted SEN by 20%, imply- +ing its efficacy in capturing localized patterns related to preterm-altered cortical +development. By comparing NeuroExplainer with its variant w/o Lconsistent, +we can see that the explanation stability-ware regularization helped stabilize +the learning of the hierarchical attention-decoding branch, leading to relatively +large improvements of overall classification ACC (by 6%). Finally, we can see +that the explanation fidelity-aware contrastive learning strategy brought overall +the largest improvements of both ACC and AUC (by 7% and 8%, respectively), + +10 +Authors Suppressed Due to Excessive Length +Table 2: Quantitative explanation results obtained by the competing post-hoc +approaches and our end-to-end NeuroExplainer. +Competing Methods +Fidelity +Sparsity Stability +CAM [49] + +SphericalCNN +0.24 +0.91 +0.77 +SphericalMoNet +0.55 +0.93 +0.58 +SubdivNet +0.06 +0.97 +0.53 +Grad-CAM [31] + +SphericalCNN +0.22 +0.99 +0.77 +SphericalMoNet +0.42 +0.98 +0.58 +SubdivNet +0.16 +0.96 +0.53 +NeuroExplainer (ours) +0.56 +0.73 +0.96 +0 +1 +Left +Right +Subject 1 +Subject 2 +Grad-CAM +Spherical +CNN +Spherical +MoNet +CAM +Spherical +MoNet +Spherical +CNN +Ours +Fine- +grained +Coarse- +scaled +Ours +Fine- +grained +Coarse- +scaled +CAM +Spherical +MoNet +Spherical +CNN +Grad-CAM +Spherical +CNN +Spherical +MoNet +Fig. 3: Typical examples of the explanation factors captured by the com- +peting post-hoc approaches (CAM and Grad-CAM) and our end-to-end +NeuroExplainer. Higher values indicate larger links to preterm birth. +implying its efficacy in transforming domain knowledge regarding atypical brain +development to capture fine-grained explanation factors that boost discrimina- +tive representation learning. +Explanation Results. The quantitative explanation results obtained by our +NeuroExplainer and other feature-based explanation methods (i.e., CAM and +Grad-CAM coupled with the trained geometric deep networks, respectively) +are summarized in Table 2. From Table 2, we can observe that our end-to-end +NeuroExplainer outperformed other post-hoc explanation approaches by a large +margin in terms of the three explainability metrics. Notably, the three metrics +should be analyzed concurrently in evaluating a network’s explainability [46], as +the isolated quantification of a single metric could be biased. For example, al- +though Grad-CAM+SphericalCNN led to the largest Sparsity value (= 0.99) in +our experiment, the corresponding Fidelity and Stability values are significantly +low (= 0.22 and 0.77, respectively), indicating that the very sparse explanation +factors captured by Grad-CAM+SphericalCNN are relatively uninformative and +random. In contrast, our NeuroExplainer led to significantly better Fidelity and +Stability, under reasonable Sparsity, suggesting that it can robustly identify lo- +calized preterm-altered cortical patterns from high-dimensional inputs to boost +discriminative representation learning for preterm infant recognition. + +NeuroExplainer +11 +-10 +10 +0 +1 +Our +NeuroExplainer +Existing Multi-Modal (dMRI + sMRI) Studies +Mean Diffusivity +Neurite Density +Cortical Thickness +Fig. 4: Comparison of the individualized preterm-altered cortical development +patterns uncovered by NeuroExplainer with the group-wise patterns identified +by existing multi-modal studies [10]. +In addition to the above quantitative evaluations, we also visually compared +the attention maps produced by different competing methods, with two typical +examples presented in Fig. 3. From Fig. 3, we can have two main observations. 1) +Compared with post-hoc explanation methods (i.e., CAM and Grad-CAM), our +end-to-end NeuroExplainer stably produced more reasonable attentions. Specif- +ically, given a subject (e.g., Subject 2 in Fig. 3), CAM and Grad-CAM could +produce very different explanation results for a trained classification model (e.g., +SphericalCNN). Similarly, a post-hoc approach (e.g., CAM) could produce dis- +tinct results for two different classification models on the same subject. Due +to the nature of end-to-end learning of explanation factors to establish classi- +fication models, our NeuroExplainer effectively avoided such a problem. More +importantly, across different subjects, our NeuroExplainer led to group-wisely +more consistent explanations than these post-hoc approaches. Also, it produced +more consistent results across hemispheres, without any related constraints dur- +ing network training. 2) We can see that the coarse attentions produced by +NeuroExplainer’s encoder are consistent with the final fine-grained outputs of +the decoder, which implies the positive effect of the cross-scale consistency reg- +ularization in enhancing explainability. +Finally, we compared the individualized preterm-altered cortical development +patterns uncovered by our NeuroExplainer with representative group-wise neu- +roimaging studies in the literature, e.g., the multi-modal (dMRI and sMRI) +quantitative analyses presented in [10]. According to the comparisons shown in +Fig. 3, we can see that our observations in this paper are consistent with [10]. +The discriminative cortical regions captured by our NeuroExplainer (using solely +morphological features) are largely overlapped with the group-wise significantly +different regions identified by [10] in terms of the mean diffusivity, neurite den- +sity, and cortical thickness, respectively. For example, they both highlighted some +specific regions in the inferior parietal, medial occipital, and superior temporal +lobe, and posterior insula, which is worth deeper evaluations in the future. + +12 +Authors Suppressed Due to Excessive Length +5 +Conclusion +In the paper, we have proposed an geometric deep network, i.e., NeuroExplainer, +to learn fine-grained explanation factors from complex cortical-surface data to +boost discriminative representation extraction and accurate classification model +construction. 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National science +review 5(1), 44–53 (2018) + diff --git a/JdAyT4oBgHgl3EQf5_qu/content/tmp_files/load_file.txt b/JdAyT4oBgHgl3EQf5_qu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5f9560883e76671a0d753ff7abd59744a82f445 --- /dev/null +++ b/JdAyT4oBgHgl3EQf5_qu/content/tmp_files/load_file.txt @@ -0,0 +1,645 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf,len=644 +page_content='NeuroExplainer: Fine-Grained Attention Decoding to Uncover Cortical Development Patterns of Preterm Infants Chenyu Xue1, Fan Wang2, Yuanzhuo Zhu2, Hui Li3, Deyu Meng1, Dinggang Shen4, and Chunfeng Lian1 1 School of Mathematics and Statistics,Xi’an Jiaotong University, Xi’an, China chunfeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='lian@xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='cn 2 Key Laboratory of Biomedical Information Engineering of Ministry of Education, School of Life Science and Technology, Xi’an Jiaotong University, Xi’an, China fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='wang@xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='cn 3 Department of Neonatology, The First Affiliated Hospital of Xi’an Jiaotong University, Xi’an, China 4 School of Biomedical Engineering, ShanghaiTech University, Shanghai, China Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Deploying reliable deep learning techniques in interdisciplinary applications needs learned models to output accurate and (even more im- portantly) explainable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Existing approaches typically expli- cate network outputs in a post-hoc fashion, under an implicit assumption that faithful explanations come from accurate predictions/classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' We have an opposite claim that explanations boost (or even determine) classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' That is, end-to-end learning of explanation factors to aug- ment discriminative representation extraction could be a more intu- itive strategy to inversely assure fine-grained explainability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', in those neuroimaging and neuroscience studies with high-dimensional data con- taining noisy, redundant, and task-irrelevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In this pa- per, we propose such an explainable geometric deep network dubbed as NeuroExplainer, with applications to uncover altered infant cortical development patterns associated with preterm birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Given fundamen- tal cortical attributes as network input, our NeuroExplainer adopts a hierarchical attention-decoding framework to learn fine-grained atten- tions and respective discriminative representations to accurately rec- ognize preterm infants from term-born infants at term-equivalent age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' NeuroExplainer learns the hierarchical attention-decoding modules un- der subject-level weak supervision coupled with targeted regularizers de- duced from domain knowledge regarding brain development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' These prior- guided constraints implicitly maximizes the explainability metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', fidelity, sparsity, and stability) in network training, driving the learned network to output detailed explanations and accurate classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Ex- perimental results on the public dHCP benchmark suggest that NeuroExplainer led to quantitatively reliable explanation results that are qualitatively consistent with representative neuroimaging studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Keywords: Geometric Deep Learning · Explainability · Infant Brain Cortical Development · AI for Neuroscience arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='00815v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='LG] 1 Jan 2023 2 Authors Suppressed Due to Excessive Length 1 Introduction Due to the capacity of learning highly nonlinear representations in task-oriented fashions, deep neural networks are showing promising applications in many in- terdisciplinary communities, including biomedical image computing, neuroimag- ing, and neuroscience studies [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' To quantitatively analyze brain develop- ment/degeneration or timely diagnose associated disorders, various deep learn- ing methods have been proposed [45,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Most of these studies focused on the designs of network architectures and learning strategies to produce accurate pre- dictions/classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' A critical challenge is that the learned models typically lack computational interpretability and results’ explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' From the aspect of practical usage, neuroimaging and neuroscience studies desire AI tools that can make accurate and (even more importantly) explainable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' For example, a key value of a brain disease diagnosis network is to identify from high-dimensional neuroimage data individualized subtle changes leading to ac- curate classification, which can be clues for human experts to analyze disease heterogeneity [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Also, the classification task to differentiate between preterm and term-born (or male and female) infants may not be practically meaningful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' in contrast, fine-grained differences on brain cortical surfaces, identified by the learned classification network, could be valuable factors for better understanding featured cortical development patterns of infants from different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Recently, explainable and interpretable deep learning is being actively stud- ied in the machine learning community, with obviously more works on gridded data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', images) [11] than on non-Euclidean data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', 3D meshes) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Accu- rate classification and faithful explanation are highly correlated and inseparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Existing methods typically adopt post-hoc techniques to explain a deep network [46], which is first trained for a specific classification task, and then the underly- ing (sparse) correlations between its input and output are analyzed offline, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', by backpropagating prediction gradients to the shallow layers [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Notably, such post-hoc approaches are established upon a common assumption that reliable ex- planations are the results caused by accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' This assumption could work in general applications that have large-scale training data, while cannot al- ways hold for neuroimaging and neuroscience research, where available data are typically small-sized and much more complex (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', high-resolution cortical sur- faces containing noisy, highly redundant, and task-irrelevant information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Distinct to existing post-hoc methods, we have an opposite claim that ex- plainability boosts or even determines classification, especially in those chal- lenging tasks related to brain cortical development analyses based on high- dimensional neuroimaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The key is how to construct an end-to-end framework, where fine-grained explanation factors can be identified in a fully learnable fashion to enhance discriminative representation extraction and finally output accurate classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' This paper presents such an explainable geometric deep network, called NeuroExplainer, with applications to uncover altered infant cortical development patterns associated with preterm birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' NeuroExplainer adopts high-resolution cortical attributes as the input to develop a hierarchi- cal attention-decoding architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In the framework of weakly supervised dis- NeuroExplainer 3 criminative localization, our NeuroExplainer is trained by minimizing general classification losses coupled with a set of constraints designed according to prior knowledge regarding brain development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' These targeted regularizers drive the network to implicitly optimize the explainability metrics from multiple aspects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', fidelity, sparsity, and stability), thus capturing fine-grained explanation fac- tors to explicitly improve classification accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Experimental results on the public dHCP benchmark suggest that our NeuroExplainer led to quantitatively reliable explanation results that are qualitatively consistent with representative neuroimaging studies, implying that it could be a practically useful AI tool for other related cortical surface-based neuroimaging studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 2 Related Work Deep Learning for Cortical Surface Analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The human cerebral cortex is a highly folded and thin sheet of gray matter [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' By leveraging structural magnetic resonance imaging (MRI), such complex topology can be rendered as a 3D mesh, with each vertex/cell presenting fundamental cortical attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', cortical thickness, mean curvature, and average convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' For brain develop- ment/degeneration analyses, advanced geometric deep learning methods can be potentially applied to learning from cortical surface data a powerful and efficient classification/prediction model, either over the original 3D mesh or after mapping it onto a spherical surface to ease the computation [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Although these existing studies suggested promising accuracies of geometric deep learning in multiple tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', parcellation [47], registration [36], and longitudinal prediction [22]), the learned models typically lack explainability and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Explainability in Deep Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The instance-level explanation approaches aim to study why a deep model makes a specific prediction for a given input, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', which part of the input contributes more to its output classification score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Such input-dependent explanation methods can be roughly categorized into four branches, including gradient/feature-based, perturbation- based, decomposition-based, and surrogate-based methods [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' As the most straightforward strategy, gradient/feature-based approaches have been actively studied in deep learning over gridded data [11], with some extensions to non- Euclidean cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', graphs) [46,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Currently, research regarding explainable geometric deep learning on 3D meshes (like brain cortical surfaces) is still lim- ited [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' More importantly, existing explanation methods (in both gridded and non-Euclidean spaces) typically investigate network outputs in a post-hoc fash- ion [11,46,23], assuming that faithful explanations are the results of accurate pre- dictions/classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In this paper, we have an opposite claim that leveraging domain knowledge to perform end-to-end learning of feature-based explanation factors for discriminative representation extraction (and classification perfor- mance enhancement) could be a more intuitive strategy to produce fine-grained explainability in challenging tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 4 Authors Suppressed Due to Excessive Length EB-1 DB-1 A EB-4 T A �𝑠�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 𝑠�} �𝑠�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 𝑠�} DB-2 A �𝑠�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 𝑠�} DB-3 A �𝑠�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 𝑠�} C EB-2 EB-3 Cortical attributes Left Right Convexity Mean curvature Cortical thickness Spherical mapping 1D Conv 1D Conv ⋯ ⋯ Input features Share weights GAP �𝑠�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 𝑠�} ⊙ Attention-gated features Spherical attention mechanism Left Right Left Right Left Right Classification score Attention maps EB : Hexagonal Conv + Pooling (except EB-4) : Cross-hemisphere Self-Attention T A : Spherical Attention Mechanism DB : Hexagonal Transposed Conv + Skip connection + Hexagonal Conv C : GAP + 1D Conv : Concatenation : Multiplication ⊙: Dot-product Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 1: The schematic diagram of our NeuroExplainer that learns to capture fine- grained explanation factors in an end-to-end attention-decoding architecture to boost discriminative representation extraction from cortical-surface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3 Method As the schematic diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 1, our NeuroExplainer works on the high-resolution spherical surfaces of both brain hemispheres (each with 10, 242 vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The inputs are fundamental vertex-wise cortical attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', thick- ness, mean curvature, and convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The architecture has two main parts, in- cluding an encoding branch to produce initial task-related attentions on down- sampled hemispheric surfaces, and a set of attention decoding blocks to hier- archically propagate such vertex-wise attentions onto higher-resolution spheres, finally capturing fine-grained explanation factors on the input high-resolution surfaces to boost the prediction task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', the differentiation between preterm and fullterm infants in our study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The whole network is trained end-to-end by minimizing multi-resolution classification losses, under the constraints provided by prior-induced regularizations to enhance explanation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='1 Spherical Attention Encoding The starting components of the encoding branch are four spherical convolution blocks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', EB-1 to EB-4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 1), with the learnable parameters shared across two hemispheric surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Each EB adopts 1-ring hexagonal convolution [47] followed by batch normalization (BN) and ReLU activation to extract vertex- wise representations, which are then downsampled by hexagonal max pooling [47] (except in EB-4) to serve as the input of the subsequent layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Based on the outputs from EB, we propose a learnable spherical attention mechanism to conduct weakly-supervised discriminative localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Specifically, let Fl and Fr ∈ R162×M0 be the vertex-wise representations (produced by EB-4) for the left and right hemispheres, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' We first concatenate them as a 324 × M0 matrix, on which a self-attention operation [38] is applied to capturing cross-hemisphere long-range dependencies to refine the vertex-wise representations from both hemispheric surfaces, resulting in a unified feature matrix denoted as F0 = [ˆFl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' ˆFr] ∈ R324×M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 10242lrvatureNeuroExplainer 5 1, F0 is further global average pooled (GAP) across all vertices to be a holistic feature vector f0 ∈ R1×M0 representing the whole cerebral cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Both F0 and f0 are then mapped by a same vertex-wise 1D convolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', W0 ∈ RM0×2, without bias) into the categorical space, denoted as A0 = [Al 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Ar 0] ∈ R324×2 and s0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Notably, so is supervised by the one-hot code of subject’s categorical label, by which Al 0 and Ar 0 highlight discriminative vertices on the (down-sampled) left and right surfaces, respectively, considering that s0[i] ∝ � 1T F0 � W0[:, i] = 1T � [ˆFl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' ˆFr]W0[:, i] � = 1T Al 0[:, i] + 1T Ar 0[:, i], (1) where so[i] (i = 0 or 1) in our study denote the prediction scores of preterm and fullterm, respectively, and 1 is an unit vector having the same row size with the subsequent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Finally, we define the hemispheric attentions as ¯Al 0 = �1 i=0 s0[i]Al 0[:, i] and ¯Ar 0 = �1 i=0 s0[i]Ar 0[:, i] ∈ R324×1, respectively, with values spatially varying and depending on the relevance to subject’s category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='2 Hierarchically Spherical Attention Decoding The explanation factors captured by the encoding branch are relatively coarse, as the receptive field of a cell on the downsampled surfaces (with 162 vertices after three pooling operations) is no smaller than a hexagonal region of 343 cells on the input surfaces (with 10, 242 vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' To tackle this challenge, we design a spherical attention decoding strategy to hierarchically propagate coarse attentions (from lower-resolution spheres) onto higher-resolution spheres, based on which fine-grained attentions are finally produced to improve classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Specifically, NeuroExplainer contains three consecutive decoding blocks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', DB-1 to DB-3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Each DB adopts both the attention-gated discriminative representations from the preceding DB (except DB-1 that uses EB-4 outputs) and the local-detailed representations from the symmetric EB (at the same reso- lution) as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Let the attention-gated representations from the preceding DB be Fl G = � ¯Al in11×Min � ⊙ ˆFl in and Fr G = � ¯Ar in11×Min � ⊙ ˆFr in, respectively, where each row of ˆFin has Min channels, and ⊙ denotes element-wise dot prod- uct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' We first upsample Fl G and Fr G to the spatial resolution of the current DB, by using hexagonal transposed convolutions [47] with learnable weights shared across hemispheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Then, the upsampled discriminative representations from each hemisphere (say ˜Fl G and ˜Fr G) are channel-wisely concatenated with the lo- cal representations from the corresponding EB (say Fl E and Fr E), followed by an 1-ring convolution to produce a unified feature matrix, such as FD = [Cθ(˜Fl G ⊕ Fl E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Cθ(˜Fr G ⊕ Fr E)], (2) where Cθ(·) denotes 1-ring conv parameterized by θ, and ⊕ stands for channel concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In terms of FD, the attention mechanism described in (1) is further applied to producing refined spherical attentions and classification scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Finally, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 1, based on the fine-grained attentions over the input surfaces (each with 10, 242 vertices), we use GAP to aggregate the attention- gated representations and apply an 1D conv to output the classification score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 6 Authors Suppressed Due to Excessive Length Original convexity … Random coarsening 15 10 Atypical subject ⋯ Feature matrix 𝐅� � Attention 𝐀� � ⋯ Feature matrix 𝐅� � Attention 𝐀� � Typical subject Pull Discriminative representation space (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 2: Brief illustrations of (a) the explanation fidelity-aware contrastive learning strategy, and (b) explanation stability-aware data augmentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='3 Domain Knowledge-Guided Explanation Enhancement In this study, we design NeuroExplainer under a central idea that task-oriented learning of explanation factors to boost discriminative representation extraction can inversely assure fine-grained explainability in challenging cases like learn- ing on complex cortical-surface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' To effectively train our network for such a purpose, we design a set of targeted regularization strategies by considering fun- damental domain knowledge regarding infant brain development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Specifically, it is reasonable to assume that human brains in infancy have generally consistent developments, while the structural/functional discrepancies between different groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', preterm and term-born) are typically localized [37,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Accordingly, we require the preterm-altered cortical development patterns captured by our NeuroExplainer to be discriminative, spatially sparse, and robust, which sug- gests the design of the following constraints that concurrently optimize fidelity, sparsity, and stability metrics [46] in deploying an explainable deep network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Explanation Fidelity-Aware Contrastive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Given the spherical attention block at a specific resolution, we have A+ i and A− j ∈ RV ×1 as the output attentions for a positive and negative subjects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', preterm and fullterm infants in our study), respectively, and F+ i and F− j ∈ RV ×M are the correspond- ing representation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Based on the prior knowledge regarding infant brain development, it is reasonable to assume that A+ i highlights atypically-developed cortical regions caused by preterm birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In contrast, the remaining part of the cerebral cortex of a preterm infant (corresponding to 1 − A+ i ) still growths nor- mally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', looking globally similar to the cortex of a term-born infant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Accordingly, as the illustration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 2(a), we design a fidelity- aware contrastive penalty to regularize the learning of the attention maps and associated representations to improve their discriminative power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Let f + i = 1T � A+ i 11×M ⊙ F+ i � and ¯f + i = 1T � {1 − A+ i }11×M ⊙ F+ i � be the holistic fea- ture vector and its inverse for the ith (positive) sample, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Similarly, f − j = 1T � A− j 11×M ⊙ F− j � denotes the holistic feature vector for the compared jth (negative) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' By pushing f + i away from both ¯f + i and f − j , while pulling Original convexitNeuroExplainer 7 ¯f + i close to f − j , we define the respective loss as Lcontrast = N � i̸=j || ¯f + i − f − j || + max(m − || ¯f + i − f + i ||, 0) + max(m − ||f − j − f + i ||, 0), (3) where i and j indicate any a pair of positive and negative cases from totally N training samples, and m is a margin setting as 1 in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Explanation Sparsity-Aware Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' According to the specified prior knowledge regarding infant brain development, the attention maps pro- duced by our NeuroExplainer should have two featured properties in terms of sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' That is, the attention map for a preterm infant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', A+ i ) should be sparse, considering that altered cortical developments are assumed to be local- ized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In contrast, the attention map for a healthy term-born infant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', A− j ) should not be spatially informative, as all brain regions growth typically without abnormality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' To this end, we design a straightforward entropy-based regulariza- tion to enhance results’ explainability, such as Lentropy = N � i̸=j 1T � A+ i ⊙ log(A+ i ) − A− j ⊙ log(A− j ) � , (4) where i and j indicate a positive and a negative cases from totally N training samples, respectively, and 1 is an unit vector to sum up the values of all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Explanation Stability-Aware Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' We enhance the explanation stability of our NeuroExplainer from two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' First, we require the spherical attention mechanisms to robustly decode from complex cortical-surface data (po- tentially containing perturbed and noisy information) fine-grained explanation factors to produce accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' To this end, a customized data augmen- tation strategy is designed to explicitly increase perturbations and variances in preprocessing the data for training such a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Specifically, the inputs of our network are cortical surfaces with 10, 242 vertices that are downsampled from the original data with 163, 842 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' We randomize this surface coarsening step by quantifying a vertex’s cortical attributes (on the downsampled surface) as the average of a random subset of the vertices from the respective hexagonal region of the highest-resolution surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' As the examples summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 2(b), such a data augmentation strategy can generate partially different training samples from one single subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Considering that the network is trained to pro- duce consistently accurate predictions for all these variants with perturbations, it inversely enhances the stability of learned explanation factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Second, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='2, the fine-grained explanation factors over high-resolution cortical surface are captured by the proposed hierarchical atten- tion decoding strategy, where coarse results from the encoder part serve as the foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' To further enhance the explanation stability, we design a cross-scale consistency regularization to refine the decoding branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Specifically, let Al i and 8 Authors Suppressed Due to Excessive Length Ah i be the spherical attentions from two different DB blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' We simply minimize Lconsistent = N � i=1 � Al i − Ah i �2, (5) which encourages spherical attention maps at different spatial resolutions to be consistent in network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In our implementation, the feature representations produced by EB-1 to EB-4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 1 have 32, 64, 128, and 256 channels, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Correspondingly, DB-1 to DB-3, and the final classification layer have 256, 128, 64, and 32 channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The network was trained end-to-end by minimizing the cross-entropy classification losses defined at three different spa- tial resolutions (overall denoted as LCE), coupled with the regularization terms introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='3, such as L = LCE + λ1Lcontrast + λ2Lentropy + λ3Lconsistent, (6) where the tuning parameters were empirically set as λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='2, λ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='5, and λ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The network parameters were updated by using Adam optimizer for 500 epochs, with the initial learning rate setting as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='001 and bath size as 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 4 Experiments Dataset & Experimental Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' We conducted experiments on the dHCP benchmark [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The structural MRIs of 700 infants scanned at term-equivalent ages (35-44 weeks postmenstrual age) were studied, including 143 preterm and 557 term-born infants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' These subjects were randomly split as a training set of 500 infants (89 preterm and 411 fullterm), and a test set of the remaining 200 infants (54 preterm and 146 fullterm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Using the data-augmentation strategy described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='3, the training set was augmented to have roughly 1, 250 subjects from each category for balanced network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The input spherical surfaces contain 10, 242 vertices, and each of them has three morphological attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', cortical thickness, mean curvature, and convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' For classification, our NeuroExplainer was compared with three representa- tive geometric deep networks, including a spherical network based on 1-ring con- volution (SphericalCNN) [47], a MoNet reimplementation working on spherical surfaces (SphericalMoNet) [36], and SubdivNet [16] working on original cor- tical meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In addition, we conducted detailed ablation studies to verify the efficacy of each prior-guided regularization introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The classifi- cation performance was quantified in terms of accuracy (ACC), area under the ROC curve (AUC), sensitivity (SEN), and specificity (SPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' On the other hand, the explanation performance of our NeuroExplainer was compared with two representative feature-based explanation approaches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', CAM [49] and Grad-CAM [31], which were coupled with the geometric net- works described above for post-hoc analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The explanation performance was NeuroExplainer 9 Table 1: Classification results obtained by the competing geometric deep net- works and different variants of our NeuroExplainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Competing Mehtods ACC AUC SEN SPE SphericalCNN [47] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='98 SphericalMoNet [36] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='92 SubdivNet [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='80 NeuroExplainer (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='95 w/o Lcontrast (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='91 w/o Lentropy (4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='97 w/o Lconsistent (5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='88 quantitatively evaluated in terms of three metrics [46], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', Fidelity that mea- sures the classification differences between the network captured explanation factors and the remaining part, Sparsity of the captured explanation factors compared to the whole surface, and Stability that measures the average clas- sification accuracy on different perturbations of a single subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Please refer to [46] for more details regarding these explainability evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Classification Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The classification results obtained by different com- peting methods are summarized in Table 1, from which we can have at least three observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 1) Compared with SubdivNet working on original meshes, and the other two networks working on spherical surfaces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', SphericalCNN and SphericalMoNet), our NeuroExplainer consistently led to better classifica- tion accuracies in terms of all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 2) The improvements brought by our method are especially significant in terms of SEN and AUC, suggesting that it can reliably identify featured development patterns associated with preterm birth to make accurate predictions in such an imbalanced learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' These results imply that our idea to capture fine-grained explanation factors in an end- to-end fashion to boost discriminative representation extraction is beneficial for deploying an accurate classification model in the task of learning on complex surface data containing noisy, redundant, and task-irrelevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3) To check the efficacy of the prior-induced regularization strategies, we orderly removed them from the loss function (6) to quantify the respective in- fluence on classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' From Table 1, we can see that all the three reg- ularizations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', Lcontrast, Lentropy, and Lconsistent) demonstrated significant but different improvements on classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Specifically, according to the com- parison between NeuroExplainer and its variant w/o Lentropy, we can see that such an explanation sparsity-aware regularization boosted SEN by 20%, imply- ing its efficacy in capturing localized patterns related to preterm-altered cortical development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' By comparing NeuroExplainer with its variant w/o Lconsistent, we can see that the explanation stability-ware regularization helped stabilize the learning of the hierarchical attention-decoding branch, leading to relatively large improvements of overall classification ACC (by 6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Finally, we can see that the explanation fidelity-aware contrastive learning strategy brought overall the largest improvements of both ACC and AUC (by 7% and 8%, respectively), 10 Authors Suppressed Due to Excessive Length Table 2: Quantitative explanation results obtained by the competing post-hoc approaches and our end-to-end NeuroExplainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Competing Methods Fidelity Sparsity Stability CAM [49] + SphericalCNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='77 SphericalMoNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='58 SubdivNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='53 Grad-CAM [31] + SphericalCNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='77 SphericalMoNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='58 SubdivNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='53 NeuroExplainer (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='96 0 1 Left Right Subject 1 Subject 2 Grad-CAM Spherical CNN Spherical MoNet CAM Spherical MoNet Spherical CNN Ours Fine- grained Coarse- scaled Ours Fine- grained Coarse- scaled CAM Spherical MoNet Spherical CNN Grad-CAM Spherical CNN Spherical MoNet Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3: Typical examples of the explanation factors captured by the com- peting post-hoc approaches (CAM and Grad-CAM) and our end-to-end NeuroExplainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Higher values indicate larger links to preterm birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' implying its efficacy in transforming domain knowledge regarding atypical brain development to capture fine-grained explanation factors that boost discrimina- tive representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Explanation Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The quantitative explanation results obtained by our NeuroExplainer and other feature-based explanation methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', CAM and Grad-CAM coupled with the trained geometric deep networks, respectively) are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' From Table 2, we can observe that our end-to-end NeuroExplainer outperformed other post-hoc explanation approaches by a large margin in terms of the three explainability metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Notably, the three metrics should be analyzed concurrently in evaluating a network’s explainability [46], as the isolated quantification of a single metric could be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' For example, al- though Grad-CAM+SphericalCNN led to the largest Sparsity value (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='99) in our experiment, the corresponding Fidelity and Stability values are significantly low (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='22 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='77, respectively), indicating that the very sparse explanation factors captured by Grad-CAM+SphericalCNN are relatively uninformative and random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In contrast, our NeuroExplainer led to significantly better Fidelity and Stability, under reasonable Sparsity, suggesting that it can robustly identify lo- calized preterm-altered cortical patterns from high-dimensional inputs to boost discriminative representation learning for preterm infant recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' NeuroExplainer 11 10 10 0 1 Our NeuroExplainer Existing Multi-Modal (dMRI + sMRI) Studies Mean Diffusivity Neurite Density Cortical Thickness Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 4: Comparison of the individualized preterm-altered cortical development patterns uncovered by NeuroExplainer with the group-wise patterns identified by existing multi-modal studies [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' In addition to the above quantitative evaluations, we also visually compared the attention maps produced by different competing methods, with two typical examples presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3, we can have two main observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 1) Compared with post-hoc explanation methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', CAM and Grad-CAM), our end-to-end NeuroExplainer stably produced more reasonable attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Specif- ically, given a subject (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', Subject 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3), CAM and Grad-CAM could produce very different explanation results for a trained classification model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', SphericalCNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Similarly, a post-hoc approach (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', CAM) could produce dis- tinct results for two different classification models on the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Due to the nature of end-to-end learning of explanation factors to establish classi- fication models, our NeuroExplainer effectively avoided such a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' More importantly, across different subjects, our NeuroExplainer led to group-wisely more consistent explanations than these post-hoc approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Also, it produced more consistent results across hemispheres, without any related constraints dur- ing network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 2) We can see that the coarse attentions produced by NeuroExplainer’s encoder are consistent with the final fine-grained outputs of the decoder, which implies the positive effect of the cross-scale consistency reg- ularization in enhancing explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Finally, we compared the individualized preterm-altered cortical development patterns uncovered by our NeuroExplainer with representative group-wise neu- roimaging studies in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', the multi-modal (dMRI and sMRI) quantitative analyses presented in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' According to the comparisons shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 3, we can see that our observations in this paper are consistent with [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The discriminative cortical regions captured by our NeuroExplainer (using solely morphological features) are largely overlapped with the group-wise significantly different regions identified by [10] in terms of the mean diffusivity, neurite den- sity, and cortical thickness, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' For example, they both highlighted some specific regions in the inferior parietal, medial occipital, and superior temporal lobe, and posterior insula, which is worth deeper evaluations in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' 12 Authors Suppressed Due to Excessive Length 5 Conclusion In the paper, we have proposed an geometric deep network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', NeuroExplainer, to learn fine-grained explanation factors from complex cortical-surface data to boost discriminative representation extraction and accurate classification model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' On the benchmark dHCP database, the applications of our NeuroExplainer to uncover preterm-altered infant cortical development patterns achieved bet- ter performance in terms of both explainability and prediction accuracy, when compared with representative post-hoc approaches coupled with state-of-the- art geometric deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' The proposed method could be a promising AI tool applied to other similar cortical surface-based neuroimage and neuroscience studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=' Ball, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} +page_content=', Seidlitz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQf5_qu/content/2301.00815v1.pdf'} 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Narayan +Shashanka B. R. * +Rohit Damodaran∗ +Dr. Chandrashekhar Jayaram∗ +Dr. M. A. Kareem +Dr. Mamta P.∗ +Dr. Saravanan K. R.∗ +Dr. Monu Krishnan∗ +Dr. Raja Indana +MFine +Abstract +The reliability of the results of self monitoring of the +vitals in the wild using medical devices or wearables or +camera based smart phone solutions is subject to variabil- +ities such as position of placement, hardware of the device +and environmental factors. In this first of its kind study, +we demonstrate that this variability in self monitoring of +Blood Pressure (BP), Blood oxygen saturation level (SpO2) +and Heart rate (HR) is statistically significant (p < 0.05) +on 203 healthy subjects by quantifying positional and hard- +ware variability. We also establish the existance of this vari- +ability in camera based solutions for self-monitoring of vi- +tals in smart phones and thus prove that the use of camera +based smart phone solutions is similar to the use of medical +devices or wearables for self-monitoring in the wild. +1. Introduction +Monitoring of vitals is important for identifying and +managing of diseases. Glasziou et. al. [11] identify five +phases of monitoring a disease from a chronic condition +perspective along with the interval for monitoring. These +include continuous monitoring of vitals to (a) detect abnor- +mality; (b) confirm abnormality; (c) establish plan of action +to treat the abnormality; (d) make adjustments to the treat- +ment; and (e) to confirm the success of treatment. These +phases can very well be extended to acute conditions with- +out loss of generality except for the fact that the monitor- +ing intervals may be much shorter than that for chronic +conditions and may totally be based on self-monitoring. +Self-monitoring refers to monitoring of conditions by pa- +tients/users by using medical devices that are available off- +*The authors were at MFine when the research was carried out. Prior +consent has been obtained from the authors to publish this work +the-shelf at a place of the patient’s convenience without the +intervention of a qualified medical professional to monitor +the condition. Self monitoring in the wild refers to monitor- +ing the vitals in uncontrolled environments such as outdoors +or house where the conditions with respect to lightihg, phys- +ical activity etc., dynamically change. +At certain points in time, when access to care is made +difficult either due to the unavailability of care providers +or due to environmental factors such as the most recent +COVID-19 pandemic where patients were forced to mon- +itor their conditions at home before approaching a health- +care provider, self-monitoring plays a crucial role in saving +human lives. Given the importance of self-monitoring of +diseases in recent times, there has been a flurry of activity in +research circles to come up with novel technologies that en- +able self-monitoring of vitals such as Blood Pressure (BP), +Blood oxygen saturation level (SpO2), Heart rate (HR), +Blood Glucose etc. While some of these have made it to +commercial devices such as wearables and mobile phones, a +vast majority of the technology is still restricted to academia +and research circles. +1.1. Prior art +Of the technologies that are being researched for self- +monitoring of vitals, computer vision has received consid- +erable amount of attention the the last decade. One of the +early attempts in this direction was by Jonathan et. al., [16] +where the authors used a Nokia device to record videos at +15 FPS of a user before and after performing a physical ac- +tivity and the change in HR calculated by employing us- +ing Photoplethysmography(PPG) Imaging. Thereafter there +have been quite a few attempts to demonstrate the capability +of using PPG signals from videos/image stream to estimate +HR in an individual [4,13,14,19,24,32]. +The PPG imaging methodology employed for HR is also +a popular method to estimate SpO2 by both signal process- +1 +arXiv:2301.12943v1 [cs.CV] 30 Jan 2023 + +ing/mathematical models [6, 21, 31] and machine learning +[2, 3, 5, 17, 18]. A majority of smart phone PPG solutions +for SpO2 consider the video of a finger tip placed against +the back camera of the phone, hereafter referred to as Fin- +ger tip (FT) PPG, as the input to extract the PPG signals +and estimate the vital. A non-contact way to monitor SpO2 +is explored in [15, 35]where a video stream of user’s face, +hereafter referred to as Face (FC) PPG, is used as input for +PPG signal extraction and SpO2 Estimation. +There has been growing interest in recent times to esti- +mate BP using PPG signals extracted either from FT PPG or +FC PPG. While, the exact relationship between PPG the sig- +nal and Blood Pressure has not been clinically established, +fitting an ML model on top of either the raw signal, or some +features extracted from the PPG signal seems to work and +is the general direction of work so far. Neural Networks +[1,9,20,33], Ensemble methods [10] and LSTM’s [27] are +some popular machine learning algorithms that have been +employed to estimate BP using either the raw PPG signal +or features extracted from the raw PPG signal as inputs to +the algorithms. Efforts towards developing contact-less so- +lutions include using a camera to capture facial videos of +the user and use deep Learning models on the extracted +rPPG(remote PPG) signals to estimate BP [29]. Transder- +mal Optical Imaging technology is another way to capture +facial blood flow changes and estimate BP in a contact-less +manner [23]. +1.2. Motivation +The algorithms discussed in Section 1.1 either use pub- +licly available datasets or data available from a limited set of +devices (usually one) to develop camera based vitals moni- +toring solutions. PhysioNet [12] is a popular database com- +monly used to develop algorithms based on Finger tip PPG. +Algorithms based on Face PPG use either [7] or [36] or +both to train the models. It is interesting to note that these +datasets are constructed under strictly controlled environ- +ment such as fixed lighting settings, fixed background, fixed +distance from camera etc., and do not account for interde- +vice variability. Thus, these datasets do not account for vari- +abilities that arise when the devices are used in the wild or +when multiple devices of different kinds are used for self- +monitoring. Some examples of variabilities that may arise +while acquiring data for self monitoring in the wild are: +• Positional Variance: (a) Wrong positioning of the BP +cuff of a digital monitor during data acquisition for +BP algorithms. Previous studies have established that +there is a significant Inter Arm BP difference when a +digital monitor is used for BP measurement [22, 26]. +Several clinical studies in the past have also estab- +lished the existence of variabilities in the technique +used to measure BP by experts and the negative im- +pact that it has while monitoring in a clinical setting +Table 1. Mobile applications and wearables that support vitals +monitoring +Technology +Name +Vitals support +Solution +BP +SpO2 +HR +Wearable +Apple Watch +Y +Y +IoT +GOQii +Y +Y +Y +IoT +boAT Xtend +Y +Y +IoT +One Plus +Y +Y +IoT +Fitbit +Y +Y +IoT +Oura ring +Y +IoT +Omron +Y +IoT +Mobile App +Careplix +Y +Y +Camera (FT) +MFine +Y +Y +Y +Camera (FT) +ICICI +Y +Y +Y +Camera (FC) +Anura +Y +Y +Camera (FC) +[28,30,34]. When such variabilities exist in measure- +ments taken by experts themselves, it is not uncom- +mon to expect the variability to exist while measuring +BP in a self monitoring setting. (b) Similar positional +variances exist while measuring SpO2 and HR using +pulseoximeters. +• Hardware Variance: Variabilities in the hardware used +to acquire Image / Video signals for PPG based vitals +analysis. It is a well known fact that manufacturers +of different brands of smart phones employ sensors of +different Original Equipment Manufacturers (OEM’s) +to have a competitive edge. This will directly result +in the variability of the quality of PPG signal obtained +from different sensors. +• Environmental Variance: the mobility aspect of smart +phones by default will introduce variances related to +lighting, motion etc. +In this paper we address the following questions quan- +titatively/statistically by undertaking a systematic clinical +study: (a) What variabilities exist in self monitoring of the +following vitals: BP, SpO2 and HR? (b) How do these vari- +abilities compare to the measurements obtained by an ex- +pert who is a qualified medical professional? (c) If the vari- +abilities between self-monitoring and the measurements by +an expert are similar, then how do these variabilities affect +the training of computer vision based solutions? for vitals +monitoring? (d) What do we need to do to minimise the +effect of this variability in AI based solutions? +The focus of this paper is on the variabilities associated +with Position and Hardware while Environmental Variance +will be picked up as an extension to this study to do justice +to the number of environmental factors that may influence +the outcome of self-monitoring in the wild. It should also +be noted that the primary goal of this paper is to validate +2 + +Figure 1. Study setup +commercially available solutions at this point in time as we +wish to evaluate that solution which is easily accessible to +a user in the current situation given the pandemic and the +global burden on healthcare infrastructure. +1.3. Contributions +To the best of our knowledge, there is no prior work that +has formally quantified the variabilities that exist in self- +monitoring of vitals. Since a majority of solutions in lit- +erature are targeted towards self-monitoring of vitals, it is +crucial to identify this variability and determine it’s signif- +icance in order to establish error bounds for measurements +of vitals on smartphones and wearables in the future. In this +first-of-a-kind study, we establish statistically that there is +a significant variability in the Vitals when measured by self +using medical devices that are available off the shelf. Ad- +ditionally, we also establish the existance of this variability +in camera based solutions for self-monitoring of vitals and +thus prove that the use of camera based smart phone solu- +tions is similar to the use of medical devices or wearables +for self-monitoring. Finally, discuss some methods that can +potentially be used to reduce this variability in camera based +solutions. +2. Materials and Method +2.1. Data +The study was conducted on 203 healthy subjects com- +prising of 112 (55%) men and 91 (45%) women in the age +range [20,55] years with an average age of 28.77 ± 5.877 +years. The sample size was estimated for a 5% error at 95% +confidence interval based on the computations in [8,22,25]. +Prior written consent was obtained from the subjects who +volunteered for the study as per the IRB guidelines and ap- +provals obtained for the study. +In order to monitor the vitals on medical devices, mobile +phones and wearables, we will require at least: +1. one pair of mobile phones/smartphones with one of +the phones pre-installed with a mobile application that +measures vitals by extracting PPG signals from finger +tip image stream/video and the other from face videos +from front camera of the phone. These mobile appli- +cations are assumed to employ any one of the methods +described in Section 1.1. The exact details on the na- +ture of the algorithm are unavailable at this point in +time as the developers of the applications have not dis- +closed it publicly as either patents or publications. +The phones of the pair should be of the same +brand to eliminate the variabilities that may arise due +to differences in the hardware and software used to ac- +quire the signals (Image/Video). Since the study also +involves observing the variabilities associated with +changes in hardware and software of the smartphones, +we employ 4 pairs of phones in this study. The cri- +teria for selection of the phone brand/model is based +on price of the phone and global availability of the +phone. Lower the price, higher is the reach to the peo- +ple who would need access to affordable healthcare. +The price range under consideration for this study is +US$100-US$250. Accordingly, the following 4 mod- +els of phones are used: 1.(a) Xiaomi Note 9 Pro (Xi +N9); 1.(b) Xiaomi Note 8 Pro (Xi N8); 1.(c) Oppo A15 +(Oppo); and 1.(d) Samsung M31 (SM31); +The mobile applications should be capable of +measuring all the vitals that are considered for this +study. Table 1 shows different commercially available +mobile applications and wearables along with the vi- +tals supported by each. We select one mobile appli- +cation each for Finger tip PPG based monitoring and +Face PPG based monitoring from the Mobile App cat- +egory in Table 1. Accordingly, the following mobile +applications are considered for this study: MFine for +Finger tip PPG based monitoring and ICICI mobile ap- +plication for Face PPG based monitoring. +2. one wearable. As with the mobile phones, the wear- +able selected for this study should cover all the vitals +considered for the study. Froom Table 1 it can be seen +that the GOQii smartwatch supports all vitals and is +thus used in the study. Since we also want to study +the variabilities between wearables, we have also in- +cluded Apple Watch Series 7 in our study even though +BP monitoring capability is absent in it; +3. a digital BP monitor. Omron HEM 7121J fully auto- +matic Digital Blood Pressure monitor is used in this +3 + +1 +Observation Station 1 +Observer 1 +Digital BP +Mercury +★ Pulse Oximeter +monitor +Sphygmomanometer +1 +Observation Station 2 +Observer 2 +Finger Tip PPG +FacePPG +Wearables +1 +Observation Station 3 +Observer 3 +Cardiac monitorFigure 2. Quantifying the (a,c) transverse variability; and (b,d) +angular variability in the placement of the sensor of the BP cuff +for self-monitoring +Figure 3. Variations in measuring SpO2 and HR by placing a +pulseoximeter on the index finger of the hand (a) resting on the +table; (b) resting on the table with index figer at maximum angle; +and (c) in the air. +study; +4. a mercury based sphygmomanometer. Diamond Mer- +curial Blood Pressure apparatus was used for this +study; +5. a pulseoximeter. BPL Smary Oxy pulseoximeter was +used for this study to measure SpO2 and Heart Rate; +and +6. a cardiac monitor that will be used as gold standard to +eliminate the interobserver variability that may arise +with manual measurements with a sphygmomanome- +ter. Yonker YK 8000C Multi-parameter patient moni- +tor was used in this study. +Figure 4. Commercially available vitals monitoring solutions that +use (a) face videos (selfie); and (b) Finger tip videos to measure +vitals such as BP, SpO2 and HR. +2.2. Methodology +The study setup comprises of three observation stations +in a well lit room manned by three independent observers +who are qualified medical practitioners for : (a) self mon- +itoring; (b) monitoring with camera based mobile applica- +tion and wearables; and (c) monitoring with a cardiac mon- +itor. Figure 1 illustrates the setup used for this study. +The first station is where the variability in self monitor- +ing is studied. The self monitoring exercise starts with the +subjects being asked to measure BP using a digital BP mon- +itor that is placed in front of him/her without any instruc- +tions given on how to operate the BP monitor (which also +includes the placement of the cuff on the arm). The observer +of the station then notes down the position of the sensor in +the cuff on the arm with respect to: (a) the displacement +along the arm which is quantified as per the grading in Fig +2 (c); and (b) the approximate angle it makes with the cen- +treline (0o) of Fig 2 (d). Once the BP measurement and the +corresponding variations are recorded by the observer, the +subjects are then asked to measure the SpO2 and HR by +placing the pulseoximeter on the index finger and starting +the measurement. The observer notes down the SpO2 and +HR readings after 15s of the start of the measurement at +each of the positions indicated in Fig 3. The last step in sta- +tion 1 involves the observer measuring the Blood pressure +of the subject using a mercury based sphygmomanometer. +The subject is then asked to proceed to the second station +where the vitals are monitored using mobile based applica- +tions and wearables. As described in Section 2.1, we use 4 +pairs. of mobile phones of and two wearables per subject to +monitor the vitals. One phone of each pair is used for vitals +monitoring using Face PPG and the other is used for vitals +monitoring using Finger Tip PPG. Phones in each pair be- +long to the same brand and measurements on both phones +4 + +Posterior (P) +Median nerve +Brachial artery +Mediannerve +Brachial artery +Anterior (A) +(a) +(b) +p +180° +202.5° +157.5° +cm +.... +2 +cm +225° +135° +1 cm +0 cm +.-1 cm +:-2 cm +247.5° +112.5° +-3 cm +270° +90° +292.5° +67.5° +..... +45° +315° +... +.... +337.5° +22.5° +0° +A +(c) +(d)1. Hand resting on +1. Hand resting on +1. Hand 5 cm +table +table +abovetable +2. Index Finger with +2. Index Finger with +Pulseoximeter +Pulseoximeter at +resting on table +maximum lift22:41网02 +72% +21:24国· +20% +22:52国网· +l 70% +Knowyourhealth +< +Blood Pressure +01:48 +124/81 +mmHg +Whatdoes itmean +Your BP appears to be within normal. However, this is merely a +ConstructingthePPG +high-level analysis. We do not claim 100% accuracy. +SYS +60 +80 +100 +120 +140 +160 +180 +200 +220 +DIA +30 +50 +70 +90 +110 +130 +150 +170 +190 +Legends +oVery Low +●Normal +·HighBP +A +MeasurementusingMFine's BPMonitorisonlyadvisedfor +general wellbeing and fitness. +sec +Itisnotintendedformedicaluse +HeartRate +OxygenSaturation +92 beats pm +99% +RespiratoryRate +Heart Rate Variability +21breaths pm +114 ms +Stress Level +Blood Pressure +Normal +Systolic -123 mm Hg +Diastolic -78 mm Hg +Hold yourfinger overthecamera inthe same +position. +STOP + Tell a friend +MeasureAgain +川I +0 +(a) +(q)Figure 5. Vitals monitoring using mobile phones and wearables in +the 2nd station +are taken simultaneously. The phone used for Face PPG is +placed on a mobile phone holder with the angle adjusted so +that the front camera points to the face of the subject (with +the full face in view). Figure 4 illustrates the face based +and finger tip based methods to measure Vitals on mobile +phones. Figure 5 shows the procedure followed to measure +the vitals in the second observation station. Subjects who +have completed measuring their vitals in the second station +are asked to proceed to the final observation station where +their vitals are monitored by a cardiac monitor. The mea- +surements of the final station are used as the gold standard +for evaluating the measurements obtained at each observa- +tion station. +3. Experimental results +In this section we demonstrate the variabilities that ex- +ist in self monitoring using medical devices, mobile phones +and wearables, respectively. We use oneway ANOVA and +one tailed Student’s t-test to establish the statistical signifi- +cance of the variabilities using p-values. Where variabilities +are concerned, we use the mean and variance of differences +between the measurements of the device under considera- +tion and the gold standard which is the cardiac monitor. +3.1. Self monitoring of vitals using medical devices +BP +The plots in Figure 6 show the variability in both +Systolic and Diastolic BP measurements when the sensor +in the cuff of the digital BP monitor is placed at different +angles around the arm and at different positions along the +arm (Left/Right). It is interesting to note that the Systole +BP measurements obtained on the Left hand showed a sta- +tistically significant difference (p < 0.05) with the read- +ings obtained from the cardiac monitor. Difference in mea- +surements between cardiac monitor and digital BP moni- +tor for Systolic BP measurement was statistically significant +(p < 0.05) while Diastolic measurements were within sta- +Table 2. Error in camera based vitals measurement on mobile +phone when readings of cardiac monitor are used as gold standard +Brand +Method +BP (S) +BP (D) +SpO2 +HR +(mm/hg) +(mm/hg) +(%) +(/min) +Xi N9 +FT +2.432 +−0.680 +−2.277 +−3.30 +±13.66 +±10.25 +±3.28 +±8.69 +FC +−2.929 +−2.596 +−1.368 +5.758 +±20.70 +±13.60 +±2.32 +±14.70 +Xi N8 +FT +1.783 +−1.082 +1.233 +−5.919 +±13.72 +±9.89 +±3.25 +±10.13 +FC +−2.929 +−2.403 +−1.526 +−2.103 +±18.76 +±13.835 +±2.406 +±13.96 +Oppo +FT +1.350 +−1.814 +−1.5 +−7.70 +±13.62 +±10.00 +±3.20 +±12.25 +FC +−2.070 +−0.824 +−1.543 +0.362 +±17.251 +±12.937 +±2.464 +±14.91 +Sm 31 +FT +1.412 +−1.412 +0.866 +−4.797 +±13.90 +±10.24 +±5.00 +±9.94 +FC +−1.964 +−3.157 +−1.105 +1.982 +±18.10 +±12.80 +±3.621 +±12.59 +Xi N9: Xiaomi Note 9 Pro; Xi N8:Xiaomi Note 8 Pro; Oppo: Oppo +A15; Sm 31: Samsung M31; FT: Finger Tip PPG; FC: Face PPG; +BP(S): Systolic Blood Pressure; BP(D): Diastolic Blood Pressure; +SpO2: Blood Oxygen Saturation; HR: Heart Rate +tistically acceptable limits (p > 0.05). Both Systolic and +Diastolic BP readings did not show statistically significant +results (p > 0.05) when compared to the measurement by +an expert using mercury based Sphygmomanometer. +SpO2 and HR +The plots in Figure 7 shows the variabil- +ity in the measurements of both SpO2 and HR when the +pulseoximeter is clamped on to the index finger of the hand +according to the variations shown in Fig. 3. +Both SpO2 and Heart Rate showed statistically signifi- +cant difference (p < 0.05) between pulseoximeter and car- +diac monitor readings at all positions. A one way ANOVA +performed on the pulseoximeter readings at different posi- +tions of hand and index finger showed no statistically signif- +icant difference (p > 0.05) between the readings for SpO2. +Heart Rate on the other hand showed a statistically signifi- +cant difference (p < 0.05) between the readings at different +positions. +3.2. Self monitoring of vitals using mobile devices +Figure 8 shows the rate of failure in measuring the vitals +on each of the phones considered for this experiment. It can +be seen that Face PPG in general has a higher rate of failure +compared to Finger tip PPG. While Finger tip PPG based +methods had average failure rates of 14.676 ± 7.383%, +15.796 ± 4.197% and 15.049 ± 9.088% for BP, SpO2 and +5 + +Select wearable and +Select phone pair +measure vitals +Record values +Measure vitals ++ +N +Are all wearables +Finger Tip +Face PPG +covered? +PPG +N +Are all phone +Y +pairs covered? +Record values +Proceed to next stationFigure 6. Variability in self monitoring of Blood Pressure (BP) using digital BP monitors. (a,d,g). Variations in the angular placement of +the sensor (in the cuff) on the arm near the synovial joint; (b,e,h). Variations in the placement of the sensor (in the cuff) along the arm near +the synovial joint; (c,f,i). Variations in the selection of the arm [Left/Right] to place the sensor for BP measurement. +HR, respectively, Face PPG based methods had average fail- +ure rates of 24.875 ± 15.216%, 24.129 ± 15.941% and +23.631±16.169%. The plots in Fig. 9 shows the number of +attempts it took for those subjects where it was possible to +successfully obtain a measurement. The percentage of users +who could get successful readings in the 1st, 2nd and 3rd +attempts for Finger tip PPG, respectively stood at 67.492 ± +18.201%, 31.451±18.504% and 1.056±1.241%, while for +Face PPG it was 73.072±15.959%, 25.199±15.324% and +1.728 ± 1.658% for 1st, 2nd and 3rd attempts, respectively. +Table 2 shows that the error between the measurements +obtained from Finger tip PPG based methods and cardiac +monitor are in general lower compared to those obtained be- +tween Face PPG and cardiac monitor. However, the results +of Finger tip PPG based measurements statistically varied +across the phones for the same user while it remained quite +similar for Face PPG across the phones as indicated by the +p-values of Table 3. +6 + +Angularvariation +Transversevariation +Choice of Hand for Measurement +Errorin measurement [Systole] +measurement [Systole] +measurement [Systole] +Errorin +Error +30 +-2.0 +-1.0 +0.0 +2.0 +3.0 +Left +Right +Distance from center (in cm) +Hand +Angle (in degrees) +(b) +(c) +(a) +Error in measurement [Diastole] +iastole]l +measurement [Diastole] +in measurement [Di +2 +20 +u! +Error +Error +60 +3.0 +2.0 +Distance from center (in cm) +Left +Right +Hand +Angle (in degrees) +(e) +(f) +(d) +20.0 +17.5 +.20 +15.0 +.00 +samples +Number of samples +12.5 +-2.0 +-1.0 +2.0 +Left +Right +Distance from center (in cm) +Hand +Angle (in degrees) +(h) +(i) +(g)Figure 7. Positional variability in measuring SpO2 and HR using +pulseoximeter +Table 3. Results of ANOVA on measurements of vitals across +phones +Method +BP (S) +BP (D) +SpO2 +HR +FT (p-value) +0.00002 +0.00002 +0.0 +0.067 +FC (p-value) +0.967 +0.397 +0.520 +0.04 +BP(S): Systolic Blood Pressure; BP(D): Diastolic Blood Pressure; +SpO2: Blood Oxygen Saturation; HR: Heart Rate +Table 4. Error in vitals measurement on wearables when readings +of cardiac monitor are used as gold standard +Brand +BP (S) +BP (D) +SpO2 +HR +(mm/hg) +(mm/hg) +(%) +(/min) +GOQii +2.015 +−0.492 +−0.940 +−4.422 +±14.73 +±11.58 +±3.725 +±10.53 +Apple watch +NA +NA +−0.744 +−4.536 +±3.884 +±9.744 +3.3. Self monitoring of vitals using wearable +There were no failures in measuring BP on GOQii and +HR on both GOQii and Apple Watch. There was however +a 2% failure rate while measuring SpO2 on Apple Watch. +The results of Table 4 show that while GOQii smart watch +has a lower error rate for HR, Apple Watch has a lower error +Table 5. Results of ANOVA on measurements of vitals across +wearables +Method +SpO2 +HR +p-value +2.14 +0.928 +SpO2: Blood Oxygen Saturation; HR: Heart Rate +Table 6. p-value of differences in measurements between a device +and the cardiac monitor using t-test +Brand +Method +BP (S) +BP (D) +SpO2 +HR +(mm/hg) +(mm/hg) +(%) +(/min) +E +- +0.457 +0.457 +0.0009 +0.0009 +DM +- +0.537 +0.537 +0.0009 +0.0009 +Xi N9 +FT +0.017 +0.780 +0.0 +0.011 +FC +0.009 +0.121 +0.0 +0.00038 +Xi N8 +FT +0.269 +0.622 +0.0 +0.0001 +FC +0.237 +0.279 +0.0 +0.784 +Oppo +FT +0.940 +0.005 +0.00001 +0.0 +FC +0.015 +0.080 +0.026 +0.001 +SM31 +FT +0.646 +0.098 +0.025 +0.002 +FC +0.085 +0.008 +0.022 +0.751 +GO +- +0.033 +0.578 +0.0002 +0.0007 +AW +- +- +- +0.007 +0.0002 +Xi N9: Xiaomi Note 9 Pro; Xi N8:Xiaomi Note 8 Pro; Oppo: Oppo +A15; Sm 31: Samsung M31; E: Expert; DM: Digital BP Monitor; GO: +GOQii Smart watch; AW: Apple Watch; FT: Finger Tip PPG; FC: +Face PPG; BP(S): Systolic Blood Pressure; BP(D): Diastolic Blood +Pressure; SpO2: Blood Oxygen Saturation; HR: Heart Rate +rate for SpO2 when the results of cardiac monitor are used +as the gold standard for comparison. The measurements +between the two wearables were not statistically significant +as is evident from Table 5. +3.4. Overall Comparison +The results of Table 6 shows that the inter-observer vari- +ability was statistically insignificant. While Finger tip PPG +based measurements were in agreement with the gold stan- +dard for BP, the rest of the methods for all vitals showed +statistically significant differences between the device mea- +surements and gold standard. +4. Discussion, Conclusion and Future work +In this study we have shown that statistically significant +variations exist in self monitoring of vitals using medical +devices. One potential solution to address this is to sen- +sitise users to the correct procedure to be followed while +performing self-monitoring. An alternative to this would be +to ask the subjects to use mobile based or wearable based +solutions where the degrees of freedom for variabilities are +fewer and easily manageable. +7 + +Positional variation +0 +5 +0 +0 +0 +[Sp02] +0 +-5 +00 +00 +0000 +-10 +8 +0 +0 +-15 +-20 +-25 +0 +0 +0 +-30 +0 +0 +-35 +Resting +Max Angle +Air +Position of index finger +(a) +30 - +8 +8 +20 - +[HR] +8 +10 + measurements +-10 +m +-20 +rror +E -30 +0 +0 +8 +-40 +Resting +Max Angle +Air +Position of index finger +(b)Figure 8. Failure rates in phones while measuring vitals using Finger tip PPG and Face PPG methods +Figure 9. The number of attempts required to get a successful +measurement of at least one vital +The variabilities in self monitoring and monitoring with +Mobiles and Wearables are similar. The ground truth mea- +surements for algorithms on mobiles and wearables are ob- +tained from experts who either use digital monitors or mer- +cury based monitors (for BP) to measure vitals. Since an in- +herent variability existis in the technique used by experts as +noted in [28,30,34], this variability creeps into the training +data for mobiles and wearables. What is even more interest- +ing is the fact that the user variability in self monitoring and +that by experts is more or less similar and thus the variabili- +ties in self monitoring and that in mobiles and wearables are +similar. A potential solution to eliminate this variability in +training data will be to use the positional variability charts +of Fig. 2 and Fig. 3 as reference while acquiring ground +truth data for vitals. From the plots of Fig. 6, it can be +seen that the position of the sensor in the BP cuff between +[315o,22.5o] and [0cm,1cm] would result in less variability +in BP measurements and thus result in consistent Ground +Truth BP readings. Whereas, measurements taken with the +hand in resting position with the index finger horizontally +placed on the table is the ideal position to obtain Ground +Truth measurements for SpO2 and HR. +Variabilities in hardware used for imaging within mo- +bile phones result in statistically significant inconsistencies +across mobile phones of different brand. Not only are the +results inconsistent, it also takes multiple attempts on cer- +tain phones with certain technologies to get the measure- +ments right. This will lead to bad user experience and loss +of trust in the solution for vitals monitoring thereby result- +ing in a loss of adoptability of mobile camera based solu- +tions for self-monitoring of vitals in the wild. One potential +solution to reduce this variability will be to use camera cali- +bration techniques which will normalize the color and white +balance of the image stream or video that is being acquired +for PPG signal construction. +Face PPG in general had a higher success rate compared +to Finger tip PPG and that the varaibilty across phones for +Face PPG was statistically insignificant compared to that of +the Finger tip PPG based methods. This is a direct result of +the experimental setup, where the phones were placed on a +mobile holder for Face PPG and the user was asked to hold +the phone for Finger tip PPG. This indicates the influence +of the following two environment factors in the consistency +of results: (a) lighting condition (the study was in a well lit +room); and (b) motion artifacts. The impact of environmen- +tal factors as contributors for variability in self monitoring +will be considered as an extension to this study. +Conclusion and Future Work +In this paper we demon- +strate the various variabilities that exist while performing +self-monitoring of vitals using smart phones, wearables and +medical devices and establish the statistical significance of +the results of each when compared to the gold standard mea- +surement obtained from a cardiac monitor. The study of en- +vironmental factors for variability in self monitoring, cam- +era calibration for minimising hardware variability and PPG +signal quality improvements will be extensions to our cur- +rent work on Self monitoring in the wild using camera based +solutions. +8 + +BP Measurement +SpO2 Measurement +HR Measurement +Finger Tip PPG +Finger Tip PPG +Finger Tip PPG +Face PPG +Face PPG +Face PPG +40 +40 +40 - +(%) +(%) +(%) +failure +30 +30 +of failure ( +30 +20 +20 +Rate +Rate +10 +10 - +10 - +0 : +0 +-0 +Xiaomi N9 Xiaomi N8 Oppo A15 Samsung M31 +Xiaomi N9 Xiaomi N8 Oppo A15 Samsung M31 +Xiaomi N9 Xiaomi N8 Oppo A15 Samsung M31 +Phone brand +Phone brand +Phone brandFingertip PPG +Face PPG +1st attempt +1st attempt +2nd Attempt +2nd Attempt +80 +>=3 attempts +80 +>=3 attempts +(%) +60 +Percentage +60 +40 +40 +20 +20 +XiaomiN9XiaomiN8OppoA15SamsungM31 +XiaomiN9 +9XiaomiN8OppoA15SamsungM31 +Phone brand +PhonebrandReferences +[1] Sanghyun Baek, Jiyong Jang, Sung-Hwan Cho, Jong Min +Choi, and Sungroh Yoon. 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In Proceed- +ings of the IEEE conference on computer vision and pattern +recognition, pages 3438–3446, 2016. 2 +10 + diff --git a/NtFOT4oBgHgl3EQf2jTj/content/tmp_files/load_file.txt b/NtFOT4oBgHgl3EQf2jTj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52ea394b7bc4713691060b04808a8b1f1feb37cd --- /dev/null +++ b/NtFOT4oBgHgl3EQf2jTj/content/tmp_files/load_file.txt @@ -0,0 +1,665 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf,len=664 +page_content='Factors that affect Camera based Self-Monitoring of Vitals in the Wild Nikhil S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Narayan Shashanka B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' * Rohit Damodaran∗ Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Chandrashekhar Jayaram∗ Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Kareem Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Mamta P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='∗ Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Saravanan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='∗ Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Monu Krishnan∗ Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Raja Indana MFine Abstract The reliability of the results of self monitoring of the vitals in the wild using medical devices or wearables or camera based smart phone solutions is subject to variabil- ities such as position of placement, hardware of the device and environmental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' In this first of its kind study, we demonstrate that this variability in self monitoring of Blood Pressure (BP), Blood oxygen saturation level (SpO2) and Heart rate (HR) is statistically significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='05) on 203 healthy subjects by quantifying positional and hard- ware variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' We also establish the existance of this vari- ability in camera based solutions for self-monitoring of vi- tals in smart phones and thus prove that the use of camera based smart phone solutions is similar to the use of medical devices or wearables for self-monitoring in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Introduction Monitoring of vitals is important for identifying and managing of diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Glasziou et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' [11] identify five phases of monitoring a disease from a chronic condition perspective along with the interval for monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' These include continuous monitoring of vitals to (a) detect abnor- mality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (b) confirm abnormality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (c) establish plan of action to treat the abnormality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (d) make adjustments to the treat- ment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and (e) to confirm the success of treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' These phases can very well be extended to acute conditions with- out loss of generality except for the fact that the monitor- ing intervals may be much shorter than that for chronic conditions and may totally be based on self-monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Self-monitoring refers to monitoring of conditions by pa- tients/users by using medical devices that are available off- The authors were at MFine when the research was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Prior consent has been obtained from the authors to publish this work the-shelf at a place of the patient’s convenience without the intervention of a qualified medical professional to monitor the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Self monitoring in the wild refers to monitor- ing the vitals in uncontrolled environments such as outdoors or house where the conditions with respect to lightihg, phys- ical activity etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=', dynamically change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' At certain points in time, when access to care is made difficult either due to the unavailability of care providers or due to environmental factors such as the most recent COVID-19 pandemic where patients were forced to mon- itor their conditions at home before approaching a health- care provider, self-monitoring plays a crucial role in saving human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Given the importance of self-monitoring of diseases in recent times, there has been a flurry of activity in research circles to come up with novel technologies that en- able self-monitoring of vitals such as Blood Pressure (BP), Blood oxygen saturation level (SpO2), Heart rate (HR), Blood Glucose etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' While some of these have made it to commercial devices such as wearables and mobile phones, a vast majority of the technology is still restricted to academia and research circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Prior art Of the technologies that are being researched for self- monitoring of vitals, computer vision has received consid- erable amount of attention the the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' One of the early attempts in this direction was by Jonathan et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=', [16] where the authors used a Nokia device to record videos at 15 FPS of a user before and after performing a physical ac- tivity and the change in HR calculated by employing us- ing Photoplethysmography(PPG) Imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Thereafter there have been quite a few attempts to demonstrate the capability of using PPG signals from videos/image stream to estimate HR in an individual [4,13,14,19,24,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The PPG imaging methodology employed for HR is also a popular method to estimate SpO2 by both signal process- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='12943v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='CV] 30 Jan 2023 ing/mathematical models [6, 21, 31] and machine learning [2, 3, 5, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' A majority of smart phone PPG solutions for SpO2 consider the video of a finger tip placed against the back camera of the phone, hereafter referred to as Fin- ger tip (FT) PPG, as the input to extract the PPG signals and estimate the vital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' A non-contact way to monitor SpO2 is explored in [15, 35]where a video stream of user’s face, hereafter referred to as Face (FC) PPG, is used as input for PPG signal extraction and SpO2 Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' There has been growing interest in recent times to esti- mate BP using PPG signals extracted either from FT PPG or FC PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' While, the exact relationship between PPG the sig- nal and Blood Pressure has not been clinically established, fitting an ML model on top of either the raw signal, or some features extracted from the PPG signal seems to work and is the general direction of work so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Neural Networks [1,9,20,33], Ensemble methods [10] and LSTM’s [27] are some popular machine learning algorithms that have been employed to estimate BP using either the raw PPG signal or features extracted from the raw PPG signal as inputs to the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Efforts towards developing contact-less so- lutions include using a camera to capture facial videos of the user and use deep Learning models on the extracted rPPG(remote PPG) signals to estimate BP [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Transder- mal Optical Imaging technology is another way to capture facial blood flow changes and estimate BP in a contact-less manner [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Motivation The algorithms discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='1 either use pub- licly available datasets or data available from a limited set of devices (usually one) to develop camera based vitals moni- toring solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' PhysioNet [12] is a popular database com- monly used to develop algorithms based on Finger tip PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Algorithms based on Face PPG use either [7] or [36] or both to train the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' It is interesting to note that these datasets are constructed under strictly controlled environ- ment such as fixed lighting settings, fixed background, fixed distance from camera etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=', and do not account for interde- vice variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Thus, these datasets do not account for vari- abilities that arise when the devices are used in the wild or when multiple devices of different kinds are used for self- monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Some examples of variabilities that may arise while acquiring data for self monitoring in the wild are: Positional Variance: (a) Wrong positioning of the BP cuff of a digital monitor during data acquisition for BP algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Previous studies have established that there is a significant Inter Arm BP difference when a digital monitor is used for BP measurement [22, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Several clinical studies in the past have also estab- lished the existence of variabilities in the technique used to measure BP by experts and the negative im- pact that it has while monitoring in a clinical setting Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Mobile applications and wearables that support vitals monitoring Technology Name Vitals support Solution BP SpO2 HR Wearable Apple Watch Y Y IoT GOQii Y Y Y IoT boAT Xtend Y Y IoT One Plus Y Y IoT Fitbit Y Y IoT Oura ring Y IoT Omron Y IoT Mobile App Careplix Y Y Camera (FT) MFine Y Y Y Camera (FT) ICICI Y Y Y Camera (FC) Anura Y Y Camera (FC) [28,30,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' When such variabilities exist in measure- ments taken by experts themselves, it is not uncom- mon to expect the variability to exist while measuring BP in a self monitoring setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (b) Similar positional variances exist while measuring SpO2 and HR using pulseoximeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Hardware Variance: Variabilities in the hardware used to acquire Image / Video signals for PPG based vitals analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' It is a well known fact that manufacturers of different brands of smart phones employ sensors of different Original Equipment Manufacturers (OEM’s) to have a competitive edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' This will directly result in the variability of the quality of PPG signal obtained from different sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Environmental Variance: the mobility aspect of smart phones by default will introduce variances related to lighting, motion etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' In this paper we address the following questions quan- titatively/statistically by undertaking a systematic clinical study: (a) What variabilities exist in self monitoring of the following vitals: BP, SpO2 and HR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (b) How do these vari- abilities compare to the measurements obtained by an ex- pert who is a qualified medical professional?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (c) If the vari- abilities between self-monitoring and the measurements by an expert are similar, then how do these variabilities affect the training of computer vision based solutions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' for vitals monitoring?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (d) What do we need to do to minimise the effect of this variability in AI based solutions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The focus of this paper is on the variabilities associated with Position and Hardware while Environmental Variance will be picked up as an extension to this study to do justice to the number of environmental factors that may influence the outcome of self-monitoring in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' It should also be noted that the primary goal of this paper is to validate 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Study setup commercially available solutions at this point in time as we wish to evaluate that solution which is easily accessible to a user in the current situation given the pandemic and the global burden on healthcare infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Contributions To the best of our knowledge, there is no prior work that has formally quantified the variabilities that exist in self- monitoring of vitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Since a majority of solutions in lit- erature are targeted towards self-monitoring of vitals, it is crucial to identify this variability and determine it’s signif- icance in order to establish error bounds for measurements of vitals on smartphones and wearables in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' In this first-of-a-kind study, we establish statistically that there is a significant variability in the Vitals when measured by self using medical devices that are available off the shelf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Ad- ditionally, we also establish the existance of this variability in camera based solutions for self-monitoring of vitals and thus prove that the use of camera based smart phone solu- tions is similar to the use of medical devices or wearables for self-monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Finally, discuss some methods that can potentially be used to reduce this variability in camera based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Materials and Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Data The study was conducted on 203 healthy subjects com- prising of 112 (55%) men and 91 (45%) women in the age range [20,55] years with an average age of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='77 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='877 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The sample size was estimated for a 5% error at 95% confidence interval based on the computations in [8,22,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Prior written consent was obtained from the subjects who volunteered for the study as per the IRB guidelines and ap- provals obtained for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' In order to monitor the vitals on medical devices, mobile phones and wearables, we will require at least: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' one pair of mobile phones/smartphones with one of the phones pre-installed with a mobile application that measures vitals by extracting PPG signals from finger tip image stream/video and the other from face videos from front camera of the phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' These mobile appli- cations are assumed to employ any one of the methods described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The exact details on the na- ture of the algorithm are unavailable at this point in time as the developers of the applications have not dis- closed it publicly as either patents or publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The phones of the pair should be of the same brand to eliminate the variabilities that may arise due to differences in the hardware and software used to ac- quire the signals (Image/Video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Since the study also involves observing the variabilities associated with changes in hardware and software of the smartphones, we employ 4 pairs of phones in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The cri- teria for selection of the phone brand/model is based on price of the phone and global availability of the phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Lower the price, higher is the reach to the peo- ple who would need access to affordable healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The price range under consideration for this study is US$100-US$250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Accordingly, the following 4 mod- els of phones are used: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (a) Xiaomi Note 9 Pro (Xi N9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (b) Xiaomi Note 8 Pro (Xi N8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (c) Oppo A15 (Oppo);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (d) Samsung M31 (SM31);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The mobile applications should be capable of measuring all the vitals that are considered for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Table 1 shows different commercially available mobile applications and wearables along with the vi- tals supported by each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' We select one mobile appli- cation each for Finger tip PPG based monitoring and Face PPG based monitoring from the Mobile App cat- egory in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Accordingly, the following mobile applications are considered for this study: MFine for Finger tip PPG based monitoring and ICICI mobile ap- plication for Face PPG based monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' one wearable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' As with the mobile phones, the wear- able selected for this study should cover all the vitals considered for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Froom Table 1 it can be seen that the GOQii smartwatch supports all vitals and is thus used in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Since we also want to study the variabilities between wearables, we have also in- cluded Apple Watch Series 7 in our study even though BP monitoring capability is absent in it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' a digital BP monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Omron HEM 7121J fully auto- matic Digital Blood Pressure monitor is used in this 3 1 Observation Station 1 Observer 1 Digital BP Mercury ★ Pulse Oximeter monitor Sphygmomanometer 1 Observation Station 2 Observer 2 Finger Tip PPG FacePPG Wearables 1 Observation Station 3 Observer 3 Cardiac monitorFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Quantifying the (a,c) transverse variability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and (b,d) angular variability in the placement of the sensor of the BP cuff for self-monitoring Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Variations in measuring SpO2 and HR by placing a pulseoximeter on the index finger of the hand (a) resting on the table;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (b) resting on the table with index figer at maximum angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and (c) in the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' a mercury based sphygmomanometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Diamond Mer- curial Blood Pressure apparatus was used for this study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' a pulseoximeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' BPL Smary Oxy pulseoximeter was used for this study to measure SpO2 and Heart Rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' a cardiac monitor that will be used as gold standard to eliminate the interobserver variability that may arise with manual measurements with a sphygmomanome- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Yonker YK 8000C Multi-parameter patient moni- tor was used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Commercially available vitals monitoring solutions that use (a) face videos (selfie);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and (b) Finger tip videos to measure vitals such as BP, SpO2 and HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Methodology The study setup comprises of three observation stations in a well lit room manned by three independent observers who are qualified medical practitioners for : (a) self mon- itoring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (b) monitoring with camera based mobile applica- tion and wearables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and (c) monitoring with a cardiac mon- itor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Figure 1 illustrates the setup used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The first station is where the variability in self monitor- ing is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The self monitoring exercise starts with the subjects being asked to measure BP using a digital BP mon- itor that is placed in front of him/her without any instruc- tions given on how to operate the BP monitor (which also includes the placement of the cuff on the arm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The observer of the station then notes down the position of the sensor in the cuff on the arm with respect to: (a) the displacement along the arm which is quantified as per the grading in Fig 2 (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and (b) the approximate angle it makes with the cen- treline (0o) of Fig 2 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Once the BP measurement and the corresponding variations are recorded by the observer, the subjects are then asked to measure the SpO2 and HR by placing the pulseoximeter on the index finger and starting the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The observer notes down the SpO2 and HR readings after 15s of the start of the measurement at each of the positions indicated in Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The last step in sta- tion 1 involves the observer measuring the Blood pressure of the subject using a mercury based sphygmomanometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The subject is then asked to proceed to the second station where the vitals are monitored using mobile based applica- tions and wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' As described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='1, we use 4 pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' of mobile phones of and two wearables per subject to monitor the vitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' One phone of each pair is used for vitals monitoring using Face PPG and the other is used for vitals monitoring using Finger Tip PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Phones in each pair be- long to the same brand and measurements on both phones 4 Posterior (P) Median nerve Brachial artery Mediannerve Brachial artery Anterior (A) (a) (b) p 180° 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5° 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5° cm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='. 2 cm 225° 135° 1 cm 0 cm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='-1 cm :-2 cm 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5° 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5° 3 cm 270° 90° 292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5° 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 45° 315° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='. 337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5° 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5° 0° A (c) (d)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Hand resting on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Hand resting on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Hand 5 cm table table abovetable 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Index Finger with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Index Finger with Pulseoximeter Pulseoximeter at resting on table maximum lift22:41网02 72% 21:24国· 20% 22:52国网· l 70% Knowyourhealth < Blood Pressure 01:48 124/81 mmHg Whatdoes itmean Your BP appears to be within normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' However, this is merely a ConstructingthePPG high-level analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' We do not claim 100% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=" SYS 60 80 100 120 140 160 180 200 220 DIA 30 50 70 90 110 130 150 170 190 Legends oVery Low Normal HighBP A MeasurementusingMFine's BPMonitorisonlyadvisedfor general wellbeing and fitness." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' sec Itisnotintendedformedicaluse HeartRate OxygenSaturation 92 beats pm 99% RespiratoryRate Heart Rate Variability 21breaths pm 114 ms Stress Level Blood Pressure Normal Systolic -123 mm Hg Diastolic -78 mm Hg Hold yourfinger overthecamera inthe same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' STOP Tell a friend MeasureAgain 川I 0 (a) (q)Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Vitals monitoring using mobile phones and wearables in the 2nd station are taken simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The phone used for Face PPG is placed on a mobile phone holder with the angle adjusted so that the front camera points to the face of the subject (with the full face in view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Figure 4 illustrates the face based and finger tip based methods to measure Vitals on mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Figure 5 shows the procedure followed to measure the vitals in the second observation station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Subjects who have completed measuring their vitals in the second station are asked to proceed to the final observation station where their vitals are monitored by a cardiac monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The mea- surements of the final station are used as the gold standard for evaluating the measurements obtained at each observa- tion station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Experimental results In this section we demonstrate the variabilities that ex- ist in self monitoring using medical devices, mobile phones and wearables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' We use oneway ANOVA and one tailed Student’s t-test to establish the statistical signifi- cance of the variabilities using p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Where variabilities are concerned, we use the mean and variance of differences between the measurements of the device under considera- tion and the gold standard which is the cardiac monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Self monitoring of vitals using medical devices BP The plots in Figure 6 show the variability in both Systolic and Diastolic BP measurements when the sensor in the cuff of the digital BP monitor is placed at different angles around the arm and at different positions along the arm (Left/Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' It is interesting to note that the Systole BP measurements obtained on the Left hand showed a sta- tistically significant difference (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='05) with the read- ings obtained from the cardiac monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Difference in mea- surements between cardiac monitor and digital BP moni- tor for Systolic BP measurement was statistically significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='05) while Diastolic measurements were within sta- Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Error in camera based vitals measurement on mobile phone when readings of cardiac monitor are used as gold standard Brand Method BP (S) BP (D) SpO2 HR (mm/hg) (mm/hg) (%) (/min) Xi N9 FT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='432 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='680 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='277 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='30 ±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='66 ±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='25 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='28 ±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='69 FC −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='929 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='596 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='368 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='758 ±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='70 ±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='60 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='32 ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='70 Xi N8 FT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='783 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='082 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='233 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='919 ±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='72 ±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='89 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='25 ±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='13 FC −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='929 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='403 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='526 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='103 ±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='76 ±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='835 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='406 ±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='96 Oppo FT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='350 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='814 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='70 ±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='62 ±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='00 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='20 ±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='25 FC −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='070 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='824 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='362 ±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='251 ±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='937 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='464 ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='91 Sm 31 FT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='412 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='866 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='797 ±13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='90 ±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='24 ±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='00 ±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='94 FC −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='964 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='157 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='982 ±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='10 ±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='80 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='621 ±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='59 Xi N9: Xiaomi Note 9 Pro;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Xi N8:Xiaomi Note 8 Pro;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Oppo: Oppo A15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Sm 31: Samsung M31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' FT: Finger Tip PPG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' FC: Face PPG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' BP(S): Systolic Blood Pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' BP(D): Diastolic Blood Pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' SpO2: Blood Oxygen Saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' HR: Heart Rate tistically acceptable limits (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Both Systolic and Diastolic BP readings did not show statistically significant results (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='05) when compared to the measurement by an expert using mercury based Sphygmomanometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' SpO2 and HR The plots in Figure 7 shows the variabil- ity in the measurements of both SpO2 and HR when the pulseoximeter is clamped on to the index finger of the hand according to the variations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Both SpO2 and Heart Rate showed statistically signifi- cant difference (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='05) between pulseoximeter and car- diac monitor readings at all positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' A one way ANOVA performed on the pulseoximeter readings at different posi- tions of hand and index finger showed no statistically signif- icant difference (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='05) between the readings for SpO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Heart Rate on the other hand showed a statistically signifi- cant difference (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='05) between the readings at different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Self monitoring of vitals using mobile devices Figure 8 shows the rate of failure in measuring the vitals on each of the phones considered for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' It can be seen that Face PPG in general has a higher rate of failure compared to Finger tip PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' While Finger tip PPG based methods had average failure rates of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='676 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='383%, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='796 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='197% and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='049 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='088% for BP, SpO2 and 5 Select wearable and Select phone pair measure vitals Record values Measure vitals + N Are all wearables Finger Tip Face PPG covered?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' PPG N Are all phone Y pairs covered?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Record values Proceed to next stationFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Variability in self monitoring of Blood Pressure (BP) using digital BP monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (a,d,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Variations in the angular placement of the sensor (in the cuff) on the arm near the synovial joint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (b,e,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Variations in the placement of the sensor (in the cuff) along the arm near the synovial joint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' (c,f,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Variations in the selection of the arm [Left/Right] to place the sensor for BP measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' HR, respectively, Face PPG based methods had average fail- ure rates of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='875 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='216%, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='129 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='941% and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='631±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='169%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 9 shows the number of attempts it took for those subjects where it was possible to successfully obtain a measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The percentage of users who could get successful readings in the 1st, 2nd and 3rd attempts for Finger tip PPG, respectively stood at 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='492 ± 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='201%, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='451±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='504% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='056±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='241%, while for Face PPG it was 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='072±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='959%, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='199±15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='324% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='728 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='658% for 1st, 2nd and 3rd attempts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Table 2 shows that the error between the measurements obtained from Finger tip PPG based methods and cardiac monitor are in general lower compared to those obtained be- tween Face PPG and cardiac monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' However, the results of Finger tip PPG based measurements statistically varied across the phones for the same user while it remained quite similar for Face PPG across the phones as indicated by the p-values of Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 6 Angularvariation Transversevariation Choice of Hand for Measurement Errorin measurement [Systole] measurement [Systole] measurement [Systole] Errorin Error 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 Left Right Distance from center (in cm) Hand Angle (in degrees) (b) (c) (a) Error in measurement [Diastole] iastole]l measurement [Diastole] in measurement [Di 2 20 u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Error Error 60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 Distance from center (in cm) Left Right Hand Angle (in degrees) (e) (f) (d) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='20 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='00 samples Number of samples 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 Left Right Distance from center (in cm) Hand Angle (in degrees) (h) (i) (g)Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Positional variability in measuring SpO2 and HR using pulseoximeter Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Results of ANOVA on measurements of vitals across phones Method BP (S) BP (D) SpO2 HR FT (p-value) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='00002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='00002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='067 FC (p-value) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='04 BP(S): Systolic Blood Pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' BP(D): Diastolic Blood Pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' SpO2: Blood Oxygen Saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' HR: Heart Rate Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Error in vitals measurement on wearables when readings of cardiac monitor are used as gold standard Brand BP (S) BP (D) SpO2 HR (mm/hg) (mm/hg) (%) (/min) GOQii 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='492 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='940 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='422 ±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='73 ±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='58 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='725 ±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='53 Apple watch NA NA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='744 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='536 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='884 ±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='744 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Self monitoring of vitals using wearable There were no failures in measuring BP on GOQii and HR on both GOQii and Apple Watch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' There was however a 2% failure rate while measuring SpO2 on Apple Watch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The results of Table 4 show that while GOQii smart watch has a lower error rate for HR, Apple Watch has a lower error Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Results of ANOVA on measurements of vitals across wearables Method SpO2 HR p-value 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='928 SpO2: Blood Oxygen Saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' HR: Heart Rate Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' p-value of differences in measurements between a device and the cardiac monitor using t-test Brand Method BP (S) BP (D) SpO2 HR (mm/hg) (mm/hg) (%) (/min) E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='784 Oppo FT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='940 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0 FC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='001 SM31 FT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='002 FC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='751 GO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0007 AW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='0002 Xi N9: Xiaomi Note 9 Pro;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Xi N8:Xiaomi Note 8 Pro;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Oppo: Oppo A15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Sm 31: Samsung M31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' E: Expert;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' DM: Digital BP Monitor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' GO: GOQii Smart watch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' AW: Apple Watch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' FT: Finger Tip PPG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' FC: Face PPG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' BP(S): Systolic Blood Pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' BP(D): Diastolic Blood Pressure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' SpO2: Blood Oxygen Saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' HR: Heart Rate rate for SpO2 when the results of cardiac monitor are used as the gold standard for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The measurements between the two wearables were not statistically significant as is evident from Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Overall Comparison The results of Table 6 shows that the inter-observer vari- ability was statistically insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' While Finger tip PPG based measurements were in agreement with the gold stan- dard for BP, the rest of the methods for all vitals showed statistically significant differences between the device mea- surements and gold standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Discussion, Conclusion and Future work In this study we have shown that statistically significant variations exist in self monitoring of vitals using medical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' One potential solution to address this is to sen- sitise users to the correct procedure to be followed while performing self-monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' An alternative to this would be to ask the subjects to use mobile based or wearable based solutions where the degrees of freedom for variabilities are fewer and easily manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 7 Positional variation 0 5 0 0 0 [Sp02] 0 5 00 00 0000 10 8 0 0 15 20 25 0 0 0 30 0 0 35 Resting Max Angle Air Position of index finger (a) 30 - 8 8 20 - [HR] 8 10 measurements 10 m 20 rror E -30 0 0 8 40 Resting Max Angle Air Position of index finger (b)Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Failure rates in phones while measuring vitals using Finger tip PPG and Face PPG methods Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The number of attempts required to get a successful measurement of at least one vital The variabilities in self monitoring and monitoring with Mobiles and Wearables are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The ground truth mea- surements for algorithms on mobiles and wearables are ob- tained from experts who either use digital monitors or mer- cury based monitors (for BP) to measure vitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Since an in- herent variability existis in the technique used by experts as noted in [28,30,34], this variability creeps into the training data for mobiles and wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' What is even more interest- ing is the fact that the user variability in self monitoring and that by experts is more or less similar and thus the variabili- ties in self monitoring and that in mobiles and wearables are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' A potential solution to eliminate this variability in training data will be to use the positional variability charts of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 3 as reference while acquiring ground truth data for vitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' From the plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' 6, it can be seen that the position of the sensor in the BP cuff between [315o,22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='5o] and [0cm,1cm] would result in less variability in BP measurements and thus result in consistent Ground Truth BP readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Whereas, measurements taken with the hand in resting position with the index finger horizontally placed on the table is the ideal position to obtain Ground Truth measurements for SpO2 and HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Variabilities in hardware used for imaging within mo- bile phones result in statistically significant inconsistencies across mobile phones of different brand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Not only are the results inconsistent, it also takes multiple attempts on cer- tain phones with certain technologies to get the measure- ments right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' This will lead to bad user experience and loss of trust in the solution for vitals monitoring thereby result- ing in a loss of adoptability of mobile camera based solu- tions for self-monitoring of vitals in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' One potential solution to reduce this variability will be to use camera cali- bration techniques which will normalize the color and white balance of the image stream or video that is being acquired for PPG signal construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Face PPG in general had a higher success rate compared to Finger tip PPG and that the varaibilty across phones for Face PPG was statistically insignificant compared to that of the Finger tip PPG based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' This is a direct result of the experimental setup, where the phones were placed on a mobile holder for Face PPG and the user was asked to hold the phone for Finger tip PPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' This indicates the influence of the following two environment factors in the consistency of results: (a) lighting condition (the study was in a well lit room);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and (b) motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The impact of environmen- tal factors as contributors for variability in self monitoring will be considered as an extension to this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Conclusion and Future Work In this paper we demon- strate the various variabilities that exist while performing self-monitoring of vitals using smart phones, wearables and medical devices and establish the statistical significance of the results of each when compared to the gold standard mea- surement obtained from a cardiac monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' The study of en- vironmental factors for variability in self monitoring, cam- era calibration for minimising hardware variability and PPG signal quality improvements will be extensions to our cur- rent work on Self monitoring in the wild using camera based solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='BP Measurement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='SpO2 Measurement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='HR Measurement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='Finger Tip PPG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content='Finger Tip PPG ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Jiyong Jang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Sung-Hwan Cho,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Jong Min Choi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' and Sungroh Yoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' Blood pressure prediction by a smartphone sensor using fully convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} +page_content=' In 2020 42nd Annual International Conference of the IEEE En- gineering 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFOT4oBgHgl3EQf2jTj/content/2301.12943v1.pdf'} diff --git a/O9FPT4oBgHgl3EQfnTX9/vector_store/index.pkl b/O9FPT4oBgHgl3EQfnTX9/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f65c7a15db29112b2e009fc1c48f7ab83f17f500 --- /dev/null +++ b/O9FPT4oBgHgl3EQfnTX9/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bd913f051eab0757593ab513a85a5ee9c6ba1ed22e8e9e519743fdfa631b8f61 +size 76923 diff --git a/PNAzT4oBgHgl3EQflP3v/content/tmp_files/2301.01546v1.pdf.txt b/PNAzT4oBgHgl3EQflP3v/content/tmp_files/2301.01546v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..23334d05829428dfb9c5e982c98b9c608b6ee70e --- /dev/null +++ b/PNAzT4oBgHgl3EQflP3v/content/tmp_files/2301.01546v1.pdf.txt @@ -0,0 +1,1638 @@ +arXiv:2301.01546v1 [math.AP] 4 Jan 2023 +On the first Robin eigenvalue of the Finsler +p-Laplace operator as p Ñ 1 +Rosa Barbato˚, Francesco Della Pietra˚, Gianpaolo Piscitelli˚ +Abstract. Let Ω be a bounded, connected, sufficiently smooth open set, +p ą 1 and β P R. In this paper, we study the Γ-convergence, as p Ñ 1`, of +the functional +Jppϕq “ +ż +Ω +F pp∇ϕqdx ` β +ż +BΩ +|ϕ|p FpνqdHN´1 +ż +Ω +|ϕ|p dx +where ϕ P W 1,ppΩqzt0u and F is a sufficientely smooth norm on Rn. We +study the limit of the first eigenvalue λ1pΩ, p, βq “ infϕPW 1,ppΩq +ϕ‰0 +Jppϕq, as +p Ñ 1`, that is: +ΛpΩ, βq “ +inf +ϕPBV pΩq +ϕı0 +|Du|F pΩq ` mintβ, 1u +ż +BΩ +|ϕ| FpνqdHN´1 +ż +Ω +|ϕ| dx +. +Furthermore, for β ą ´1, we obtain an isoperimetric inequality for ΛpΩ, βq +depending on β. +The proof uses an interior approximation result for BV pΩq functions by +C8pΩq functions in the sense of strict convergence on Rn and a trace inequal- +ity in BV with respect to the anisotropic total variation. +MSC 2020: 28A75, 35J25, 35P15. +Keywords and phrases: +Finsler p-Laplace eigenvalues; Γ-convergence; +Isoperimetric inequalities; Trace inequalities; Strict interior approximation. +∗Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universit`a degli studi di Napoli Federico +II, Via Cintia, Monte S. Angelo - 80126 Napoli, Italia. +Email: f.dellapietra@unina.it (corresponding author), gianpaolo.piscitelli@unina.it +1 + +Contents +1 +Introduction +2 +2 +Notation and preliminaries +4 +2.1 +The Finsler norm . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Anisotropic curvatures +. . . . . . . . . . . . . . . . . +6 +2.3 +The anisotropic total variation +. . . . . . . . . . . . . +7 +3 +An anisotropic trace inequality +8 +4 +Interior approximation +11 +5 +The first Robin eigenvalue of the Finsler p-Laplacian as p Ñ 1 14 +5.1 +The case p “ 1 +. . . . . . . . . . . . . . . . . . . . . +15 +5.2 +Γ-convergence of Jp . . . . . . . . . . . . . . . . . . . +19 +5.3 +An isoperimetric inequality . . . . . . . . . . . . . . . +22 +1 Introduction +Let Ω be a bounded, connected, sufficiently smooth open set, p ą 1 and β P R. In +this paper, we study the asympthotic behaviour, as p Ñ 1`, of the following minimum +problem +λ1pΩ, p, βq “ +inf +ϕPW 1,ppΩq +ϕ‰0 +Jppϕq +(1.1) +where +Jppϕq “ +ż +Ω +F pp∇ϕqdx ` β +ż +BΩ +|ϕ|p FpνqdHN´1 +ż +Ω +|ϕ|p dx +, +(1.2) +ν is the outer normal to BΩ and F is a sufficiently smooth norm on Rn. If u P W 1,ppΩq +is a minimizer of (1.1), then it solves the following Robin eigenvalue problem +# +´Qpu “ λ1pΩ, p, βq |u|p´2 u +in Ω +F p´1p∇uqFξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0 +on BΩ, +where Qpu is the anisotropic p´Laplace operator +Qpu :“ div +ˆ1 +p∇ξrF psp∇uq +˙ +. +From the point of view of finding optimal domains for λ1pΩ, pβq, there is a significant +difference from the case of β ą 0 to the case β ă 0. It is known that the optimal shape +2 + +with a volume constraint for (1.1) depends on the sign of β. For positive values of the +Robin parameter, the so-called Wulff shape (see Section 2 for details) is a minimizer [9]: +λ1pΩ, p, βq ě λ1pW, p, βq +where |W| “ |Ω|. +If β ă 0, the problem is not completely solved, even in the Euclidean case (that is when +Fpξq “ +bř +i ξ2 +i ). Indeed, in 1977 Bareket [5] conjectured that, the first eigenvalue is +maximized by a ball in the class of the smooth bounded domains of given volume. In +[13] it has been showed that it is true for domains close, in certain sense, to a ball. +Subsequently, the authors in [15] (see [19] for the p-Laplacian case) have disproved the +conjecture for |β| large enough and have showed that it is true for small values of |β| +in suitable class of domains. In the Finsler setting, this problem has been addressed in +[23]. +Our final aim is to obtain optimal shapes for the limiting functional of (1.1), as +p Ñ 1`. To do that, we first study the limit of λ1pΩ, pβq. In particular, we prove that +when β ą ´1, the functional Jp, defined in (1.2), Γ´converges, as p Ñ 1`, to +Jpϕq “ +|Dϕ|F pΩq ` mint1, βu +ż +BΩ +|ϕ| FpνqdHN´1 +ż +Ω +|ϕ| dx +, +where |Dϕ|F pΩq is the anisotropic total variation of ϕ (see Section 2 for the precise +definition). This will imply that +lim +pÑ1` λ1pΩ, p, βq “ Λpβ, Ωq :“ +inf +ϕPBV pΩq Jpϕq. +(1.3) +Then, we prove an isoperimetric inequality for ΛpΩ, βq. In particular, we obtain that +keeping the volume of Ω fixed, the Wulff shape minimises ΛpΩ, βq when β ě 0, and +maximises it when ´1 ă β ă 0. +The proof of the convergence result, and then of the two isoperimetric inequalities, +relies on two results on the anisotropic total variation, which are also of independent +interest. The first one is a trace inequality in the BV space: +ż +BΩ +|u| Fpνq dHN´1 ď c1 |Du|F pΩq ` c2 +ż +Ω +|u| dx, +@u P BV pΩq, +(1.4) +where c1 and c2 are two constants which depend on the geometry of the domain. This +inequality has been revealed very useful in capillarity problems, and it has been studied +for example in [4, 16, 17]. +The second key result is an interior approximation for BV functions by smooth func- +tions with compact support. It is well known that if Ω is an open set, then the total +variation of a function u P BV pΩq can be approximated with the corresponding total +variation of a sequence in C8pΩq. +Actually, an analogous result is not true in gen- +eral if one need to axpproximate |Du| with a sequence of C8 function with compact +3 + +support in Ω. In order to do that, more regularity is needed on Ω. In the Euclidean +setting, this problem has been addressed in [20, 24]. In this paper, we show that for any +u P BV pΩq X LppΩq, for some p P r1, 8q, there exists a sequence tukukPN Ď C8 +0 pΩq such +that, for any q P r1, ps, +uk Ñ u in LqpΩq +and +|Duk|F pRNq Ñ |Du|F pRNq. +We finally stress that the problem we deal with is strictly related to capillarity prob- +lems. We refer the reader, for example, to [16, 17] for the Euclidean case and to [8] for +the anisotropic case. +The structure of the paper is the following. In Section 2, we review some useful tools +on the Finsler norm, the anisotropic curvature and functions of bounded variation. In +Section 3 we prove the anisotropic trace inequality for general domains, and for smooth +domains. In Section 4, we give the strict approximation result and finally, in Section 5 +we prove the Γ´convergence results and the isoperimetric inequality for ΛpΩ, βq. +2 Notation and preliminaries +In this Section we give several definitions and properties related the Finsler norm. In +particular, we review some basic facts on the anisotropic total variation of a BV function, +and on the anisotropic curvatures. +2.1 The Finsler norm +Throughout the paper we will assume that F is a convex, even, 1´homogeneous function +ξ P RN ÞÑ Fpξq P r0, `8r, +such that +Fptξq “ |t|Fpξq, +t P R, ξ P RN, +(2.1) +and such that +a|ξ| ď Fpξq, +ξ P RN, +(2.2) +for some constant a ą 0. It is easily seen that this hypothesis assure the existence of a +positive constant b ě a such that +Fpξq ď b|ξ|, +ξ P RN. +Throughout the paper, we will also assume that F belongs to C2pRNzt0uq and that +∇2 +ξrF 2spξq is positive definite in RNzt0u. +(2.3) +The assumption (2.3) on F ensures that the operator +Qpu :“ div +ˆ1 +p∇ξrF psp∇uq +˙ +4 + +is elliptic, therefore there exists a positive constant γ such that +nÿ +i,j“1 +∇2 +ξiξjrF pspηqξiξj ě γ|η|p´2|ξ|2 +@η P RNzt0u, @ξ P RN. +The polar function F o : RN Ñ r0, `8r of F is +F opvq “ sup +ξ‰0 +ξ ¨ v +Fpξq. +It is easily seen that also F o is a convex function satisfying the properties (2.1) and (2.2). +Furthermore, we have +Fpvq “ sup +ξ‰0 +ξ ¨ v +F opξq, +and from this follows that +|ξ ¨ η| ď FpξqF opηq +@ξ, η P RN. +(2.4) +The Wulff shape centered at the origin is the set denoted by +W “ tξ P RN : F opξq ă 1u. +We denote κN “ |W|, where |W| is the Lebesgue measure of W. More generally, +the set Wrpx0q indicates rW ` x0, that is the Wulff shape centered at x0 with measure +κNrN. If no ambiguity occurs, we will write Wr instead of Wrp0q. +The functions F and F o enjoy the following properties: +Fξpξq ¨ ξ “ Fpξq, +F o +ξ pξq ¨ ξ “ F opξq +@ξ P RNzt0u, +(2.5) +FpF o +ξ pξqq “ F opFξpξqq “ 1 +@ξ P RNzt0u, +(2.6) +F opξqFξpF o +ξ pξqq “ FpξqF o +ξ pFξpξqq “ ξ +@ξ P RNzt0u, +(2.7) +where Fξ “ ∇Fpξq. +Given a bounded domain Ω, the anisotropic distance of x P Ω to BΩ is defined as +dF pxq :“ inf +yPBΩ F opx ´ yq, +x P Ω. +We highlight that, when Fpξq “ +bř +i ξ2 +i , then dF “ dE is the Euclidean distance +function from the boundary. +The function dF is a uniform Lipschitz function in Ω, and +Fp∇dF pxqq “ 1 +a.e. in Ω. +We have that dF P W 1,8 +0 +pΩq. Many properties of the anisotropic distance function are +studied in [7]. +Finally, the anisotropic inradius of Ω is +RF pΩq “ maxtdF pxq, x P Ωu, +that is the radius of the largest Wulff shape Wrpxq contained in Ω. +5 + +2.2 Anisotropic curvatures +Here we recall some properties of the anisotropic mean curvature, as well as an integra- +tion formula in anisotropic normal coordinates. We refer to [7] for further details. +If Ω has a C2 boundary, the anisotropic outer normal to BΩ is defined as +nF pyq “ Fξpνpyqq, +y P BΩ, +where νpyq is the Euclidean outer normal to BΩ at y. Moreover, by (2.6) it holds that +F opnFpyqq “ 1. +Let us denote by TyBΩ the tangent space to BΩ at y; the anisotropic Weingarten map is +defined as +dnF : TyBΩ Ñ TnpyqW. +The eigenvalues κF +1 ď κF +2 ď . . . ď κF +N´1 of this map are called the anisotropic principal +curvatures at y (see also [25]). The anisotropic mean curvature of BΩ at a point y is +defined as +HF pyq “ κF +1 pyq ` . . . κF +N´1pyq, +y P BΩ. +The anisotropic distance dF is a C2 function in a tubular neighborhood of BΩ; hence +we are in position to define the matrix-valued function +Wpyq “ ´Fξξp∇dF pyqq∇2dF pyq, +y P BΩ. +Based on this function, it is possible to give a different definition of the anisotropic +principal curvatures [7, Remark 5.9]. Since Wpyqv P TyBΩ, for any v P Rn, it remains +defined the map Wpyq: Ty Ñ Ty, as Wpyqw “ Wpyqw, w P Ty. The matrix Wpyq (that is, +in general, non-symmetric) admits the real eigenvalues κF +1 pyq ď κF +2 pyq ď . . . ď κF +N´1pyq. +Actually, the definition is equivalent to the preceding one. Moreover, it holds that +HF pyq “ div rFξ p´∇dFpyqqs “ TrpWpyqq +(see also [25, Sec. 3]). +To state the change of variable formula in anisotropic normal coordinates, we need +some preliminary definitions. +Let +Φpy, tq “ y ´ tFξpνpyqq, +y P BΩ, +t P R +and for y P BΩ, +ℓpyq “ suptdF pzq, z P Ω and y P Πpzqqu +where +Πpzq “ tη P BΩ: dF pzq “ F opz ´ ηqu +(2.8) +is the set of the anisotropic projections of a point z P Ω on BΩ. +Then, we recall the following +6 + +Theorem 2.1. ([7, Theorem 7.1]) For every h P L1pΩq, it holds +ż +Ω +hpxqdx “ +ż +BΩ +Fpνpyqq +ż ℓpyq +0 +hpΦpy, tqqJpy, tq dt dHN´1pyq, +where +Jpy, tq “ +N´1 +ź +i“1 +p1 ´ tκF +i pyqq. +(2.9) +Since 1 ´ tκF +i pyq ą 0 for any i “ 1, . . . , N ´ 1 ([7, Lemma 5.4]), Jpy, tq is positive. +Moreover it holds that +´ d +dt rJpy, tqs +Jpy, tq +“ +N´1 +ÿ +i“1 +κF +i pyq +1 ´ tκF +i pyq. +(2.10) +Finally, we conclude this section, by recalling that, for any x P Ω such that Πpxq “ tyu, +it holds that ([7, Lemma 4.3]): +∇dF pxq “ ´ +νpyq +Fpνpyqq. +(2.11) +2.3 The anisotropic total variation +Let u P BV pΩq, the total variation of u with respect to F is defined as +|Du|F pΩq “ sup +"ż +Ω +u divpgq dx +: g P C1 +0pΩ; RNq, F opgq ď 1 +* +and the perimeter of a set E with respect to F is: +PF pE; Ωq “ |DχE|F pΩq “ sup +"ż +E +divpgq dx +: g P C1 +0pΩ; RNq, F opgq ď 1 +* +. +Moreover, +PF pE; Ωq “ +ż +ΩXB˚E +FpνEqdHN´1 +where B˚E is the reduced boundary of Ω and νE is the Euclidean normal to BE. +Let us fix u P BV pΩq and assume that u ” 0 in RNzΩ. Then u P BV pRNq and +|Du|F pRNq “ |Du|F pΩq ` +ż +BΩ +|u|FpνqdHN´1 +(2.12) +(see for example [6, Lemma 3.9]). +For the anisotropic perimeter, an isoperimetric inequality holds. More precisely, +PF pΩq ě PF pWRq, +(2.13) +where WR is the Wulff shape with the same measure of Ω (see for example [14, Theorem +2.10]). +The following approximation results in BV hold (refer to [4], [18, Theorem 1.17] for +the Euclidean case and to [1, Proposition 2.1] for the Finsler case). +7 + +Proposition 2.2. Let f P BV pΩq, then there exists a sequence tfkukPN Ď C8pΩq such +that: +lim +kÑ`8 +ż +Ω +|fk ´ f| dx “ 0 +and +lim +kÑ`8 |Dfk|F pΩq “ |Df|F pΩq. +Proposition 2.3. Let E be a set of finite perimeter in Ω. A sequence of C8 sets tEkuk +exists, such that: +lim +kÑ`8 +ż +Ω +|χEk ´ χE| dx “ 0 +and +lim +kÑ`8 |DχEk|F pΩq “ PF pE; Ωq. +3 An anisotropic trace inequality +In this section we prove a trace inequality in BV with respect to the anisotropic total +variation. We first give the result in a general case (Proposition 3.1), then we refine the +constants involved in the inequality by requiring more regularity on the boundary of Ω +(Proposition 3.2). +Firstly, let us set +qpyq “ lim +ρÑ0` sup +$ +’ +’ +& +’ +’ +% +ż +BΩ +χAFpνqdHN´1 +|DχA|F pΩq +: A Ă Ω X Bρpyq, |A| ą 0, PF pA; Ωq ă `8 +, +/ +/ +. +/ +/ +- +and Q “ supyPBΩ qpyq. The following inequality generalizes the trace inequality given in +[4, Theorem 4]. +Proposition 3.1. Let Ω be a bounded open set with HN´1pΩq ă `8, and let u be a +function in BV pΩq. Then for any ε ą 0, it holds +ż +BΩ +|u|FpνqdHN´1 ď pQ ` εq|Du|F pΩq ` cpΩ, εq +ż +Ω +|u|dx, +(3.1) +where cpΩ, εq does not depend on u. +Proof. Let us fix y P BΩ and ρpyq ą 0 such that +ż +BΩ +χBFpνqdHN´1 ď pQ ` εq|DχB|F pΩq, +8 + +for any B Ă Ω X Bρpyqpyq with PF pB; Ωq ă `8. +If sptpuq Ă Bρpyqpyq, then +ż +BΩ +|u|FpνqdHN´1 “ +ż +BΩ +Fpνq +ˆż `8 +0 +χt|u|ątupyqdt +˙ +dHN´1 +“ +ż `8 +0 +ˆż +BΩ +Fpνqχt|u|ątupyqdHN´1 +˙ +dt ď pQ ` εq +ż `8 +0 +|Dχt|u|ątu|F pΩqdt. +By using the coarea formula, we have +ż +BΩ +|u|FpνqdHN´1 ď pQ ` εq|D|u||F pΩq ď pQ ` εq|Du|F pΩq. +Let tBρpyquyPBΩ be a cover of BΩ and let us extract a finite sub-cover B1, . . . , Bk. +Now, considering a partition of unity ϕ1, . . . , ϕk such that +0 ď ϕi ď 1, +ϕi P C1 +0pBiq, +kÿ +i“1 +ϕipyq “ 1 +if y P BΩ. +If f P BV pΩq, then +ż +BΩ +|u| F pνq dHN´1 ď pQ ` εq +ˇˇˇˇˇD +˜ kÿ +i“1 +ϕiu +¸ˇˇˇˇˇ +F +pΩq +ď pQ ` εq +kÿ +i“1 +p|ϕiDu|F pΩq ` |uDϕi|F pΩqq +“ pQ ` εq +kÿ +i“1 +ˆż +Ω +ϕid|Du|F ` +ż +Ω +ud |Dϕi|F pΩq +˙ +ď pQ ` εq |Du|F pΩq ` cpΩ, εq +ż +Ω +|u| dx. +If the boundary of Ω is sufficilently smooth, we can show that Q can be taken equal +to 1 and ε “ 0. More precisely, we have the following. +Proposition 3.2. Let Ω be a bounded open connected set of class C2. Then there exists +a positive constant c such that +ż +BΩ +|u| Fpνq dHN´1 ď |Du|F pΩq ` c +ż +Ω +|u| dx, +@u P BV pΩq. +(3.2) +Proof. Since Ω is C2, then a uniform sphere condition of radius r ą 0 holds, in the sense +that for every point y P BΩ there exists z P Ω such that y P Brpzq Ă Ω. Let R Ps0, `8r +be the maximum of the principal radii of curvature of BW. Such maximum exists being +9 + +F (and F o) strongly convex. If κF +1 , . . . , κF +N´1 are the anisotropic principal curvatures, +we have that +κiF pyq ď 1 +µ, +i “ 1, . . . , N ´ 1, +with µ “ +r +R ([7, Lemma 5.4]). Therefore, in the set Ω µ +2 :“ +␣ +x P Ω : dF pxq ă µ +2 +( +, it +holds that dF is C2 ([7, Lemma 4.1 and Theorem 4.16]) and +κF +i pyq +1 ´ κF +i pyqdF pxq ď +$ +& +% +0 +if κF +i ď 0 +2 +µ +if 0 ă κF +i ď 1 +µ, +where y P BΩ is the anisotropic projection of x P Ω on BΩ. Then +N´1 +ÿ +i“1 +κiFpyq +1 ´ κiFpyqdF pxq ď 2N ´ 1 +µ +(3.3) +for x P Ω µ +2 . We may restrict ourselves to the case u is nonnegative and smooth. Inte- +grating by parts and recalling that Fp∇dF pxqq “ 1 in Ω, it holds that +ż +Ω +´∆FdF pxq upxq +´µ +2 ´ dF pxq +¯` +dx +“ +ż +Ω +Fξp∇dF pxqq ¨ ∇upxq +´µ +2 ´ dF pxq +¯` +dx +´ +ż +Ω µ +2 +upxqFξp∇dF pxqq ¨ ∇dF pxq dx ´ µ +2 +ż +BΩ +upxqFξp∇dF pxqq ¨ ν dx +“ +ż +Ω +Fξp∇dF pxqq¨∇upxq +´µ +2 ´ dF pxq +¯` +dx´ +ż +Ω µ +2 +upxq dx`µ +2 +ż +BΩ +upyq FpνqdHN´1. +Now, we estimate the term +ż +Ω +Fξp∇dF pxqq ¨ ∇upxq +´µ +2 ´ dF pxq +¯` +dx. +By (2.4) and (2.6) it holds that +ż +Ω +Fξp∇dF pxqq ¨ ∇upxq +´µ +2 ´ dF pxq +¯` +dx ě ´µ +2 +ż +Ω +Fp∇upxqq dx. +(3.4) +On the other hand, the change of variable formula (2.1) gives that +ż +Ω +Fξp∇dF pxqq ¨ ∇upxq +´µ +2 ´ dF pxq +¯` +dx +“ +ż +BΩ +Fpνq +ż µ +2 +0 +´µ +2 ´ t +¯ d +dtrupφpy, tqqsJpy, tqdt dHN´1. +10 + +Integrating by parts and using the fact that Jpy, 0q “ 1, the above integral becomes +´ µ +2 +ż +BΩ +upyqFpνqdHN´1pyq ´ +ż +BΩ +Fpνq +ż µ +2 +0 +upφpy, tqq +´µ +2 ´ t +¯ dJ +dt dt dHN´1pyq +` +ż +BΩ +Fpνq +ż µ +2 +0 +upφpy, tqqJpy, tqdt dHN´1pyq +ď ´µ +2 +ż +BΩ +upyqFpνqdHN´1pyq ` pN ´ 1q +ż +BΩ +Fpνq +ż µ +2 +0 +upyqJpy, tq dt dHN´1pyq +` +ż +BΩ +Fpνq +ż µ +2 +0 +upφpy, tqqJpy, tqdt dHN´1pyq +“ ´µ +2 +ż +BΩ +upyqFpνqdHN´1pyq ` N +ż +Ω +u dx +where in the inequality we have used (2.10) and the bound (3.3). Hence, joining with +(3.4) it holds that +ż +BΩ +upyqFpνqdHN´1 ď |Du|F pΩq ` 2N +µ +ż +Ω +upxq dx. +Remark 3.3. Using the notation of the above theorem, we explicitly observe that the +constant in (3.2) is +c “ 2N +µ . +4 Interior approximation +Now we provide an approximation result for BV -functions by smooth functions with +compact support in Ω. +We preliminary state two useful lemmas. Firstly, we recall from [20, Lemma 3.2] the +following result on diffeomorphic perturbations of sets Ω with Lipschitz boundary. We +denote by ι and I the identical vector and matrix function, respectively. +Lemma 4.1. Let Ω Ă RN be a bounded open set with Lipschitz boundary. Then there +exists τ0 ą 0 and, for 0 ď τ ď τ0, a family of C8´diffeomorphisms Φτ : RN Ñ RN with +inverses Ψτ such that +• Φ0 “ Ψ0 “ ι; +• Φτ Ñ ι and Ψτ Ñ ι as τ Ñ 0 uniformly on RN; +• ∇Φτpxq Ñ I and ∇Ψτpxq Ñ I as τ Ñ 0 uniformly with respect to x on RN; +11 + +• ΦτpΩq Ť Ω for all τ P p0, τ0s. +Now, we give the anisotropic version of the change of coordinates formula for BV - +functions, stated in [18, Lemma 10.1]. +Lemma 4.2. Let u be a function in BVlocpΩq, Φ : RN Ñ RN be a diffeomorphism and +A Ť Ω. Then +ˇˇDpu ˝ Φ´1q +ˇˇ +F pΦpAqq “ |HDu|F pAq, +(4.1) +where H “ | det ∇Φ|r∇Φs´1. +Proof. Let us consider u P C1pΩq and g P C1 +0pA; RNq, then the following change of area +formula holds +ż +ΦpAq +pg ˝ Φ´1q ¨ ∇pu ˝ Φ´1qdx “ +ż +ΦpAq +pg ˝ Φ´1q ¨ pp∇u ˝ Φ´1q∇Φ´1qdx +“ +ż +A +g ¨ p∇up∇Φ´1 ˝ Φqq| det ∇Φ|dz +“ +ż +A +g ¨ pH∇uqdz. +(4.2) +Thus, the thesis (4.1) holds for u in C1pΩq, that is +ż +ΦpAq +Fp∇pu ˝ Φ´1qqdx “ +ż +A +FpH∇uqdx. +Suppose now that u P BVlocpΩq. +By Proposition 2.2 we can approximate u by a +sequence tuiu Ă C8. Moreover, the corresponding functions ui˝Φ´1 converge to u˝Φ´1 +in L1pAq. +Hence, we can pass to the limit in (4.2), obtaining +ż +ΦpAq +pg ˝ Φ´1q ¨ dDpu ˝ Φ´1q “ +ż +A +g ¨ H dDu “ +ż +A +g ¨ pHνqd |Du| +(4.3) +where ν is obtained by differentiating Du with respect to |Du|. +If F opgq ď 1, then also F opg ˝ Φ´1q ď 1 and sptpg ˝ Φ´1qq Ď ΦpAq. Therefore, by +definition of total variation with respect to F, we have +ż +A +g ¨ pHνqd |Du| ď +ˇˇDpu ˝ Φ´1q +ˇˇ +F pΦpAqq, +(4.4) +The inequality (2.4) implies that +sup +F opgqď1 +ż +A +g ¨ pHνq d |Du| “ |HDu|F pAq. +Hence, taking the supremum on the left hand side in (4.4), it holds +|HDu|F pAq ď +ˇˇDpu ˝ Φ´1q +ˇˇ +F pΦpAqq. +12 + +For the reverse inequality, we consider g “ γ ˝ Φ P C1 +0pA; RNq, where γ P C1 +0pΦpAq; RNq +and F opγq ď 1. Therefore g ˝ Φ´1 “ γ and by (4.3), we have +ż +ΦpAq +γ ¨ dDpu ˝ Φ´1q “ +ż +A +pγ ˝ Φq ¨ pHνqd |Du| pAq +ď +ż +A +F opγ ˝ Φq|H|d|Du|F ď +ż +A +|H| d |Du|F . +Hence, we have +ˇˇDpu ˝ Φ´1q +ˇˇ +F pΦpAqq ď |HDu|F pAq. +At this stage, we are in position to state the main approximation result. +Theorem 4.3. Let Ω Ă RN be an open bounded set with Lipschitz boundary and let +u P BV pΩq X LppΩq for some p P r1, 8q. Then there exists a sequence tukukPN Ď C8 +0 pΩq +such that, for any q P r1, ps, +uk Ñ u in LqpΩq +and +|Duk|F pRNq Ñ |Du|F pRNq. +Proof. Let us fix a family pΦτq0ďτďτ0 of diffeomorphisms fron RN to RN with inverses +pΨτq0ďτďτ0 according to Lemma 4.1, and consider +uτ :“ u ˝ Ψτ +for +τ P r0, τ0s. +By construction uτ “ 0 a.e. outside a compact subset of Ω. By [20, Theorem 3.1], we +know that uτ P LppΩq for all τ P r0, τ0s, uτ Ñ u in LppΩq and also in LppRNq. +The change of coordinates formula in Lemma 4.2 implies that +|Duτ|F pRNq “ +ż +RN +ˇˇp∇ΨτqT ˇˇ |detp∇Φτq| d |Du|F . +Thus uτ P BV pRNq and, since the integrand on the right hand side uniformly converges +to 1, +lim +τÑ0` |Duτ|F pRNq “ |Du|F pRNq +It remains only to prove that there exists vτ P C8 +0 pΩq such that +||uτ ´ vτ||p ă τ +and +ˇˇ|Duτ|F pRNq ´ |Dvτ|F pRNq +ˇˇ ă τ. +The first convergence (in Lp) has been proved in [20, Theorem 3.1]; meanwhile the +convergence of the total variation is based on the following argument. +For any ε ą 0, let us consider the mollification uε :“ uτ ˚ ηε, where ηεpxq “: +ε´nηpε´1xq, for the standard mollifier η. Hence uε Ñ uτ in LppΩq and for the zero +extensions, in L1pRNq [3, Proposition 3.2.c]. +13 + +Then, by the lower semicontinuity of the anisotropic total variation, we have +|Duτ|F pRNq ď lim +εÑ0` inf |Duε|F pRNq. +Therefore, it remains to prove the opposite inequality +lim +εÑ0` sup |Duε|F pRNq ď |Duτ|F pRNq. +(4.5) +Let us choose ϕ P C8 +0 pRN, RNq with F opϕq ď 1 and calculate +ż +RN uε ˚ divpϕqdx +ż +RN uτpηε ˚ div ϕqdx “ +ż +RN uτ divpηε ˚ ϕqdx ď |Duτ|F pRNq, (4.6) +where the inequality in the last term holds since F opηε ˚ ϕq ď 1. Indeed, by using the +1-homogeneity of F o and Jensen’s Inequality (see, for instance, [22, Lemma 1.8.2]), we +gain that +F o +ˆż +RN ηǫpx ´ yqϕpyqdy +˙ +ď +ż +RN F opηǫpx´yqϕpyqqdy “ +ż +RN ηεpx´yqF opϕpyqqdy ď 1. +Hence, by passing to the limit in (4.5), we reach the inequality (4.6) by the arbitrariness +of ϕ. +5 The first Robin eigenvalue of the Finsler p-Laplacian as p Ñ 1 +In this Section, we give an application of the results proved above to a Robin eigenvalue +problem. More precisely, our aim is analyze the Γ-limit of the functional +Jppϕq “ +ż +Ω +F pp∇ϕqdx ` β +ż +BΩ +|ϕ|p FpνqdHN´1 +ż +Ω +|ϕ|p dx +, +ϕ P W 1,ppΩqzt0u, +(5.1) +where Ω is a bounded, connected, sufficiently smooth open set, p ą 1 and β P R, and +prove an isoperimetric inequality for the limit, as p Ñ 1`, of the first eigenvalue +λ1pΩ, p, βq “ +inf +ϕPW 1,ppΩq +ϕ‰0 +Jppϕq, +(5.2) +depending on the value of the parameter β. A key point for proving this result is the +convergence of the functional Jp. +We first recall the following existence result for (5.2) holds. +Theorem 5.1 ([9, 12]). Let p ą 1, β P R and Ω bounded Lipschitz domain. Then there +exists a minimum u P C1,αpΩq X CpΩq of (5.2) that satisfies +# +´Qpu “ λpΩ, p, βq |u|p´2 u +in Ω +F p´1p∇uqFξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0 +on BΩ. +(5.3) +Moreover, u does not change sign in Ω. Finally, λ1pΩ, p, βq is positive if β ą 0, while is +negative if β ă 0. +14 + +5.1 The case p “ 1 +In order to study the limit case of Jp as p goes to 1, we consider the functional +Jpϕq “ +|Dϕ|F pΩq ` mintβ, 1u +ż +BΩ +|ϕ| FpνqdHN´1 +ż +Ω +|ϕ| dx +, +(5.4) +where ϕ P BV pΩq and u ı 0. Hence, we study the associated minimum problem +ΛpΩ, βq “ +inf +ϕPBV pΩq +ϕı0 +Jpϕq. +(5.5) +Depending on β, we will impose different assumptions on the regularity of the domain. +Indeed: +• if β ě 0, we will suppose that BΩ is Lipschitz; +• if ´1 ă β ă 0, we will assume that BΩ is C2. +In particular, this difference depends on the fact that in the case β ă 0 we use the trace +inequality, studied in Section 3. +Finally, if β ď ´1 the problem is not well posed; indeed if β ă ´1, then ΛpΩ, βq “ ´8 +while if β “ ´1, Λ is finite but can be not achieved, also in the case of smooth domains. +For further details, we refer the reader to the Euclidean case treated in [10]. +Let us discuss the presence of the term mintβ, 1u in (5.4). For any value of β, it could +seem more natural to study the problem +λpΩ, 1, βq “ +inf +ϕPBV pΩq +ϕ‰0 +|Dϕ|F pΩq ` β +ż +BΩ +|ϕ| FpνqdHN´1 +ż +Ω +|ϕ| dx +. +(5.6) +Actually, we have that for β ě 1 it holds +λpΩ, 1, βq “ ΛpΩ, βq “ hF pΩq, +where hF pΩq is the first Cheeger constant of Ω in the Finsler setting (see e.g. [6]): +hF pΩq “ +inf +ϕPBV pΩq +ϕı0 +|Dϕ|F pRNq +ż +Ω +|ϕ| dx +“ inf +EĎΩ +PF pE; Ωq +|E| +. +(5.7) +Indeed, in this case, it is immediate to see that +λpΩ, 1, βq ě hF pΩq. +15 + +On the other hand, if u is a minimizer of (5.6), then, by Theorem 4.3, there exists +uk P C8 +0 pΩq such that +uk +Lq +ÝÑ u, +||∇uk||L1pΩq +L1 +ÝÑ |Du|F pRNq, +for any q ď +N +N ´ 1. Therefore +λpΩ, 1, βq ď +lim +kÑ`8 +ż +Ω +Fp∇ukqdx +ż +Ω +|uk| dx +“ hF pΩq. +Now we focus on the possibility of studying the minimization problem (5.5) restricting +our analysis to characteristic functions. Hence if E Ď Ω, we have +JpχEq “ +PF pE; Ωq ` mint1, βu +ż +BΩXB˚E +FpνEqdHN´1 +|E| +. +By denoting +RpE, βq :“ JpχEq, +we consider the minimization problem +ℓpΩ, βq “ inf +EĎΩ RpE, βq. +(5.8) +Before proving the equivalence between problems (5.5) and (5.8), we need the following +result on the lower semicontinuity of the numerator of the functional J. +Lemma 5.2. Let β ě ´1. The functional +Gpuq “ |Du|F pΩq ` mint1, βu +ż +BΩ +|u|FpνqdHN´1 +is lower semicontinuous on BV pΩq with respect to the topology of L1pΩq. +Proof. If β ě 0, the lower semicontinuity (with Ω Lipschitz) follows immediately by the +lower semicontinuity of each term. Then we assume β ă 0 (and Ω in C2). The proof is +an adaptation of [21, Proposition 1.2] to the Finsler case. +Let u P BV pΩq, and let us consider a sequence tukukPN Ď BV pΩq converging to u in +L1pΩq; we have the following estimate +Gpuq ´ Gpukq ď |Du|F pΩq ´ |Duk|F pΩq ` +ż +BΩ +|u ´ uk|FpνqdHN´1. +(5.9) +Now, for a fixed δ ą 0, let us define Ωδ “ tx P Ω : dEpxq ă δu, where dE is the standard +Euclidean distance to the boundary of Ω ; moreover let us consider vδ “ p1´χδqpu´ukq, +16 + +where χδ is a cut-off function such that χδ “ 1 in ΩzΩδ and |∇χδ| ď 2 +δ in Ω. The trace +inequality (3.2) applied to vδ gives +ż +BΩ +|u´uk|FpνqdHN´1 ď |Dpu ´ ukq|F pΩδq` 2b +δ +ż +Ωδ +|u´uk|dx`c +ż +Ωδ +|u´uk|dx. (5.10) +Moreover, we have +|Dpu ´ ukq|F pΩδq ď |Du|F pΩδq ` |Duk|F pΩδq ` |Dpu ´ ukq|F pBpΩzΩδqq, +(5.11) +but last term is zero on a set of δ’s of positive measure because u ´ uk P BV pΩq, for all +k P N. Hence, by (5.9)-(5.10)-(5.11), we gain: +Gpuq ´ Gpukq ď |Du|F pΩq ` |Du|F pΩδq ´ |Duk|F pΩzΩδq ` +ˆ2b +δ ` c +˙ ż +Ωδ +|u ´ uk|dx. +By the lower semicontinuity of the functional |Duk|F pΩzΩδq in L1pΩzΩδq, we have that +lim sup +kÑ`8 +rGpuq ´ Gpukqs ď 2|Du|F pΩδq. +The conclusion follows by sending δ Ñ 0`. +At this stage, we state the main existence result of the minimum problem (5.5). +Theorem 5.3. For any β ą ´1, there exists a minimum to problem (5.5). In particular, +it holds +ΛpΩ, βq “ ℓpΩ, βq. +Moreover, if u P BV pΩq is a minimum of (5.5), then +ΛpΩ, βq “ Rptu ą tu, βq, +for some t P R. +Proof. Let un be a minimizing sequence in BV pΩq of (5.5), such that }un}L1pΩq “ 1. If +β ą 0, then un is bounded in BV pΩq and hence +un +˚á u in BV pΩq +and +un +L1 +ÝÑ u. +In particular, if β ě 1, Jpunq “ |Dun|F pRNq, by using the lower semicontinuity of the +anisotropic total variation [2], we obtain that +Jpuq ď lim inf +n +Jpunq. +Hence u is the minimum of the functional J. +If 0 ă β ă 1, let Ωδ “ tx P Ω | dEpxq ă δu, with δ ą 0. We have +|Dun|F pΩq “ |Dun|F pΩzΩδq ` |Dun|F pΩδq ě |Dun|F pΩzΩδq ` β |Dun|F pΩδq +17 + +and hence +Jpunq ě |Dun|F pΩzΩδq ` β +„ +|Dun|F pΩδq ` +ż +BΩ +|un| FpνqdHN´1 + +. +Moreover, by the lower semicontinuity of J, we have +lim inf +n +Jpunq ě |Dun|F pΩzΩδq ` β |Dun|F pRNzpΩzΩδqq. +By using the fact that u P BV pΩq, we obtain that +lim inf +n +Jpunq ě Jpuq, +as δ Ñ 0. +Now, let us take ´1 ă β ă 0. It easily seen that Jpunq ď C. Using the trace inequality +(3.2), we obtain +Jpunq ě p1 ` βq |Dun|F pΩq ` cβ ě cβ +and +|Dun|F pΩq ď +C +1 ` β ´ +βc +1 ` β . +Being un P BV pΩq and by the fact that the functional J is lower semicontinuous (proved +in Lemma 5.2), we have that u is a minimum of J. +Now, we want to prove last part of the Theorem. Obviously, we have +ΛpΩ, βq ď ℓpΩ, βq. +To prove the reverse inequality, we take u P BV pΩq a minimizer of (5.5). By using the +coarea formula +|Du|F pΩq “ +ż `8 +´8 +PF ptu ą tu, Ωqdt, +we have +ΛF pΩ, βq “ +ż `8 +´8 +PFptu ą tu, Ωqdt ` mintβ, 1u +ż `8 +´8 +HN´1pBΩ X Btu ą tuqFpνqdt +ż `8 +´8 +|tu ą tu| dt +“ +ż `8 +´8 +Rptu ą tu, βq |tu ą tu| dt +ż `8 +´8 +|tu ą tu| dt +ě inf +EĎΩ RpE, βq +“ ℓpΩ, βq. +18 + +This shows that ΛpΩ, βq “ ℓpΩ, βq and, in particular, we have that +ż `8 +´8 +tRptu ą tu, βq ´ ℓpΩ, βqu |tu ą tu| dt “ 0 +and using the definition of ℓpΩ, βq we observe that the integrand is nonnegative. +In +particular, u ı 0 and we have that ΛpΩ, βq “ Rptu ą tu, βq. +5.2 Γ-convergence of Jp +Now we will prove that the functional Jp Γ´converges to the functional J, as p Ñ 1`. +Definition 5.4. A functional Jp Γ-converges to J as p Ñ 1` in the weak˚ topology of +BV pΩq if, for any u P BV pΩq, the following hold: +(i) For any sequence up P BV pΩq which converges to u weak˚ in BV pΩq as p Ñ 1`, +then +lim inf +pÑ1` Jppupq ě Jpuq. +(5.12) +(ii) There exists a sequence up P W 1,ppΩq which converges to u weak˚ in BV pΩq as +p Ñ 1`, such that +lim sup +pÑ1` Jppupq ď Jpuq. +(5.13) +Now, we are in position to prove the convergence theorem for the functional Jp. +Theorem 5.5. Let β ą ´1, then Jp Γ-converges to J as p Ñ 1`. +Proof. Let us suppose up P W 1,ppΩq and ||up||LppΩq “ 1. +We give the proof by dis- +tinguishing the possible values of β. +In any cases, we will have to prove (5.12) and +(5.13). +The case β ě 1. Let us fix a sequence up P W 1,ppΩq weak˚ converging to u in BV pΩq, +as p Ñ 1`. By using the H¨older-type inequality contained for example in [11, Proposition +A.1], we have: +ˆż +Ω +Fp∇upqdx ` +ż +BΩ +|up|FpνqdHN´1 +˙p +ď +ˆż +Ω +Fp∇upqpdx ` +ż +BΩ +|up|pFpνqdHN´1 +˙ +p|Ω| ` PF pΩqqp´1 . +Hence we have +lim inf +pÑ1` Jppupq ě |Du|F pΩq ` +ż +BΩ +|u|FpνqdHN´1 “ |Du|F pRNq “ Jpuq. +This proves (5.12). +19 + +To give the proof of (5.13), we observe that, by Theorem 4.3, there exists a se- +quence tukukPN Ď C8 +0 pΩq such that, for any q P r1, ps, uk converges to u in LqpΩq +and ||Fp∇ukq||L1pΩq converges to |Du|F pRNq as k Ñ `8. Moreover, it easily seen that +that ||Fp∇ukq||LppΩq converges to ||Fp∇ukq||L1pΩq and hence we have that there exists a +subsequence pk Ñ 1`, as k Ñ `8, such that ||FpDukq||pk +Lpk pΩq converges to |Du|F pRNq +as k Ñ `8. This implies that lim supkÑ`8 Jpkpukq ě Jpuq, that concludes the proof of +(5.13). +The case 0 ď β ă 1. Let us consider a sequence up weak˚ converging to u in BV pΩq, +as p Ñ 1`. A simple application of the Young inequality ap ě pab ´ pp ´ 1qb +p +p´1 with +b “ 1 +p, yields to +Jppupq “ +ż +Ω +F pp∇upqdx ` β +ż +BΩ +|up|pFpνqdHN´1 +ě +ż +Ω +Fp∇upqdx ` β +ż +BΩ +|up|FpνqdHN´1 ´ p ´ 1 +p +p|Ω| ` PF pΩqq , +Therefore, the conclusion (5.12) follows by applying the Proposition 5.2. +In order to get the second claim, Proposition 2.2 assures the existence of a sequence +uk P C8pΩq strongly converging to u in L1pΩq and }FpDukq}L1pΩq converges to |Du|F pΩq, +as k Ñ `8. Moreover ukFpνq converges to uFpνq in L1pBΩ, HN´1q. An argument sim- +ilar to the previous case leads us to say that }FpDukq}LpkpΩq converges to |Du|F pΩq and +ż +BΩ +|uk|pFpνqdHN´1 converges to +ż +BΩ +|u|FpνqdHN´1, as k Ñ `8. Hence the sequence +tukukPN satisfies (5.13). +The case ´1 ă β ă 0. Let us consider a sequence up P W 1,ppΩq weak˚ converging to +u in BV pΩq, as p Ñ 1`. +For any δ ą 0, let us set Ωδ “ tx P Ω : dEpxq ă δu and consider a smooth function ψ +equal to zero in ΩzΩδ and to one on BΩ, such that |∇ψ| ď 2 +δ . +The trace inequality (3.2) applied to the function v “ pu ´ |up|p´1upqψ gives +ż +BΩ +|u ´ |up|p´1up|FpνqdHN´1 +ď |Dpu ´ |up|p´1upq|F pΩδq ` +ˆ2b +δ ` c +˙ ż +Ωδ +|u ´ |up|p´1up|dx +ď |Du|F pΩδq ` +ż +Ωδ +Fp∇p|up|p´1upqqdx ` +ˆ2b +δ ` c +˙ ż +Ωδ +|u ´ |up|p´1up|dx, +(5.14) +where we have used that |Dpu´|up|p´1upq|F pBΩδq “ 0 for a set of δ’s of positive measure +because u ´ |up|p´1up P BV pΩq. +20 + +By using (5.14), we have +Jpuq ´ Jppupq “ |Du|F pΩq ´ +ż +Ω +F pp∇upqdx ` β +ż +BΩ +p|u| ´ |up|p´1upqFpνqdHN´1 +ď |Du|FpΩq ´ +ż +Ω +F pp∇upqdx ` |β||Du|F pΩδq +` |β| +ż +Ωδ +Fp∇p|up|p´1upqqdx ` |β| +ˆ2b +δ ` c +˙ ż +Ωδ +|u ´ |up|p´1up|dx :“ A. +(5.15) +Since +1 +|β| ą 1, we have +A ď 2|Du|F pΩδq ` |Du|F pΩzΩδq ´ +ż +Ω +Fp∇upqpdx ` +ż +ΩzΩδ +Fp∇p|up|p´1upqqdx +` +ˆK +δ ` c +˙ ż +Ωδ +|u ´ |up|p´1up|dx. +(5.16) +The Young inequality gives that +ż +Ω +Fp∇p|up|p´1upqqdx “ +ż +Ω +p|up|p´1Fp∇upqdx +ď +ż +Ω +F pp∇upqdx ` pp ´ 1q +ż +Ω +|up|pdx. +(5.17) +Furthermore by (5.15), (5.16) and (5.17), we have that +Jpuq ´ Jppupq ď 2|Du|F pΩδq ` |Du|F pΩzΩδq ´ +ż +ΩzΩδ +Fp∇p|up|p´1upqqdx +` pp ´ 1q +ż +Ω +|up|pdx ` +ˆK +δ ` c +˙ ż +Ωδ +|u ´ |up|p´1up|dx. +Since up converges to u in LqpΩq, then |up|p´1up converges to u in L1pΩq, as p Ñ 1`. +Hence, by taking p Ñ 1`, we have +lim sup +pÑ1` rJpuq ´ Jppupqs ď 2|Du|F pΩδq. +By sending δ Ñ 0`, we obtain (5.12). +Finally, the inequality (5.13) is obtained as in the previous case. +The proof of the Γ-convergence of the functional Jp is useful to prove the convergence +of the eigenvalues and eigenfunction, as p Ñ 1`. +Proposition 5.6. For any β ą ´1, it holds +lim +pÑ1` λ1pΩ, p, βq “ ΛpΩ, βq. +Moreover, the minimizers up P W 1,ppΩq of (5.1), with }u}LppΩq “ 1, weak˚ converge to +a minimizer u P BV pΩq of (5.4) as p Ñ 1`. +21 + +Proof. The Theorem 5.5 assures the existence of a sequence wp converging to a fixed +minimizer ¯u of (5.4). Let us consider the sequence of minimizers up of (5.4), we have: +lim sup +pÑ1` Jppupq ď lim sup +pÑ1` Jppwpq ď Jp¯uq “ ΛpΩ, βq. +(5.18) +This means that Jppupq is upper bounded for any p ą 1. If β ă 0, by the trace inequality +(3.2), we have that +ż +Ω +F pp∇upqdx ď ΛpΩ, βq ´ β|Dpup +pq|F pΩq ´ βc, +and, by (5.17) and (2.1), that +p1 ` βqap +ż +Ω +|∇up|pdx ď ΛpΩ, βq ´ βc ´ βpp ´ 1q. +Hence, by the compactness, we have that up is upper bounded in BV pΩq. If β ě 0, this +directly follows from (5.18). Therefore up weak˚ converges to u in BV pΩq. +Finally, by (5.12) and (5.18), we have that +ΛpΩ, βq “ Jp¯uq ď Jpuq ď lim inf +pÑ1` Jppupq ď lim sup +pÑ1` Jppupq ď ΛpΩ, βq, +and hence the conclusion by observing that λ1pΩ, p, βq “ Jppupq. +5.3 An isoperimetric inequality +Here we treat the shape optimization problem for ΛpΩ, βq. To this aim, we briefly recall +the properties of the eigenvalue problem λ1pWR, p, βq and then we prove an explicit +computation for ΛpWR, βq. By Theorem 5.1, a minimizer of (5.2) solves the following +problem: +# +´Qpu “ λ1pWR, p, βq |u|p´2 u +in WR +pFp∇uqqp´1Fξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0 +on BWR. +(5.19) +In particular, the following result holds (refer to in [9] for the positive values of the Robin +parameter). +Theorem 5.7. If up P C1,αpWRq X CpWRq is a positive solution of (5.19), then there +exists a monotone function ϕp “ ϕpprq, r P r0, Rs, such that ϕp P C8p0, RqXC1pr0, Rsq, +and +$ +’ +& +’ +% +uppxq “ ϕppF opxqq +in WR +ϕ +1 +pp0q “ 0 +|ϕ +1 +ppRq|p´2ϕ1 +ppRq ` βϕppRqp´1 “ 0. +(5.20) +Moreover, ϕp is decreasing if β ą 0, while is increasing if β ă 0. +22 + +We first compute ΛpWR, βq. +Proposition 5.8. If β ą ´1, then +ΛpWR, βq “ ˆβhF pWRq “ ˆβ N +R , +(5.21) +where ˆβ “ mintβ, 1u. +Proof. If β ě 0, we recall that +ΛpWR, βq “ +inf +EĎWR JpχEq. +By using Theorem 5.3 and the isoperimetric inequality, we have +RpE, βq “ JpχEq +“ +PF pE, WRq ` ˆβ +ż +BWRXBE +FpνEqdHN´1 +|E| +ě ˆβ PF pEq +|E| +ě ˆβ PF pWrq +|Wr| +ě ˆβ PF pWRq +|WR| +“ ˆβ N +R , +where Wr is the wulff shape of radius r ă R, with |Wr| “ |E|. +This proves ΛpWR, βq ě ˆβ N +R . For the reverse inequality we take E “ WR, hence +ΛpWR, βq “ ℓpWR, βq ď RpWR, βq “ ˆβ N +R . +Now, we study the case ´1 ă β ă 0 and we will make use of the Γ-convergence. +Hence, let up P W 1,ppWRq a minimizer of (5.2). We know, thanks to the Proposition +5.6, that +lim +pÑ1` λ1pWR, p, βq “ ΛpWR, βq +and we take up “ ϕp as in (5.20). So, the minimizer converges strongly in L1pWRq to +u P BV pWRq, almost everywhere in WR and up +˚á u in BV pΩq for p Ñ 1`. Moreover, +up is radially increasing, hence u is nondecreasing and this implies that its superlevel +sets tu ą tu are concentric Wulff shapes tr ă F opxq ă Ru and, by Theorem 5.3, it holds +that +ΛpWR, βq “ N +R +p r +RqN´1 ` β +1 ´ p r +RqN´1 +for some r P r0, Rr. Therefore, by minimizing the function +fptq “ tN´1 ` β +1 ´ tN +t P r0, 1r, +we observe that the minimum is attained at t “ 0. Hence, the thesis follows. +23 + +Finally, we prove an isoperimetric inequality for ΛpΩ, βq when a volume constraint +holds: if β ě 0, the Wulff shape is a minimizer and, if β ă 0, it is a maximizer for +ΛpΩ, βq. +Proposition 5.9. If β ě 0 and WR is the wulff shape of radius R and |WR| “ |Ω|, then +ΛpWR, βq ď ΛpΩ, βq. +If ´1 ă β ă 0, then +ΛpWR, βq ě ΛpΩ, βq. +Proof. If β ě 0, by using the same argument of the Proposition 5.8, we have that then +RpE, βq “ JpχEq ě ˆβ PF pWRq +|WR| +“ ˆβ N +R “ ΛpWR, βq, +for any E Ď Ω. The conclusion follows by passing to the infimum on the set E Ď Ω and +using Theorem 5.3. +If ´1 ă β ă 0, then using the isoperimetric inequality [14] and (5.21), we have that +ΛpΩ, βq ď β PF pΩq +|Ω| +ď β PF pWRq +|WR| +“ ΛpWR, βq. +Acknowledgement +This work has been partially supported by the MIUR-PRIN 2017 grant “Qualitative +and quantitative aspects of nonlinear PDE’s”, by GNAMPA of INdAM, by the FRA +Project (Compagnia di San Paolo and Universit`a degli studi di Napoli Federico II) +000022--ALTRI_CDA_75_2021_FRA_PASSARELLI. +References +[1] A. Alvino, V. Ferone, G. Trombetti, and P.-L. Lions. Convex symmetrization and +applications. Ann. Inst. H. Poincar´e C Anal. Non Lin´eaire, 14(2):275–293, 1997. 7 +[2] M. Amar and G. Bellettini. A notion of total variation depending on a metric with +discontinuous coefficients. In Annales de l’Institut Henri Poincar´e C, Analyse non +lin´eaire, volume 11, pages 91–133. Elsevier, 1994. 17 +[3] L. Ambrosio, N. Fusco, and D. Pallara. 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Annales de l’IHP Analyse non lin´eaire, 30(6):983–996, 2013. 6 +26 + diff --git a/PNAzT4oBgHgl3EQflP3v/content/tmp_files/load_file.txt b/PNAzT4oBgHgl3EQflP3v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be0fc63c89d8e7d2b0e0b58cb38a56fb337d6678 --- /dev/null +++ b/PNAzT4oBgHgl3EQflP3v/content/tmp_files/load_file.txt @@ -0,0 +1,820 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf,len=819 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='01546v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='AP] 4 Jan 2023 On the first Robin eigenvalue of the Finsler p-Laplace operator as p Ñ 1 Rosa Barbato˚, Francesco Della Pietra˚, Gianpaolo Piscitelli˚ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let Ω be a bounded, connected, sufficiently smooth open set, p ą 1 and β P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In this paper, we study the Γ-convergence, as p Ñ 1`, of the functional Jppϕq “ ż Ω F pp∇ϕqdx ` β ż BΩ |ϕ|p FpνqdHN´1 ż Ω |ϕ|p dx where ϕ P W 1,ppΩqzt0u and F is a sufficientely smooth norm on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We study the limit of the first eigenvalue λ1pΩ, p, βq “ infϕPW 1,ppΩq ϕ‰0 Jppϕq, as p Ñ 1`, that is: ΛpΩ, βq “ inf ϕPBV pΩq ϕı0 |Du|F pΩq ` mintβ, 1u ż BΩ |ϕ| FpνqdHN´1 ż Ω |ϕ| dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Furthermore, for β ą ´1, we obtain an isoperimetric inequality for ΛpΩ, βq depending on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The proof uses an interior approximation result for BV pΩq functions by C8pΩq functions in the sense of strict convergence on Rn and a trace inequal- ity in BV with respect to the anisotropic total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' MSC 2020: 28A75, 35J25, 35P15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Keywords and phrases: Finsler p-Laplace eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Γ-convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Isoperimetric inequalities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Trace inequalities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Strict interior approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' ∗Dipartimento di Matematica e Applicazioni “R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Caccioppoli”, Universit`a degli studi di Napoli Federico II, Via Cintia, Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Angelo - 80126 Napoli, Italia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Email: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='dellapietra@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='it (corresponding author), gianpaolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='piscitelli@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='it 1 Contents 1 Introduction 2 2 Notation and preliminaries 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1 The Finsler norm .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3 The anisotropic total variation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 7 3 An anisotropic trace inequality 8 4 Interior approximation 11 5 The first Robin eigenvalue of the Finsler p-Laplacian as p Ñ 1 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1 The case p “ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2 Γ-convergence of Jp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3 An isoperimetric inequality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 22 1 Introduction Let Ω be a bounded, connected, sufficiently smooth open set, p ą 1 and β P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In this paper, we study the asympthotic behaviour, as p Ñ 1`, of the following minimum problem λ1pΩ, p, βq “ inf ϕPW 1,ppΩq ϕ‰0 Jppϕq (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) where Jppϕq “ ż Ω F pp∇ϕqdx ` β ż BΩ |ϕ|p FpνqdHN´1 ż Ω |ϕ|p dx , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) ν is the outer normal to BΩ and F is a sufficiently smooth norm on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If u P W 1,ppΩq is a minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1), then it solves the following Robin eigenvalue problem # ´Qpu “ λ1pΩ, p, βq |u|p´2 u in Ω F p´1p∇uqFξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0 on BΩ, where Qpu is the anisotropic p´Laplace operator Qpu :“ div ˆ1 p∇ξrF psp∇uq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' From the point of view of finding optimal domains for λ1pΩ, pβq, there is a significant difference from the case of β ą 0 to the case β ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' It is known that the optimal shape 2 with a volume constraint for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) depends on the sign of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For positive values of the Robin parameter, the so-called Wulff shape (see Section 2 for details) is a minimizer [9]: λ1pΩ, p, βq ě λ1pW, p, βq where |W| “ |Ω|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ă 0, the problem is not completely solved, even in the Euclidean case (that is when Fpξq “ bř i ξ2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Indeed, in 1977 Bareket [5] conjectured that, the first eigenvalue is maximized by a ball in the class of the smooth bounded domains of given volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In [13] it has been showed that it is true for domains close, in certain sense, to a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Subsequently, the authors in [15] (see [19] for the p-Laplacian case) have disproved the conjecture for |β| large enough and have showed that it is true for small values of |β| in suitable class of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In the Finsler setting, this problem has been addressed in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Our final aim is to obtain optimal shapes for the limiting functional of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1), as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' To do that, we first study the limit of λ1pΩ, pβq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In particular, we prove that when β ą ´1, the functional Jp, defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2), Γ´converges, as p Ñ 1`, to Jpϕq “ |Dϕ|F pΩq ` mint1, βu ż BΩ |ϕ| FpνqdHN´1 ż Ω |ϕ| dx , where |Dϕ|F pΩq is the anisotropic total variation of ϕ (see Section 2 for the precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' This will imply that lim pÑ1` λ1pΩ, p, βq “ Λpβ, Ωq :“ inf ϕPBV pΩq Jpϕq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3) Then, we prove an isoperimetric inequality for ΛpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In particular, we obtain that keeping the volume of Ω fixed, the Wulff shape minimises ΛpΩ, βq when β ě 0, and maximises it when ´1 ă β ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The proof of the convergence result, and then of the two isoperimetric inequalities, relies on two results on the anisotropic total variation, which are also of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The first one is a trace inequality in the BV space: ż BΩ |u| Fpνq dHN´1 ď c1 |Du|F pΩq ` c2 ż Ω |u| dx, @u P BV pΩq, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) where c1 and c2 are two constants which depend on the geometry of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' This inequality has been revealed very useful in capillarity problems, and it has been studied for example in [4, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The second key result is an interior approximation for BV functions by smooth func- tions with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' It is well known that if Ω is an open set, then the total variation of a function u P BV pΩq can be approximated with the corresponding total variation of a sequence in C8pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Actually, an analogous result is not true in gen- eral if one need to axpproximate |Du| with a sequence of C8 function with compact 3 support in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In order to do that, more regularity is needed on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In the Euclidean setting, this problem has been addressed in [20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In this paper, we show that for any u P BV pΩq X LppΩq, for some p P r1, 8q, there exists a sequence tukukPN Ď C8 0 pΩq such that, for any q P r1, ps, uk Ñ u in LqpΩq and |Duk|F pRNq Ñ |Du|F pRNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We finally stress that the problem we deal with is strictly related to capillarity prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We refer the reader, for example, to [16, 17] for the Euclidean case and to [8] for the anisotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The structure of the paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In Section 2, we review some useful tools on the Finsler norm, the anisotropic curvature and functions of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In Section 3 we prove the anisotropic trace inequality for general domains, and for smooth domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In Section 4, we give the strict approximation result and finally, in Section 5 we prove the Γ´convergence results and the isoperimetric inequality for ΛpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 2 Notation and preliminaries In this Section we give several definitions and properties related the Finsler norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In particular, we review some basic facts on the anisotropic total variation of a BV function, and on the anisotropic curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1 The Finsler norm Throughout the paper we will assume that F is a convex, even, 1´homogeneous function ξ P RN ÞÑ Fpξq P r0, `8r, such that Fptξq “ |t|Fpξq, t P R, ξ P RN, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) and such that a|ξ| ď Fpξq, ξ P RN, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) for some constant a ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' It is easily seen that this hypothesis assure the existence of a positive constant b ě a such that Fpξq ď b|ξ|, ξ P RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Throughout the paper, we will also assume that F belongs to C2pRNzt0uq and that ∇2 ξrF 2spξq is positive definite in RNzt0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3) The assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3) on F ensures that the operator Qpu :“ div ˆ1 p∇ξrF psp∇uq ˙ 4 is elliptic, therefore there exists a positive constant γ such that nÿ i,j“1 ∇2 ξiξjrF pspηqξiξj ě γ|η|p´2|ξ|2 @η P RNzt0u, @ξ P RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The polar function F o : RN Ñ r0, `8r of F is F opvq “ sup ξ‰0 ξ ¨ v Fpξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' It is easily seen that also F o is a convex function satisfying the properties (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Furthermore, we have Fpvq “ sup ξ‰0 ξ ¨ v F opξq, and from this follows that |ξ ¨ η| ď FpξqF opηq @ξ, η P RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) The Wulff shape centered at the origin is the set denoted by W “ tξ P RN : F opξq ă 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We denote κN “ |W|, where |W| is the Lebesgue measure of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' More generally, the set Wrpx0q indicates rW ` x0, that is the Wulff shape centered at x0 with measure κNrN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If no ambiguity occurs, we will write Wr instead of Wrp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The functions F and F o enjoy the following properties: Fξpξq ¨ ξ “ Fpξq, F o ξ pξq ¨ ξ “ F opξq @ξ P RNzt0u, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5) FpF o ξ pξqq “ F opFξpξqq “ 1 @ξ P RNzt0u, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6) F opξqFξpF o ξ pξqq “ FpξqF o ξ pFξpξqq “ ξ @ξ P RNzt0u, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='7) where Fξ “ ∇Fpξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Given a bounded domain Ω, the anisotropic distance of x P Ω to BΩ is defined as dF pxq :“ inf yPBΩ F opx ´ yq, x P Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We highlight that, when Fpξq “ bř i ξ2 i , then dF “ dE is the Euclidean distance function from the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The function dF is a uniform Lipschitz function in Ω, and Fp∇dF pxqq “ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We have that dF P W 1,8 0 pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Many properties of the anisotropic distance function are studied in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Finally, the anisotropic inradius of Ω is RF pΩq “ maxtdF pxq, x P Ωu, that is the radius of the largest Wulff shape Wrpxq contained in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2 Anisotropic curvatures Here we recall some properties of the anisotropic mean curvature, as well as an integra- tion formula in anisotropic normal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We refer to [7] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If Ω has a C2 boundary, the anisotropic outer normal to BΩ is defined as nF pyq “ Fξpνpyqq, y P BΩ, where νpyq is the Euclidean outer normal to BΩ at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6) it holds that F opnFpyqq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us denote by TyBΩ the tangent space to BΩ at y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' the anisotropic Weingarten map is defined as dnF : TyBΩ Ñ TnpyqW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The eigenvalues κF 1 ď κF 2 ď .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' ď κF N´1 of this map are called the anisotropic principal curvatures at y (see also [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The anisotropic mean curvature of BΩ at a point y is defined as HF pyq “ κF 1 pyq ` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' κF N´1pyq, y P BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The anisotropic distance dF is a C2 function in a tubular neighborhood of BΩ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' hence we are in position to define the matrix-valued function Wpyq “ ´Fξξp∇dF pyqq∇2dF pyq, y P BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Based on this function, it is possible to give a different definition of the anisotropic principal curvatures [7, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Since Wpyqv P TyBΩ, for any v P Rn, it remains defined the map Wpyq: Ty Ñ Ty, as Wpyqw “ Wpyqw, w P Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The matrix Wpyq (that is, in general, non-symmetric) admits the real eigenvalues κF 1 pyq ď κF 2 pyq ď .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' ď κF N´1pyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Actually, the definition is equivalent to the preceding one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, it holds that HF pyq “ div rFξ p´∇dFpyqqs “ TrpWpyqq (see also [25, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' To state the change of variable formula in anisotropic normal coordinates, we need some preliminary definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let Φpy, tq “ y ´ tFξpνpyqq, y P BΩ, t P R and for y P BΩ, ℓpyq “ suptdF pzq, z P Ω and y P Πpzqqu where Πpzq “ tη P BΩ: dF pzq “ F opz ´ ηqu (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='8) is the set of the anisotropic projections of a point z P Ω on BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then, we recall the following 6 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' ([7, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1]) For every h P L1pΩq, it holds ż Ω hpxqdx “ ż BΩ Fpνpyqq ż ℓpyq 0 hpΦpy, tqqJpy, tq dt dHN´1pyq, where Jpy, tq “ N´1 ź i“1 p1 ´ tκF i pyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='9) Since 1 ´ tκF i pyq ą 0 for any i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' , N ´ 1 ([7, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4]), Jpy, tq is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover it holds that ´ d dt rJpy, tqs Jpy, tq “ N´1 ÿ i“1 κF i pyq 1 ´ tκF i pyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='10) Finally, we conclude this section, by recalling that, for any x P Ω such that Πpxq “ tyu, it holds that ([7, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3]): ∇dF pxq “ ´ νpyq Fpνpyqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3 The anisotropic total variation Let u P BV pΩq, the total variation of u with respect to F is defined as |Du|F pΩq “ sup "ż Ω u divpgq dx : g P C1 0pΩ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' RNq, F opgq ď 1 and the perimeter of a set E with respect to F is: PF pE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Ωq “ |DχE|F pΩq “ sup "ż E divpgq dx : g P C1 0pΩ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' RNq, F opgq ď 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, PF pE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Ωq “ ż ΩXB˚E FpνEqdHN´1 where B˚E is the reduced boundary of Ω and νE is the Euclidean normal to BE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us fix u P BV pΩq and assume that u ” 0 in RNzΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then u P BV pRNq and |Du|F pRNq “ |Du|F pΩq ` ż BΩ |u|FpνqdHN´1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='12) (see for example [6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For the anisotropic perimeter, an isoperimetric inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' More precisely, PF pΩq ě PF pWRq, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='13) where WR is the Wulff shape with the same measure of Ω (see for example [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The following approximation results in BV hold (refer to [4], [18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='17] for the Euclidean case and to [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1] for the Finsler case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let f P BV pΩq, then there exists a sequence tfkukPN Ď C8pΩq such that: lim kÑ`8 ż Ω |fk ´ f| dx “ 0 and lim kÑ`8 |Dfk|F pΩq “ |Df|F pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let E be a set of finite perimeter in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' A sequence of C8 sets tEkuk exists, such that: lim kÑ`8 ż Ω |χEk ´ χE| dx “ 0 and lim kÑ`8 |DχEk|F pΩq “ PF pE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 3 An anisotropic trace inequality In this section we prove a trace inequality in BV with respect to the anisotropic total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We first give the result in a general case (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1), then we refine the constants involved in the inequality by requiring more regularity on the boundary of Ω (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Firstly, let us set qpyq “ lim ρÑ0` sup $ ’ ’ & ’ ’ % ż BΩ χAFpνqdHN´1 |DχA|F pΩq : A Ă Ω X Bρpyq, |A| ą 0, PF pA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Ωq ă `8 , / / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' / / and Q “ supyPBΩ qpyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The following inequality generalizes the trace inequality given in [4, Theorem 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let Ω be a bounded open set with HN´1pΩq ă `8, and let u be a function in BV pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then for any ε ą 0, it holds ż BΩ |u|FpνqdHN´1 ď pQ ` εq|Du|F pΩq ` cpΩ, εq ż Ω |u|dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) where cpΩ, εq does not depend on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us fix y P BΩ and ρpyq ą 0 such that ż BΩ χBFpνqdHN´1 ď pQ ` εq|DχB|F pΩq, 8 for any B Ă Ω X Bρpyqpyq with PF pB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Ωq ă `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If sptpuq Ă Bρpyqpyq, then ż BΩ |u|FpνqdHN´1 “ ż BΩ Fpνq ˆż `8 0 χt|u|ątupyqdt ˙ dHN´1 “ ż `8 0 ˆż BΩ Fpνqχt|u|ątupyqdHN´1 ˙ dt ď pQ ` εq ż `8 0 |Dχt|u|ątu|F pΩqdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By using the coarea formula, we have ż BΩ |u|FpνqdHN´1 ď pQ ` εq|D|u||F pΩq ď pQ ` εq|Du|F pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let tBρpyquyPBΩ be a cover of BΩ and let us extract a finite sub-cover B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' , Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Now, considering a partition of unity ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' , ϕk such that 0 ď ϕi ď 1, ϕi P C1 0pBiq, kÿ i“1 ϕipyq “ 1 if y P BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If f P BV pΩq, then ż BΩ |u| F pνq dHN´1 ď pQ ` εq ˇˇˇˇˇD ˜ kÿ i“1 ϕiu ¸ˇˇˇˇˇ F pΩq ď pQ ` εq kÿ i“1 p|ϕiDu|F pΩq ` |uDϕi|F pΩqq “ pQ ` εq kÿ i“1 ˆż Ω ϕid|Du|F ` ż Ω ud |Dϕi|F pΩq ˙ ď pQ ` εq |Du|F pΩq ` cpΩ, εq ż Ω |u| dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If the boundary of Ω is sufficilently smooth, we can show that Q can be taken equal to 1 and ε “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' More precisely, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let Ω be a bounded open connected set of class C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then there exists a positive constant c such that ż BΩ |u| Fpνq dHN´1 ď |Du|F pΩq ` c ż Ω |u| dx, @u P BV pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Since Ω is C2, then a uniform sphere condition of radius r ą 0 holds, in the sense that for every point y P BΩ there exists z P Ω such that y P Brpzq Ă Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let R Ps0, `8r be the maximum of the principal radii of curvature of BW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Such maximum exists being 9 F (and F o) strongly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If κF 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' , κF N´1 are the anisotropic principal curvatures, we have that κiF pyq ď 1 µ, i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' , N ´ 1, with µ “ r R ([7, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Therefore, in the set Ω µ 2 :“ ␣ x P Ω : dF pxq ă µ 2 ( , it holds that dF is C2 ([7, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='16]) and κF i pyq 1 ´ κF i pyqdF pxq ď $ & % 0 if κF i ď 0 2 µ if 0 ă κF i ď 1 µ, where y P BΩ is the anisotropic projection of x P Ω on BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then N´1 ÿ i“1 κiFpyq 1 ´ κiFpyqdF pxq ď 2N ´ 1 µ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3) for x P Ω µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We may restrict ourselves to the case u is nonnegative and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Inte- grating by parts and recalling that Fp∇dF pxqq “ 1 in Ω, it holds that ż Ω ´∆FdF pxq upxq ´µ 2 ´ dF pxq ¯` dx “ ż Ω Fξp∇dF pxqq ¨ ∇upxq ´µ 2 ´ dF pxq ¯` dx ´ ż Ω µ 2 upxqFξp∇dF pxqq ¨ ∇dF pxq dx ´ µ 2 ż BΩ upxqFξp∇dF pxqq ¨ ν dx “ ż Ω Fξp∇dF pxqq¨∇upxq ´µ 2 ´ dF pxq ¯` dx´ ż Ω µ 2 upxq dx`µ 2 ż BΩ upyq FpνqdHN´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Now, we estimate the term ż Ω Fξp∇dF pxqq ¨ ∇upxq ´µ 2 ´ dF pxq ¯` dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6) it holds that ż Ω Fξp∇dF pxqq ¨ ∇upxq ´µ 2 ´ dF pxq ¯` dx ě ´µ 2 ż Ω Fp∇upxqq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) On the other hand, the change of variable formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) gives that ż Ω Fξp∇dF pxqq ¨ ∇upxq ´µ 2 ´ dF pxq ¯` dx “ ż BΩ Fpνq ż µ 2 0 ´µ 2 ´ t ¯ d dtrupφpy, tqqsJpy, tqdt dHN´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 10 Integrating by parts and using the fact that Jpy, 0q “ 1, the above integral becomes ´ µ 2 ż BΩ upyqFpνqdHN´1pyq ´ ż BΩ Fpνq ż µ 2 0 upφpy, tqq ´µ 2 ´ t ¯ dJ dt dt dHN´1pyq ` ż BΩ Fpνq ż µ 2 0 upφpy, tqqJpy, tqdt dHN´1pyq ď ´µ 2 ż BΩ upyqFpνqdHN´1pyq ` pN ´ 1q ż BΩ Fpνq ż µ 2 0 upyqJpy, tq dt dHN´1pyq ` ż BΩ Fpνq ż µ 2 0 upφpy, tqqJpy, tqdt dHN´1pyq “ ´µ 2 ż BΩ upyqFpνqdHN´1pyq ` N ż Ω u dx where in the inequality we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='10) and the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, joining with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) it holds that ż BΩ upyqFpνqdHN´1 ď |Du|F pΩq ` 2N µ ż Ω upxq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Using the notation of the above theorem, we explicitly observe that the constant in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) is c “ 2N µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 4 Interior approximation Now we provide an approximation result for BV -functions by smooth functions with compact support in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We preliminary state two useful lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Firstly, we recall from [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2] the following result on diffeomorphic perturbations of sets Ω with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We denote by ι and I the identical vector and matrix function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let Ω Ă RN be a bounded open set with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then there exists τ0 ą 0 and, for 0 ď τ ď τ0, a family of C8´diffeomorphisms Φτ : RN Ñ RN with inverses Ψτ such that Φ0 “ Ψ0 “ ι;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Φτ Ñ ι and Ψτ Ñ ι as τ Ñ 0 uniformly on RN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' ∇Φτpxq Ñ I and ∇Ψτpxq Ñ I as τ Ñ 0 uniformly with respect to x on RN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 11 ΦτpΩq Ť Ω for all τ P p0, τ0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Now, we give the anisotropic version of the change of coordinates formula for BV - functions, stated in [18, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let u be a function in BVlocpΩq, Φ : RN Ñ RN be a diffeomorphism and A Ť Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then ˇˇDpu ˝ Φ´1q ˇˇ F pΦpAqq “ |HDu|F pAq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) where H “ | det ∇Φ|r∇Φs´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us consider u P C1pΩq and g P C1 0pA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' RNq, then the following change of area formula holds ż ΦpAq pg ˝ Φ´1q ¨ ∇pu ˝ Φ´1qdx “ ż ΦpAq pg ˝ Φ´1q ¨ pp∇u ˝ Φ´1q∇Φ´1qdx “ ż A g ¨ p∇up∇Φ´1 ˝ Φqq| det ∇Φ|dz “ ż A g ¨ pH∇uqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) Thus, the thesis (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) holds for u in C1pΩq, that is ż ΦpAq Fp∇pu ˝ Φ´1qqdx “ ż A FpH∇uqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Suppose now that u P BVlocpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2 we can approximate u by a sequence tuiu Ă C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, the corresponding functions ui˝Φ´1 converge to u˝Φ´1 in L1pAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, we can pass to the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2), obtaining ż ΦpAq pg ˝ Φ´1q ¨ dDpu ˝ Φ´1q “ ż A g ¨ H dDu “ ż A g ¨ pHνqd |Du| (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3) where ν is obtained by differentiating Du with respect to |Du|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If F opgq ď 1, then also F opg ˝ Φ´1q ď 1 and sptpg ˝ Φ´1qq Ď ΦpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Therefore, by definition of total variation with respect to F, we have ż A g ¨ pHνqd |Du| ď ˇˇDpu ˝ Φ´1q ˇˇ F pΦpAqq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) implies that sup F opgqď1 ż A g ¨ pHνq d |Du| “ |HDu|F pAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, taking the supremum on the left hand side in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4), it holds |HDu|F pAq ď ˇˇDpu ˝ Φ´1q ˇˇ F pΦpAqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 12 For the reverse inequality, we consider g “ γ ˝ Φ P C1 0pA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' RNq, where γ P C1 0pΦpAq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' RNq and F opγq ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Therefore g ˝ Φ´1 “ γ and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3), we have ż ΦpAq γ ¨ dDpu ˝ Φ´1q “ ż A pγ ˝ Φq ¨ pHνqd |Du| pAq ď ż A F opγ ˝ Φq|H|d|Du|F ď ż A |H| d |Du|F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, we have ˇˇDpu ˝ Φ´1q ˇˇ F pΦpAqq ď |HDu|F pAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' At this stage, we are in position to state the main approximation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let Ω Ă RN be an open bounded set with Lipschitz boundary and let u P BV pΩq X LppΩq for some p P r1, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then there exists a sequence tukukPN Ď C8 0 pΩq such that, for any q P r1, ps, uk Ñ u in LqpΩq and |Duk|F pRNq Ñ |Du|F pRNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us fix a family pΦτq0ďτďτ0 of diffeomorphisms fron RN to RN with inverses pΨτq0ďτďτ0 according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1, and consider uτ :“ u ˝ Ψτ for τ P r0, τ0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By construction uτ “ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' outside a compact subset of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By [20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1], we know that uτ P LppΩq for all τ P r0, τ0s, uτ Ñ u in LppΩq and also in LppRNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The change of coordinates formula in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2 implies that |Duτ|F pRNq “ ż RN ˇˇp∇ΨτqT ˇˇ |detp∇Φτq| d |Du|F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Thus uτ P BV pRNq and, since the integrand on the right hand side uniformly converges to 1, lim τÑ0` |Duτ|F pRNq “ |Du|F pRNq It remains only to prove that there exists vτ P C8 0 pΩq such that ||uτ ´ vτ||p ă τ and ˇˇ|Duτ|F pRNq ´ |Dvτ|F pRNq ˇˇ ă τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The first convergence (in Lp) has been proved in [20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' meanwhile the convergence of the total variation is based on the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For any ε ą 0, let us consider the mollification uε :“ uτ ˚ ηε, where ηεpxq “: ε´nηpε´1xq, for the standard mollifier η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence uε Ñ uτ in LppΩq and for the zero extensions, in L1pRNq [3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 13 Then, by the lower semicontinuity of the anisotropic total variation, we have |Duτ|F pRNq ď lim εÑ0` inf |Duε|F pRNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Therefore, it remains to prove the opposite inequality lim εÑ0` sup |Duε|F pRNq ď |Duτ|F pRNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5) Let us choose ϕ P C8 0 pRN, RNq with F opϕq ď 1 and calculate ż RN uε ˚ divpϕqdx ż RN uτpηε ˚ div ϕqdx “ ż RN uτ divpηε ˚ ϕqdx ď |Duτ|F pRNq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6) where the inequality in the last term holds since F opηε ˚ ϕq ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Indeed, by using the 1-homogeneity of F o and Jensen’s Inequality (see, for instance, [22, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2]), we gain that F o ˆż RN ηǫpx ´ yqϕpyqdy ˙ ď ż RN F opηǫpx´yqϕpyqqdy “ ż RN ηεpx´yqF opϕpyqqdy ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, by passing to the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5), we reach the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6) by the arbitrariness of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 5 The first Robin eigenvalue of the Finsler p-Laplacian as p Ñ 1 In this Section, we give an application of the results proved above to a Robin eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' More precisely, our aim is analyze the Γ-limit of the functional Jppϕq “ ż Ω F pp∇ϕqdx ` β ż BΩ |ϕ|p FpνqdHN´1 ż Ω |ϕ|p dx , ϕ P W 1,ppΩqzt0u, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1) where Ω is a bounded, connected, sufficiently smooth open set, p ą 1 and β P R, and prove an isoperimetric inequality for the limit, as p Ñ 1`, of the first eigenvalue λ1pΩ, p, βq “ inf ϕPW 1,ppΩq ϕ‰0 Jppϕq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) depending on the value of the parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' A key point for proving this result is the convergence of the functional Jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We first recall the following existence result for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1 ([9, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let p ą 1, β P R and Ω bounded Lipschitz domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then there exists a minimum u P C1,αpΩq X CpΩq of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) that satisfies # ´Qpu “ λpΩ, p, βq |u|p´2 u in Ω F p´1p∇uqFξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0 on BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3) Moreover, u does not change sign in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Finally, λ1pΩ, p, βq is positive if β ą 0, while is negative if β ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1 The case p “ 1 In order to study the limit case of Jp as p goes to 1, we consider the functional Jpϕq “ |Dϕ|F pΩq ` mintβ, 1u ż BΩ |ϕ| FpνqdHN´1 ż Ω |ϕ| dx , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) where ϕ P BV pΩq and u ı 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, we study the associated minimum problem ΛpΩ, βq “ inf ϕPBV pΩq ϕı0 Jpϕq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5) Depending on β, we will impose different assumptions on the regularity of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Indeed: if β ě 0, we will suppose that BΩ is Lipschitz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' if ´1 ă β ă 0, we will assume that BΩ is C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In particular, this difference depends on the fact that in the case β ă 0 we use the trace inequality, studied in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Finally, if β ď ´1 the problem is not well posed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' indeed if β ă ´1, then ΛpΩ, βq “ ´8 while if β “ ´1, Λ is finite but can be not achieved, also in the case of smooth domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For further details, we refer the reader to the Euclidean case treated in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us discuss the presence of the term mintβ, 1u in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For any value of β, it could seem more natural to study the problem λpΩ, 1, βq “ inf ϕPBV pΩq ϕ‰0 |Dϕ|F pΩq ` β ż BΩ |ϕ| FpνqdHN´1 ż Ω |ϕ| dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6) Actually, we have that for β ě 1 it holds λpΩ, 1, βq “ ΛpΩ, βq “ hF pΩq, where hF pΩq is the first Cheeger constant of Ω in the Finsler setting (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' [6]): hF pΩq “ inf ϕPBV pΩq ϕı0 |Dϕ|F pRNq ż Ω |ϕ| dx “ inf EĎΩ PF pE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Ωq |E| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='7) Indeed, in this case, it is immediate to see that λpΩ, 1, βq ě hF pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 15 On the other hand, if u is a minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6), then, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3, there exists uk P C8 0 pΩq such that uk Lq ÝÑ u, ||∇uk||L1pΩq L1 ÝÑ |Du|F pRNq, for any q ď N N ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Therefore λpΩ, 1, βq ď lim kÑ`8 ż Ω Fp∇ukqdx ż Ω |uk| dx “ hF pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Now we focus on the possibility of studying the minimization problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5) restricting our analysis to characteristic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence if E Ď Ω, we have JpχEq “ PF pE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Ωq ` mint1, βu ż BΩXB˚E FpνEqdHN´1 |E| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By denoting RpE, βq :“ JpχEq, we consider the minimization problem ℓpΩ, βq “ inf EĎΩ RpE, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='8) Before proving the equivalence between problems (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='8), we need the following result on the lower semicontinuity of the numerator of the functional J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let β ě ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The functional Gpuq “ |Du|F pΩq ` mint1, βu ż BΩ |u|FpνqdHN´1 is lower semicontinuous on BV pΩq with respect to the topology of L1pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ě 0, the lower semicontinuity (with Ω Lipschitz) follows immediately by the lower semicontinuity of each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Then we assume β ă 0 (and Ω in C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The proof is an adaptation of [21, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2] to the Finsler case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let u P BV pΩq, and let us consider a sequence tukukPN Ď BV pΩq converging to u in L1pΩq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' we have the following estimate Gpuq ´ Gpukq ď |Du|F pΩq ´ |Duk|F pΩq ` ż BΩ |u ´ uk|FpνqdHN´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='9) Now, for a fixed δ ą 0, let us define Ωδ “ tx P Ω : dEpxq ă δu, where dE is the standard Euclidean distance to the boundary of Ω ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' moreover let us consider vδ “ p1´χδqpu´ukq, 16 where χδ is a cut-off function such that χδ “ 1 in ΩzΩδ and |∇χδ| ď 2 δ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The trace inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) applied to vδ gives ż BΩ |u´uk|FpνqdHN´1 ď |Dpu ´ ukq|F pΩδq` 2b δ ż Ωδ |u´uk|dx`c ż Ωδ |u´uk|dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='10) Moreover, we have |Dpu ´ ukq|F pΩδq ď |Du|F pΩδq ` |Duk|F pΩδq ` |Dpu ´ ukq|F pBpΩzΩδqq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='11) but last term is zero on a set of δ’s of positive measure because u ´ uk P BV pΩq, for all k P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='9)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='10)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='11), we gain: Gpuq ´ Gpukq ď |Du|F pΩq ` |Du|F pΩδq ´ |Duk|F pΩzΩδq ` ˆ2b δ ` c ˙ ż Ωδ |u ´ uk|dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By the lower semicontinuity of the functional |Duk|F pΩzΩδq in L1pΩzΩδq, we have that lim sup kÑ`8 rGpuq ´ Gpukqs ď 2|Du|F pΩδq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The conclusion follows by sending δ Ñ 0`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' At this stage, we state the main existence result of the minimum problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For any β ą ´1, there exists a minimum to problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In particular, it holds ΛpΩ, βq “ ℓpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, if u P BV pΩq is a minimum of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5), then ΛpΩ, βq “ Rptu ą tu, βq, for some t P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let un be a minimizing sequence in BV pΩq of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5), such that }un}L1pΩq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ą 0, then un is bounded in BV pΩq and hence un ˚á u in BV pΩq and un L1 ÝÑ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In particular, if β ě 1, Jpunq “ |Dun|F pRNq, by using the lower semicontinuity of the anisotropic total variation [2], we obtain that Jpuq ď lim inf n Jpunq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence u is the minimum of the functional J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If 0 ă β ă 1, let Ωδ “ tx P Ω | dEpxq ă δu, with δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We have |Dun|F pΩq “ |Dun|F pΩzΩδq ` |Dun|F pΩδq ě |Dun|F pΩzΩδq ` β |Dun|F pΩδq 17 and hence Jpunq ě |Dun|F pΩzΩδq ` β „ |Dun|F pΩδq ` ż BΩ |un| FpνqdHN´1 \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, by the lower semicontinuity of J, we have lim inf n Jpunq ě |Dun|F pΩzΩδq ` β |Dun|F pRNzpΩzΩδqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By using the fact that u P BV pΩq, we obtain that lim inf n Jpunq ě Jpuq, as δ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Now, let us take ´1 ă β ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' It easily seen that Jpunq ď C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Using the trace inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2), we obtain Jpunq ě p1 ` βq |Dun|F pΩq ` cβ ě cβ and |Dun|F pΩq ď C 1 ` β ´ βc 1 ` β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Being un P BV pΩq and by the fact that the functional J is lower semicontinuous (proved in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2), we have that u is a minimum of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Now, we want to prove last part of the Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Obviously, we have ΛpΩ, βq ď ℓpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' To prove the reverse inequality, we take u P BV pΩq a minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By using the coarea formula |Du|F pΩq “ ż `8 ´8 PF ptu ą tu, Ωqdt, we have ΛF pΩ, βq “ ż `8 ´8 PFptu ą tu, Ωqdt ` mintβ, 1u ż `8 ´8 HN´1pBΩ X Btu ą tuqFpνqdt ż `8 ´8 |tu ą tu| dt “ ż `8 ´8 Rptu ą tu, βq |tu ą tu| dt ż `8 ´8 |tu ą tu| dt ě inf EĎΩ RpE, βq “ ℓpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 18 This shows that ΛpΩ, βq “ ℓpΩ, βq and, in particular, we have that ż `8 ´8 tRptu ą tu, βq ´ ℓpΩ, βqu |tu ą tu| dt “ 0 and using the definition of ℓpΩ, βq we observe that the integrand is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In particular, u ı 0 and we have that ΛpΩ, βq “ Rptu ą tu, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2 Γ-convergence of Jp Now we will prove that the functional Jp Γ´converges to the functional J, as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' A functional Jp Γ-converges to J as p Ñ 1` in the weak˚ topology of BV pΩq if, for any u P BV pΩq, the following hold: (i) For any sequence up P BV pΩq which converges to u weak˚ in BV pΩq as p Ñ 1`, then lim inf pÑ1` Jppupq ě Jpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='12) (ii) There exists a sequence up P W 1,ppΩq which converges to u weak˚ in BV pΩq as p Ñ 1`, such that lim sup pÑ1` Jppupq ď Jpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='13) Now, we are in position to prove the convergence theorem for the functional Jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let β ą ´1, then Jp Γ-converges to J as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us suppose up P W 1,ppΩq and ||up||LppΩq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We give the proof by dis- tinguishing the possible values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In any cases, we will have to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='12) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The case β ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us fix a sequence up P W 1,ppΩq weak˚ converging to u in BV pΩq, as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By using the H¨older-type inequality contained for example in [11, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1], we have: ˆż Ω Fp∇upqdx ` ż BΩ |up|FpνqdHN´1 ˙p ď ˆż Ω Fp∇upqpdx ` ż BΩ |up|pFpνqdHN´1 ˙ p|Ω| ` PF pΩqqp´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence we have lim inf pÑ1` Jppupq ě |Du|F pΩq ` ż BΩ |u|FpνqdHN´1 “ |Du|F pRNq “ Jpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' This proves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 19 To give the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='13), we observe that, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3, there exists a se- quence tukukPN Ď C8 0 pΩq such that, for any q P r1, ps, uk converges to u in LqpΩq and ||Fp∇ukq||L1pΩq converges to |Du|F pRNq as k Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, it easily seen that that ||Fp∇ukq||LppΩq converges to ||Fp∇ukq||L1pΩq and hence we have that there exists a subsequence pk Ñ 1`, as k Ñ `8, such that ||FpDukq||pk Lpk pΩq converges to |Du|F pRNq as k Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' This implies that lim supkÑ`8 Jpkpukq ě Jpuq, that concludes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The case 0 ď β ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us consider a sequence up weak˚ converging to u in BV pΩq, as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' A simple application of the Young inequality ap ě pab ´ pp ´ 1qb p p´1 with b “ 1 p, yields to Jppupq “ ż Ω F pp∇upqdx ` β ż BΩ |up|pFpνqdHN´1 ě ż Ω Fp∇upqdx ` β ż BΩ |up|FpνqdHN´1 ´ p ´ 1 p p|Ω| ` PF pΩqq , Therefore, the conclusion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='12) follows by applying the Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' In order to get the second claim, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2 assures the existence of a sequence uk P C8pΩq strongly converging to u in L1pΩq and }FpDukq}L1pΩq converges to |Du|F pΩq, as k Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover ukFpνq converges to uFpνq in L1pBΩ, HN´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' An argument sim- ilar to the previous case leads us to say that }FpDukq}LpkpΩq converges to |Du|F pΩq and ż BΩ |uk|pFpνqdHN´1 converges to ż BΩ |u|FpνqdHN´1, as k Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence the sequence tukukPN satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The case ´1 ă β ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us consider a sequence up P W 1,ppΩq weak˚ converging to u in BV pΩq, as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For any δ ą 0, let us set Ωδ “ tx P Ω : dEpxq ă δu and consider a smooth function ψ equal to zero in ΩzΩδ and to one on BΩ, such that |∇ψ| ď 2 δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The trace inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) applied to the function v “ pu ´ |up|p´1upqψ gives ż BΩ |u ´ |up|p´1up|FpνqdHN´1 ď |Dpu ´ |up|p´1upq|F pΩδq ` ˆ2b δ ` c ˙ ż Ωδ |u ´ |up|p´1up|dx ď |Du|F pΩδq ` ż Ωδ Fp∇p|up|p´1upqqdx ` ˆ2b δ ` c ˙ ż Ωδ |u ´ |up|p´1up|dx, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='14) where we have used that |Dpu´|up|p´1upq|F pBΩδq “ 0 for a set of δ’s of positive measure because u ´ |up|p´1up P BV pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 20 By using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='14), we have Jpuq ´ Jppupq “ |Du|F pΩq ´ ż Ω F pp∇upqdx ` β ż BΩ p|u| ´ |up|p´1upqFpνqdHN´1 ď |Du|FpΩq ´ ż Ω F pp∇upqdx ` |β||Du|F pΩδq ` |β| ż Ωδ Fp∇p|up|p´1upqqdx ` |β| ˆ2b δ ` c ˙ ż Ωδ |u ´ |up|p´1up|dx :“ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='15) Since 1 |β| ą 1, we have A ď 2|Du|F pΩδq ` |Du|F pΩzΩδq ´ ż Ω Fp∇upqpdx ` ż ΩzΩδ Fp∇p|up|p´1upqqdx ` ˆK δ ` c ˙ ż Ωδ |u ´ |up|p´1up|dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='16) The Young inequality gives that ż Ω Fp∇p|up|p´1upqqdx “ ż Ω p|up|p´1Fp∇upqdx ď ż Ω F pp∇upqdx ` pp ´ 1q ż Ω |up|pdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='17) Furthermore by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='15), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='17), we have that Jpuq ´ Jppupq ď 2|Du|F pΩδq ` |Du|F pΩzΩδq ´ ż ΩzΩδ Fp∇p|up|p´1upqqdx ` pp ´ 1q ż Ω |up|pdx ` ˆK δ ` c ˙ ż Ωδ |u ´ |up|p´1up|dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Since up converges to u in LqpΩq, then |up|p´1up converges to u in L1pΩq, as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, by taking p Ñ 1`, we have lim sup pÑ1` rJpuq ´ Jppupqs ď 2|Du|F pΩδq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By sending δ Ñ 0`, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Finally, the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='13) is obtained as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The proof of the Γ-convergence of the functional Jp is useful to prove the convergence of the eigenvalues and eigenfunction, as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For any β ą ´1, it holds lim pÑ1` λ1pΩ, p, βq “ ΛpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, the minimizers up P W 1,ppΩq of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1), with }u}LppΩq “ 1, weak˚ converge to a minimizer u P BV pΩq of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4) as p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='5 assures the existence of a sequence wp converging to a fixed minimizer ¯u of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Let us consider the sequence of minimizers up of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='4), we have: lim sup pÑ1` Jppupq ď lim sup pÑ1` Jppwpq ď Jp¯uq “ ΛpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='18) This means that Jppupq is upper bounded for any p ą 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ă 0, by the trace inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2), we have that ż Ω F pp∇upqdx ď ΛpΩ, βq ´ β|Dpup pq|F pΩq ´ βc, and, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1), that p1 ` βqap ż Ω |∇up|pdx ď ΛpΩ, βq ´ βc ´ βpp ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, by the compactness, we have that up is upper bounded in BV pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ě 0, this directly follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Therefore up weak˚ converges to u in BV pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Finally, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='12) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='18), we have that ΛpΩ, βq “ Jp¯uq ď Jpuq ď lim inf pÑ1` Jppupq ď lim sup pÑ1` Jppupq ď ΛpΩ, βq, and hence the conclusion by observing that λ1pΩ, p, βq “ Jppupq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3 An isoperimetric inequality Here we treat the shape optimization problem for ΛpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' To this aim, we briefly recall the properties of the eigenvalue problem λ1pWR, p, βq and then we prove an explicit computation for ΛpWR, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='1, a minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2) solves the following problem: # ´Qpu “ λ1pWR, p, βq |u|p´2 u in WR pFp∇uqqp´1Fξp∇uq ¨ ν ` βFpνq |u|p´2 u “ 0 on BWR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='19) In particular, the following result holds (refer to in [9] for the positive values of the Robin parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If up P C1,αpWRq X CpWRq is a positive solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='19), then there exists a monotone function ϕp “ ϕpprq, r P r0, Rs, such that ϕp P C8p0, RqXC1pr0, Rsq, and $ ’ & ’ % uppxq “ ϕppF opxqq in WR ϕ 1 pp0q “ 0 |ϕ 1 ppRq|p´2ϕ1 ppRq ` βϕppRqp´1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='20) Moreover, ϕp is decreasing if β ą 0, while is increasing if β ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 22 We first compute ΛpWR, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ą ´1, then ΛpWR, βq “ ˆβhF pWRq “ ˆβ N R , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='21) where ˆβ “ mintβ, 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ě 0, we recall that ΛpWR, βq “ inf EĎWR JpχEq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' By using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3 and the isoperimetric inequality, we have RpE, βq “ JpχEq “ PF pE, WRq ` ˆβ ż BWRXBE FpνEqdHN´1 |E| ě ˆβ PF pEq |E| ě ˆβ PF pWrq |Wr| ě ˆβ PF pWRq |WR| “ ˆβ N R , where Wr is the wulff shape of radius r ă R, with |Wr| “ |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' This proves ΛpWR, βq ě ˆβ N R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' For the reverse inequality we take E “ WR, hence ΛpWR, βq “ ℓpWR, βq ď RpWR, βq “ ˆβ N R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Now, we study the case ´1 ă β ă 0 and we will make use of the Γ-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, let up P W 1,ppWRq a minimizer of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' We know, thanks to the Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='6, that lim pÑ1` λ1pWR, p, βq “ ΛpWR, βq and we take up “ ϕp as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' So, the minimizer converges strongly in L1pWRq to u P BV pWRq, almost everywhere in WR and up ˚á u in BV pΩq for p Ñ 1`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Moreover, up is radially increasing, hence u is nondecreasing and this implies that its superlevel sets tu ą tu are concentric Wulff shapes tr ă F opxq ă Ru and, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3, it holds that ΛpWR, βq “ N R p r RqN´1 ` β 1 ´ p r RqN´1 for some r P r0, Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Therefore, by minimizing the function fptq “ tN´1 ` β 1 ´ tN t P r0, 1r, we observe that the minimum is attained at t “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Hence, the thesis follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' 23 Finally, we prove an isoperimetric inequality for ΛpΩ, βq when a volume constraint holds: if β ě 0, the Wulff shape is a minimizer and, if β ă 0, it is a maximizer for ΛpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ě 0 and WR is the wulff shape of radius R and |WR| “ |Ω|, then ΛpWR, βq ď ΛpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If ´1 ă β ă 0, then ΛpWR, βq ě ΛpΩ, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If β ě 0, by using the same argument of the Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='8, we have that then RpE, βq “ JpχEq ě ˆβ PF pWRq |WR| “ ˆβ N R “ ΛpWR, βq, for any E Ď Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' The conclusion follows by passing to the infimum on the set E Ď Ω and using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' If ´1 ă β ă 0, then using the isoperimetric inequality [14] and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content='21), we have that ΛpΩ, βq ď β PF pΩq |Ω| ď β PF pWRq |WR| “ ΛpWR, βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQflP3v/content/2301.01546v1.pdf'} +page_content=' Acknowledgement This work has been partially supported by the MIUR-PRIN 2017 grant “Qualitative and quantitative aspects of nonlinear PDE’s”, by GNAMPA of INdAM, by the FRA Project (Compagnia di San Paolo and Universit`a degli studi di Napoli Federico II) 000022--ALTRI_CDA_75_2021_FRA_PASSARELLI.' metadata={'source': 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0000000000000000000000000000000000000000..273cdc8793722d67f2e7c9768ec619ebaf2e5858 --- /dev/null +++ b/QdAyT4oBgHgl3EQftvmX/content/tmp_files/2301.00601v1.pdf.txt @@ -0,0 +1,1552 @@ +Naturality of the ∞-Categorical Enriched Yoneda Embedding +Shay Ben Moshe +Abstract +We make Hinich’s ∞-categorical enriched Yoneda embedding natural. To do so, we exhibit +it as the unit of a partial adjunction between the functor taking enriched presheaves and +Heine’s functor taking a tensored category to an enriched category. Furthermore, we study a +finiteness condition of objects in a tensored category called being atomic, and show that the +partial adjunction restricts to a (non-partial) adjunction between taking enriched presheaves +and taking atomic objects. +Contents +1 +Introduction +2 +1.1 +Overview +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +1.2 +Relation to Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.3 +Further Questions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.4 +Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2 +Generalities on Enriched Categories +6 +2.1 +Enriched Categories and Tensored Categories . . . . . . . . . . . . . . . . . . . . . . +7 +2.2 +Enriched Yoneda Lemma and Weighted Colimits . . . . . . . . . . . . . . . . . . . . +7 +3 +Partial Adjunction +10 +4 +2-Categorical Structures +11 +4.1 +Adjunctions in O-Monoidal Categories . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +4.2 +Tensored Categories +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.3 +Tensored Categories to Enriched Categories . . . . . . . . . . . . . . . . . . . . . . . +16 +4.4 +Evaluation and Enriched Hom in Tensored Categories +. . . . . . . . . . . . . . . . . +17 +5 +Atomic Objects, Presheaves and Yoneda +19 +5.1 +Internally Left Adjoints +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +5.2 +Atomic Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +5.3 +Atomics–Presheaves Adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +6 +Heine’s Enriched Yoneda Embedding +24 +References +25 +1 +arXiv:2301.00601v1 [math.CT] 2 Jan 2023 + +1 +Introduction +1.1 +Overview +1.1.1 +Partial Adjunction +The main goal of this paper is to make Hinich’s enriched Yoneda embedding in the ∞-categorical +setting from [Hin20] into a natural transformation. We work in the framework of enriched ∞- +categories as developed in [Hin20, Hin21, GH15, Hei20]. These source differ in notation, and we +introduce our notation along the paper. For brevity, henceforth we use the term (enriched) category +to mean an (enriched) ∞-category, and a 2-category to mean an (∞, 2)-category. Throughout this +paper we fix a presentably monoidal category V ∈ Alg(PrL). +For every V-enriched category C0 ∈ CatV and a presentably V-tensored category D ∈ LModV(PrL), +Hinich defined an (unenriched) category of V-functors FunV(C0, D) ∈ PrL. Using this, he defined a +presentably V-tensored category of enriched presheaves PV(C0) := FunVrev(Cop +0 , V) ∈ LModV(PrL). +One of the main results of [Hin20] is the construction of a V-enriched Yoneda embedding +よV : C0 → PV(C0), +namely, an object of FunV(C0, PV(C0)) ∈ PrL, satisfying the enriched Yoneda lemma. However, +this map is not shown to be natural in C0, which is the central question of the present paper. +In a sequel paper [Hin21], Hinich shows that his enriched Yoneda embedding enjoys the following +universal property: +Theorem 1.1 ([Hin21, 6.4.7]). Let C0 ∈ CatV. Then, for every D ∈ LModV(PrL), composition +with the enriched Yoneda embedding induces an equivalence +(よV)∗ : FunL +V(PV(C0), D) ∼ +−→ FunV(C0, D). +This almost exhibits よV as a unit of an adjunction, but there are two problems: +(1) FunV(C0, D) is not the hom in any category (note that C0 is V-enriched while D is V-tensored). +(2) C0 is small while D is large. +To solve Problem (1), we use the fact that presentably V-tensored categories produce V-enriched +categories, as was first proven in [GH15, Corollary 7.4.9], and made functorial in the work of [Hei20], +a special case of which reads as follows: +Theorem 1.2 ([Hei20, Corollary 7.16]). There is an functor +χ: LModV(PrL) → � +CatV +witnessing the source as a (non-full non-wide) subcategory of the target. +We shall also use [Hei20, Theorem 5.3], showing that for every C0 ∈ CatV and D ∈ LModV(PrL) +there is an equivalence FunV(C0, D)≃ ∼= homV(C0, χ(D)), where the right hand side denotes the +space of morphisms in � +CatV. Combining these result, in Proposition 2.12 we conclude that there is +an isomorphism of spaces +homL +V(PV(C0), D) ∼ +−→ homV(C0, χ(D)), +2 + +where the left hand side denotes the underlying space of FunL +V(PV(C0), D). +To deal with Problem (2), we recall that one can define adjoints partially, namely an adjoint +and a (co)unit map defined only on a full subcategory (or, more generally, relative to a functor), see +Definition 3.1. Furthermore, we recall the folklore result that (partial) adjoints can be constructed +point-wise, as we show in Proposition 3.5. +With these results in mind, we deduce our first main result: +Theorem A (Theorem 3.6). The functor χ: LModV(PrL) → � +CatV has a partial left adjoint +PV : CatV → LModV(PrL) with partial unit agreeing with the enriched Yoneda embedding. +In particular, for every f : C0 → D0 in CatV, we get an induced V-linear left adjoint f! : PV(C0) → +PV(D0), and an isomorphism f!よV ∼= よVf, as explained in Corollary 3.8. +1.1.2 +Atomics–Presheaves Adjunction +In a somewhat different direction, we study atomic objects, a finiteness condition on objects of +presentably V-tensored categories. We show that this condition is closely related to the enriched +Yoneda embedding, and, in particular, gives another solution for Problem (2), leading to a (non- +partial) adjunction. +Let C ∈ LModV(PrL), and let X ∈ C. The functor − ⊗ X : V → C is a V-linear left adjoint +functor. We say that X is atomic if the right adjoint homV(X, −) is itself a left adjoint and the +canonical lax V-linear structure on it is strong (see Definition 5.4). These objects form a small +full V-subcategory Cat ⊂ χ(C). We show that the construction C �→ Cat is functorial in internally +left adjoint functors, that is, V-linear left adjoint functors L: C → D ∈ LModV(PrL) whose right +adjoint is itself a left adjoint and the canonical lax V-linear structure on it is strong. Namely, we +obtain a functor +(−)at : LModV(PrL)iL → CatV. +As a key example, in Proposition 5.17 we show that for any C0 ∈ CatV and X ∈ C0, the +image よV(X) ∈ PV(C0) is atomic. +To see this, recall that by the enriched Yoneda lemma +homV(よV(X), −) is given by evaluation at X, which preserves (co)limits and is V-linear, as we +show in Proposition 4.24. Therefore, we get a factorization of the enriched Yoneda embedding +through the atomics よV : C0 → PV(C0)at. +Furthermore, we show that the functoriality of PV +obtained in Theorem A sends V-functors to internally left adjoints, namely, restricts to a functor +PV : CatV → LModV(PrL)iL. +Our second main result is that the partial adjunction of Theorem A restricts accordingly: +Theorem B (Theorem 5.20). There is an adjunction +PV : CatV ⇄ LModV(PrL)iL :(−)at +with unit agreeing with the enriched Yoneda embedding. +In particular, for every f : C0 → D0 in CatV, the V-linear left adjoint f! : PV(C0) → PV(D0), +is internally left adjoint, and thus admits a right adjoint f ⊛ : PV(D0) → PV(C0) which is itself +a V-linear left adjoint. Using this, in Corollary 5.23 we show that f ⊛(G)(X) ∼= G(f(X)). We +warn the reader that this does not imply that f ⊛ is the functor given by pre-composition, see +Subsection 1.3.1 for further discussion. +3 + +1.1.3 +2-Categorical Structures +In our discussion of atomic objects and their relation to the enriched Yoneda embedding, we use +certain 2-categorical aspects of the theory of enriched categories and tensored categories. Notably, +we have considered the condition for a V-linear left adjoint functor to be internally left adjoint, +which we show has a particularly simple 2-categorical interpretation. +In Definition 4.12 we recall that the category of V-tensored categories and lax V-linear functors +enhances to a 2-category LModlax +V . We show that a 1-morphism in that 2-category is a left adjoint, if +and only if it is strong V-linear and left adjoint (on the underlying category). This result is a special +case of a more general result we deduce from Lurie’s work on relative adjunctions and [HHLN20a], +concerning the 2-category Monlax +O +of O-monoidal categories and lax O-monoidal functors for some +operad O, which may be of independent interest. +Corollary C (Corollary 4.11). The left adjoints in Monlax +O +are the lax O-monoidal functors that +are strong and fiber-wise left adjoint. +The category LModV(PrL) also enhances into a 2-category, which by the above result is a full 2- +subcategory LModV(PrL) ⊂ LModlax +V ( � +Cat)L of the left adjoints in the 2-category LModlax +V ( � +Cat). +Furthermore, using the above result again, we see that internally left adjoints are precisely the left +adjoint morphisms in the 2-category LModV(PrL). +As explained in [Hei20, 7.10], by [GH15, Proposition 5.7.16], the category of V-enriched cate- +gories also enhances to a 2-category CatV. Furthermore, by [Hei20, Theorem 7.11], the functor χ +of Theorem 1.2 enhances to a 2-functor. Using these results we also deduce the following result +which may be of independent interest. Here a 2-functor is called 2-fully faithful if it induces an +isomorphism on hom categories, i.e. if it exhibits the source as a full 2-subcategory of the target. +Corollary D (Corollary 4.19, Corollary 5.3). The functor χ enhances to a 2-fully faithful 2-functor +χ: LModV(PrL) → ( � +CatV)L. +Taking left adjoints again we get the 2-fully faithful 2-functor +χ: LModV(PrL)iL → ( � +CatV)LL. +In particular, for f : C0 → D0 in CatV, since f! : PV(C0) → PV(D0) is internally left adjoint, we +get a corresponding double adjunction χ(f!) ⊣ χ(f ⊛) ⊣ χ(f⊛) in � +CatV. +1.2 +Relation to Previous Work +This paper shares many of the ideas on atomic objects developed in our previous paper with +Tomer Schlank, most notably [BMS21, Theorem D]. To avoid the usage of enriched categories, +in our previous paper we work over a mode M, that is, an idempotent algebra in PrL. The key +feature of modes is that LModM(PrL) is a full subcategory of PrL, namely, a left adjoint functor is +automatically (and uniquely) M-linear. This allowed us, for example, to simplify the definition of +X being atomic to having homM(X, −): C → M commute with colimits (and thus automatically +M-linear). Particularly, in the previous paper we have ignored the M-enriched structure on the +category of atomic objects. In addition, we worked with unenriched presheaves and the unenriched +Yoneda embedding, further tensored with M. +In particular, [BMS21, Theorem D] is a weaker +statement than the main results of the present paper in these respects. +4 + +On the other hand, in the unenriched context, the two different functorialities of presheaves +and naturalities of the Yoneda embedding were shown coincide, contrary to the enriched case of +the present paper (see the discussion in Subsection 1.3.1). In addition, our goal in the previous +paper was different. The unenriched presheaves functor is endowed with a symmetric monoidal +structure, and through the adjunction, this makes the atomics functor lax symmetric monoidal, +and the Yoneda a symmetric monoidal natural transformation. In the present paper we do not deal +with the multiplicative structure (though see the discussion in Subsection 1.3.4). +1.3 +Further Questions +We now list several further questions left open, which we expect have a positive answer, but we do +not know how to approach. +1.3.1 +Naturality for Pre-composition +The first question is closely related to [Ram22] and [HHLN20b, Corollary F], which deal with the +case V = S. +Recall that using the universal property of PV(C0), we deduced Theorem A, assembling PV into +a functor, and the enriched Yoneda embedding into a natural transformation. In addition, we saw +that f! has a right adjoint f ⊛, and we showed that f ⊛(G)(X) ∼= G(f(X)). +The construction of PV(C0) as enriched functors from Cop +0 +to V shows that it admits an- +other functoriality in C0. More specifically, [Hin20, 6.1.4] implies that it assembles into a functor +PV : CatV → (PrR)op, sending f to f ∗ : PV(D0) → PV(C0) given by pre-composition, admitting a +left adjoint f?, commonly spelled “f lower what”. Note that this functoriality does not a priori +give the V-tensored structure. Also note that f ⊛(G)(X) ∼= G(f(X)) ∼= f ∗(G)(X), however it is not +clear that this holds naturally in X, G or C0. +Furthermore, Hinich’s construction of the enriched Yoneda embedding, as described for example +in [Hin21, 7.3.1 and 7.3.2], seems to interact with the f ∗, and thus f?, functoriality, but we don’t +know how to extract naturality from his results. +Question 1.3. Can the f? functoriality be extended to V-modules? +Can the enriched Yoneda +embedding be made natural for the f? functoriality? Do these agree with the f! functoriality and +naturality of the enriched Yoneda embedding? +1.3.2 +Heine’s and Hinich’s Enriched Yoneda Embeddings +In [Hei20], Heine also defines an enriched Yoneda embedding よV +Heine : C0 → χ(PV(C0)) by different +means. As we explain in Section 6, the main results of this paper hold for this version as well, +producing an adjunction +PV +Heine : CatV ⇄ LModV(PrL)iL :(−)at +with unit agreeing with Heine’s enriched Yoneda embedding. The uniqueness of adjoints implies that +there is a natural isomorphism ψ: PV +Heine +∼ +−→ PV, together with an isomorphism ψよV +Heine ∼= よV. On +the other hand, by the construction of the adjunction, for every C0 ∈ CatV there is an isomorphism +PV +Heine(C0) ∼= PV(C0). We thus get an automorphism PV(C0) +ψC0 +−−→ PV +Heine(C0) ∼= PV(C0). It is not +clear to us that this automorphism is the identity. Showing this is equivalent to showing that these +two versions of the enriched Yoneda embedding coincide. +5 + +Question 1.4. Do the enriched Yoneda embeddings constructed by Hinich and Heine coincide? +Namely, is the above automorphism the identity? +1.3.3 +� +Cat-Enrichment +Recall that χ enhances to a 2-functor. Furthermore, the universal property of the enriched Yoneda +embedding of Theorem 1.1 is originally stated for functor categories, though one of the sides is not +constructed as the � +Cat-enriched hom in a category. This leads to the following question: +Question 1.5. Can the (partial) adjunctions be made � +Cat-enriched? +1.3.4 +Multiplicative Structure +For simplicity, assume that V is presentably symmetric monoidal. [Hei20, Corollary 7.16] enhances +χ into a symmetric monoidal functor. Via the adjunction, it may be possible to endow PV with an +oplax symmetric monoidal structure, either by making the subfunctor (−)at lax symmetric monoidal +and using the main result of [HHLN20a], or by proving a version of it for partial adjunctions +and directly applying to PV. Analogously to the case of V = S, one would expect the resulting +oplax symmetric monoidal structure on PV to be strong, and thus make the adjunction symmetric +monoidal. Assuming this, for any operad O, we get that if C0 is O-monoidal, then PV(C0) and the +enriched Yoneda embedding are endowed with an O-monoidal structure. +Separately, in [Hin21, 7.2], Hinich studies the multiplicative structure of enriched presheaves and +the enriched Yoneda embedding. In particular, under the above assumptions, he endows PV(C0) and +the enriched Yoneda embedding with an O-monoidal structure. He further proves an O-monoidal +version of the universal property of Theorem 1.1. +Question 1.6. Can the adjunction be made symmetric monoidal? Does the induced O-monoidal +structure on PV(C0) and the enriched Yoneda embedding agree with those constructed by Hinich? +1.4 +Acknowledgments +We would like to thank Lior Yanovski for numerous useful conversations about atomic objects, +enriched categories and weighted colimits. We also thank Shai Keidar, Shaul Ragimov, Maxime +Ramzi and Noam Zimhoni for comments on earlier drafts of this paper. Particularly, we thank +Maxime for pointing several subtle missing components in earlier drafts, and suggesting corrections +for some of them. +2 +Generalities on Enriched Categories +In this section we review some generalities on enriched categories, their relationship to tensored +categories, and the enriched Yoneda embedding. We shall not delve into the details of the con- +structions, as most of them will not play a role in the present paper, but rather only the formal +properties of the resulting objects. +6 + +2.1 +Enriched Categories and Tensored Categories +Definition 2.1. We denote the (large) category of V-enriched categories, defined in [Hin20, 7.1.2], +by CatV. For C0, D0 ∈ CatV, we denote the space of V-functors between them by homV(C0, D0) ∈ S. +Similarly, we let � +CatV be the (huge) category of large V-enriched categories. +V-enriched categories are closely related to categories tensored over V, as was first proven in +[GH15, Corollary 7.4.9], and made functorial in the work of [Hei20]. +Indeed, Heine constructs +the category ωLModlax +V +(denoted ωLModV there) of weakly V-tensored categories and lax V-linear +functors, and a full subcategory ωLModcl,lax +V +thereof on the closed weakly V-tensored categories. +Theorem 2.2 ([Hei20, Theorem 7.3 and Proposition 7.8]). There is an equivalence +χ: ωLModcl,lax +V +∼ +−→ CatV. +Considering the large version of this equivalence, one can restrict the source to the subcategory +with objects presentable categories with V-action commuting with colimits (which are automatically +closed) and morphisms the (strong) V-linear left adjoint functors. +Definition 2.3. We define the category of presentably V-tensored categories to be PrL +V := LModV(PrL). +For C, D ∈ PrL +V, we denote the space of V-linear left adjoint functors between them by homL +V(C, D) ∈ +�S, which is the space of objects of the category FunL +V(C, D) ∈ � +Cat. +Corollary 2.4 ([Hei20, Corollary 7.16]). There is a functor +χ: PrL +V → � +CatV. +witnessing the source as a (non-full non-wide) subcategory of the target. +For C ∈ PrL +V, this constructs χ(C) ∈ � +CatV, both of which have the same underlying category +and thus space of objects. +2.2 +Enriched Yoneda Lemma and Weighted Colimits +In [Hin20, Hin21] Hinich constructs enriched presheaves and the enriched Yoneda embedding, which +we recall in this subsection. +We begin with Hinich’s model for the category of V-functors from a V-enriched category to a +V-tensored category. In Hinich’s model, a V-enriched category C0 ∈ CatV is an algebra in some +operad constructed from the space of objects C≃ +0 (see [Hin20, 3.1.1]). For a presentably V-tensored +category D ∈ PrL +V, Hinich endows the (unenriched) category of functors Fun(C≃ +0 , D) with a left +module structure over this operad (see [Hin20, 6.1.1]). In particular, one can take left C0-modules +inside. +Definition 2.5 ([Hin20, 6.1.3]). Let C0 ∈ CatV and D ∈ PrL +V, then the category of V-functors from +C0 to D is defined to be +FunV(C0, D) := LModC0(Fun(C≃ +0 , D)) ∈ PrL. +We note that FunV(C0, D) is indeed presentable. To see this, note that Fun(C≃ +0 , D) is an (unen- +riched) presheaf category, and thus is presentable by [Lur09, Proposition 5.5.3.6]. Then, the module +category FunV(C0, D) is also presentable by [Lur17, Corollary 4.2.3.7]. +7 + +Remark 2.6. We warn the reader that in [Hei20], FunV(C0, D) is denoted by FunV(C0, χ(D)). +Definition 2.7 ([Hin20, 6.2.2]). Let C0 ∈ CatV. The V-tensored category of V-enriched presheaves +is defined to be +PV(C0) := FunVrev(Cop +0 , V) ∈ PrL +V. +Remark 2.8. V is a V-V-bimodule in PrL. One of the V-module structures is used to define the +presentable category of Vrev-enriched functors, and the other is used to endow the resulting category +with a V-module structure. +We now record the main results about Hinich’s enriched Yoneda embedding. We remark that +in [Hin20, 6.2.7], Hinich proves his version of the enriched Yoneda lemma, very closely related to +(1) of Theorem 2.9. +However, the form described below is somewhat different, and relies on a +definition of homV and the evaluation at X that were not introduced thus far. We postpone these +definitions to Definition 4.20 and Definition 4.23, and the proof to Proposition 4.25, but include +the statement here for completeness of the exposition. Note that (1) (as well as (2)) will not be +used before Section 5. +Theorem 2.9 ([Hin20, Hin21]). Let C0 ∈ CatV. Then, there is a V-enriched Yoneda embedding +V-functor +よV : C0 → PV(C0), +that is, an object of FunV(C0, PV(C0)). For every D ∈ PrL +V, there is a weighted colimit functor +colim(−) +C0 (−): PV(C0) × FunV(C0, D) → D. +These satisfy the following properties: +(1) For every X ∈ C0, the functor homV(よV(X), −): PV(C0) → V agrees with evaluation at X +as V-linear functors. +(2) colim(−) +C0 (−) commutes with colimits in both arguments separately, and commutes with the +V-action in the first argument. +(3) There is an equivalence +(よV)∗ : FunL +V(PV(C0), D) ⇄ FunV(C0, D) :colim(−) +C0 (−). +Proof. よV is constructed in [Hin20, 6.2.4]. (1) is deferred to Proposition 4.25, though see [Hin20, +6.2.7]. colim(−) +C0 (−) is constructed in [Hin21, 6.2.2], where (2) is explained. (3) is [Hin21, 6.3.3 and +6.4.6]. +In [Hei20, Theorem 5.3, see also the discussion in the beginning of Section 5], Heine shows that +FunV(C0, D) is closely related to the hom in V-enriched categories via χ (see also [Hin20, 6.3.6]). +We recall a special case of Heine’s results as follows: +Proposition 2.10 ([Hei20, Theorem 5.3]). Let C0 ∈ CatV and D ∈ PrL +V, then there is an equiva- +lence +FunV(C0, D)≃ ∼= homV(C0, χ(D)) +8 + +natural in C0 and D. Namely, there is a natural isomorphism between the functors +(CatV)op × PrL +V +i×χ +−−→ (� +CatV)op × � +CatV +homV +−−−−→ �S +and +(CatV)op × PrL +V +FunV(−,−) +−−−−−−−→ Cat +(−)≃ +−−−→ S ⊂ �S +We use this to transform the enriched Yoneda embedding to a morphism in � +CatV, and deduce a +universal property similar to (3) of Theorem 2.9 where both the source and the target are the hom +in some category. +Definition 2.11. We denote by the same notation the enriched Yoneda embedding V-functor +よV : C0 → χ(PV(C0)), corresponding to the enriched Yoneda embedding under the equivalence +FunV(C0, PV(C0))≃ ∼= homV(C0, χ(PV(C0))). +Proposition 2.12. Let C0 ∈ CatV and D ∈ PrL +V, then the composition +homL +V(PV(C0), D) +χ−→ homV(χ(PV(C0)), χ(D)) +(よV)∗ +−−−−→ homV(C0, χ(D)) +is an equivalence. +Proof. Naturality in the PrL +V coordinate of Proposition 2.10 shows that for C0 ∈ CatV and D, E ∈ +PrL +V we get a commutative square: +homL +V(E, D) +homV(χ(E), χ(D)) +hom(FunV(C0, E)≃, FunV(C0, D)≃) +hom(homV(C0, χ(E)), homV(C0, χ(D))) +χ +∼ +Using the exponential adjunction we get a commutative square: +homL +V(E, D) × FunV(C0, E)≃ +homV(χ(E), χ(D)) × homV(C0, χ(E)) +FunV(C0, D)≃ +homV(C0, χ(D)) +χ +◦ +◦ +∼ +Taking E = PV(C0), and picking the point よV ∈ FunV(C0, PV(C0))≃, we get a commutative square: +homL +V(PV(C0), D) +homV(χ(PV(C0)), χ(D)) +FunV(C0, D)≃ +homV(C0, χ(D)) +χ +(よV)∗ +(よV)∗ +∼ +The left morphism is an equivalence by applying (−)≃ to (3) of Theorem 2.9. Thus, the left-bottom +composition is an equivalence, and by the commutativity of the diagram, so is the upper-right +composition, concluding the proof. +9 + +3 +Partial Adjunction +In this section we prove Theorem A, namely the naturality of the enriched Yoneda embedding. +To do so, we first define (unenriched) partial adjunctions. Then, we show the folklore result that +(partial) adjoints can be constructed point-wise, from which the naturality of the enriched Yoneda +embedding follows immediately. +Definition 3.1. Let L: C0 → D0 be a functor, and i: E0 → D0 another functor. The data of +a partial right adjoint (of L relative to i) is a functor R: E0 → C0 and a natural transformation +ε: LR ⇒ i called the partial counit, such that for every X ∈ C0 and Y ∈ E0 the composition +homC0(X, RY ) +L−→ homD0(LX, LRY ) +εY ◦− +−−−→ homD0(LX, iY ) +is an isomorphism. A partial left adjoint with a partial unit is defined dually, by taking (−)op. +Remark 3.2. What we call a partial adjunction is typically called a relative adjunction. However, in +Section 4 we use the distinct concept of relative adjunctions developed by Lurie. To avoid confusion, +we call what is typically called a relative adjunction a partial adjunction. +Remark 3.3. In this paper we only use the notion of a partial adjunction when i: E0 → D0 is an +inclusion of a full subcategory. In this case, one can think of R as an adjoint defined only partially +on D0, justifying the name. +Remark 3.4. Unlike in a standard adjunction, a partial adjunction is not a symmetric concept. +Particularly, note that there is no partial unit (in fact, L and R can not be composed in the other +direction). +Proposition 3.5. Let L: C0 → D0 be a functor, and i: E0 → D0 another functor. Assume that +for every Y ∈ E0 we are given an object RY ∈ C0 and a morphism εY : LRY → iY , such that for +every X ∈ C0 the composition +homC0(X, RY ) +L−→ homD0(LX, LRY ) +εY ◦− +−−−→ homD0(LX, iY ) +is an isomorphism. Then, R and ε assemble into the data of a partial right adjoint. +Proof. Consider the functor ˜R: D0 → P(C0) given by the composition +D0 +よD0 +−−−→ P(D0) +L∗ +−−→ P(C0), +which sends Z ∈ D0 to homD0(L(−), Z): Cop +0 → S. +Let Y ∈ E0. Consider the natural transformation +homC0(−, RY ) +L−→ homD0(L(−), LRY ) +εY ◦− +−−−→ homD0(L(−), iY ) = ˜R(iY ). +(1) +By assumption, it is an isomorphism at every X ∈ C0, and thus it is a natural isomorphism. +Namely, the presheaf ˜R(iY ) ∈ P(C0) is representable by RY ∈ C0. In other words, the composition +˜Ri: E0 → P(C0) lands in the essential image of the (unenriched) Yoneda embedding よC0 : C0 → +P(C0). As the Yoneda embedding is fully faithful, we get an induced functor R: E0 → C0, together +with a natural isomorphism よC0R ∼= ˜Ri = L∗よD0i of functors E0 → P(C0). +10 + +By construction, R agrees with RY at every Y ∈ E0. Furthermore, the natural isomorphism at +Y is given by (1), constructed via εY . We now extract the partial counit ε: LR ⇒ i. Consider the +following composition: +よD0LR +(1) +∼= (LR)!よE0 +(2) +⇒ (LR)!R∗よC0R +(3) +∼= (LR)!R∗L∗よD0i +∼= (LR)!(LR)∗よD0i +(4) +⇒ よD0i. +(1) is naturality of the (unenriched) Yoneda embedding. (2) is the fact that R is a functor, giving +homE0(−, −) ⇒ homC0(R(−), R(−)), which by exponential adjunction is the same as よE0 ⇒ +R∗よC0R. (3) is the isomorphism よC0R ∼= L∗よD0i. (4) is the counit of the adjunction (LR)! ⊣ +(LR)∗. Thus, we have constructed a natural transformation よD0LR ⇒ よD0i. Since よD0 is fully +faithful, this induces a natural transformation ε: LR ⇒ i. Recall that the isomorphism used at +step (3) is given at Y ∈ E0 by (1), which shows that ε indeed agrees with εY . +Theorem 3.6. The functor χ: PrL +V → � +CatV has a partial left adjoint PV : CatV → PrL +V with partial +unit よV : id|CatV → χ PV, agreeing with enriched presheaves and enriched Yoneda embedding. +Proof. This follows immediately from Proposition 2.12 and (the dual statement to) Proposition 3.5. +Definition 3.7. For f : C0 → D0 a morphism in CatV, we denote the induced morphism in PrL +V +by f! : PV(C0) → PV(D0). +An instance of the naturality of the enriched Yoneda embedding is the following: +Corollary 3.8. Let f : C0 → D0. Then, よV +D0f ∼= χ(f!)よV +C0 in homV(C0, χ(PV(D0))). Similarly, +よV +D0f ∼= f!よV +C0 in FunV(C0, PV(D0)) of Definition 2.5. +Proof. The first part is an immediate application of the adjunction of Theorem 3.6. For the sec- +ond part, recall from Proposition 2.10 that there is a natural isomorphism homV(−, χ(−)) ∼= +FunV(−, −)≃. +Recall that in Definition 2.11 we have defined よV +C0 ∈ homV(C0, χ(PV(C0))) to +be the map corresponding to よV +C0 ∈ FunV(C0, PV(C0)) via this isomorphism, and similarly for +D0. By applying the naturality of the isomorphism in the target to the morphism f!, we get that +χ(f!)よV +C0 ∈ homV(C0, χ(PV(D0))) corresponds to f!よV +C0 ∈ FunV(C0, PV(D0)). Similarly, by ap- +plying the naturality of the isomorphism in the source to the morphism f, we get that よV +D0f ∈ +homV(C0, χ(PV(D0))) corresponds to よV +D0f ∈ FunV(C0, PV(D0)). Thus, under the natural isomor- +phism, the isomorphism よV +D0f ∼= χ(f!)よV +C0 corresponds to an isomorphism よV +D0f ∼= f!よV +C0. +4 +2-Categorical Structures +In this section we study certain 2-categorical aspects of enriched categories and tensored categories. +The first main result of this section is Corollary 4.11, showing that a lax O-monoidal functor between +11 + +O-monoidal categories is a left adjoint in the 2-category Monlax +O , if and only if it is strong O-monoidal +and fiber-wise left adjoint. This is proven by applying Lurie’s [Lur17, §7.3.2 Relative Adjunctions] to +the 2-categories constructed in [HHLN20a]. From this, we deduce Corollary 4.13, the corresponding +result for lax V-linear functors between V-tensored categories. Then, after recalling that in [Hei20, +Theorem 7.11] Heine shows that χ is a 2-functor, we deduce Corollary 4.19, our second main +result, which says that χ enhances to a 2-fully faithful 2-functor χ: PrL +V → ( � +CatV)L. Finally, in +Proposition 4.25 we finish the proof of (1) of Theorem 2.9. +4.1 +Adjunctions in O-Monoidal Categories +Definition 4.1 ([HHLN20a, 3.1.7]). Let B be a a category. Denote by Cocartlax +B +the 2-category +of cocartesian fibrations over B and functors over B. +We shall not recall the precise definition of this 2-category (for which we refer the reader to +[HHLN20a, 3.1.7]), rather, let us recall its objects, 1-morphisms and 2-morphisms. +Proposition 4.2. Cocartlax +B +has +• objects: cocartesian fibrations q: C → B, +• 1-morphisms: functors over B, namely, a 1-morphism from q: C → B to p: D → B is a +functor F : C → D and a natural isomorphism pF ∼= q, +• 2-morphisms: natural transformations over B, namely, a 2-morphism from F to G is a natural +transformation α: F ⇒ G and an identification of the natural transformation q ∼= pF +pα +==⇒ +pG ∼= q with idq, i.e. exhibiting the following square as commutative: +pF +pG +q +q +pα +≀ +≀ +Proof. By construction, the underlying category of the 2-category Cocartlax +B is the full subcategory +of Cat/B on the cocartesian fibrations, which explains the objects and the 1-morphisms. For the 2- +morphisms, we recall that as explained in [HHLN20a, 3.1.9], a 2-morphism is commutative diagram +of the form: +C × [1] +D +B +q +p +¯α +Under the exponential adjunction, ¯α: C × [1] → D corresponds to the natural transformation +α: F ⇒ G, and the isomorphism p¯α ∼= q corresponds to the identification pα ∼= idq. +Lemma 4.3. A 2-morphism in Cocartlax +B +given by α: F ⇒ G together with an identification +pα ∼= idq is invertible if and only if α is invertible. +12 + +Proof. Clearly, if the 2-morphism is invertible then in particular α is. +For the other direction, +assume that α has an inverse α−1 : G ⇒ F. +We shall enhance it to an inverse 2-morphism in +Cocartlax +B . Consider the following diagram: +pF +pG +pF +q +q +q +≀ +≀ +≀ +pα +pα−1 +The identification pα ∼= idq is precisely an (invertible) 3-morphism in Cat making the left square +commute. The outer square commutes as pα−1pα ∼= idpF ∼= idq. Thus, by composing the outer +square with the inverse of the left square, we get commutativity data for the right square. This +makes α−1 into a 2-morphism in Cocartlax +B , which by construction is the required inverse. +Our next goal is to understand left adjoints in the 2-category Cocartlax +B , achieved in Proposi- +tion 4.8. To that end, we shall employ Lurie’s [Lur17, §7.3.2 Relative Adjunctions]. The definitions +and results in that section are phrased for cartesian fibrations and right adjoints, which by taking +(−)op correspond to our case of cocartesian fibrations and left adjoints, as we present here. +First, we recall that in a 2-category, we say that a 1-morphism L: X → Y is a left adjoint +if there exists a 1-morphism R: Y → X and two 2-morphisms u: idX ⇒ RL and c: LR ⇒ idY +satisfying the zigzag identities, namely +(R +uR +==⇒ RLR +Rc +==⇒ R) ∼= idR, +(L +Lu +==⇒ LRL +cL +==⇒ L) ∼= idL. +We also recall that to check that L is a left adjoint, we can check a weaker condition: +Lemma 4.4 ([RV22, 2.1.11]). In any 2-category, a 1-morphism L: X → Y is a left adjoint if and +only if there exist a 1-morphism R: Y → X and 2-morphisms u: idX ⇒ RL and c: LR ⇒ idY such +that the zigzag morphisms +R +uR +==⇒ RLR +Rc +==⇒ R, +L +Lu +==⇒ LRL +cL +==⇒ L +are invertible (in which case there exists a possibly different 2-morphism ˜c: LR ⇒ idY for which +the zigzag identities hold). +Proof. The proof of [RV22, 2.1.11] works in an arbitrary 2-category, although stated in the context +of functors between (∞-)categories. +Definition 4.5 ([Lur17, Definition 7.3.2.2]). Let +C +D +B +q +p +L +be a commutative diagram of categories. We say that L admits a relative right adjoint if L has +a (non-relative) right adjoint R: D → C, and the counit map c: LR ⇒ idD satisfies the condition +that pc is equivalent to idp (and in particular qR ∼= p). +13 + +Remark 4.6. The definition of [Lur17, Definition 7.3.2.2] is via the two equivalent conditions of +[Lur17, Proposition 7.3.2.1]. First, we note that these are phrased for admitting a relative left +adjoint, but the theory is symmetric via taking (−)op. Second, Lurie assumes that p and q are +categorical fibrations, but every functor is equivalent to a categorical fibration, so this assumption +can be dropped. Finally, and most importantly, we note that our definition clearly implies condition +(1) and is implied by condition (2) in [Lur17, Proposition 7.3.2.1], and thus is also equivalent to +them. +Proposition 4.7. A 1-morphism L: C → D over B in Cocartlax +B +is a left adjoint if and only if L +admits a relative right adjoint. +Proof. If L is a left adjoint in Cocartlax +B then it clearly admits a relative right adjoint. For the other +direction, assume that L admits a relative right adjoint, and we shall show that it is a left adjoint +in Cocartlax +B . Let R: D → C, u: idC ⇒ RL and c: LR ⇒ idD be the (non-relative) adjunction +data, satisfying that pc ∼= idp. The identification pc ∼= idp makes c into a 2-morphism in Cocartlax +B . +This also identifies qR ∼= p, making R into a 1-morphism in Cocartlax +B . In the rest of the proof we +make u into a 2-morphism as well, and show that L is a left adjoint using Lemma 4.4. +We begin by making u into a 2-morphism. Consider the following diagram: +pL +pLRL +pL +q +q +q +pLu +pcL +≀ +≀ +≀ +The upper composition is p(−) applied the zigzag morphism, which is equivalent to idpL, thus +making the outer square commute. The assumption that pc is equivalent to idp makes the right +square commute. Thus, by composing the outer square with the inverse of the right square, we get +that the left square commutes as well. In other words, we obtained an identification of pLu with +idq. The identification pL ∼= q thus shows that we got an identification of qu with idq, making u +into a 2-morphism. +By Lemma 4.4, to see that L is a left adjoint in Cocartlax +B +it suffices to check that the zigzag +morphisms corresponding to these 2-morphisms are invertible. By Lemma 4.3, a 2-morphism in +Cocartlax +B is invertible if and only if the underlying natural transformations are invertible. Indeed, +the underlying natural transformations are simply +R +uR +==⇒ RLR +Rc +==⇒ R, +L +Lu +==⇒ LRL +cL +==⇒ L +which are invertible (in fact, equivalent to the identities) by the assumption that u and c are a unit +and a counit for a (non-relative) adjunction L ⊣ R. +The following is a recast of [Lur17, Proposition 7.3.2.6] in an appropriate 2-category. +Proposition 4.8. The left adjoints in Cocartlax +B +are those functors over B that preserve cocarte- +sian morphisms and are fiber-wise left adjoint +(Cocartlax +B )L = CocartL-fw +B +. +Proof. By Proposition 4.7, a 1-morphism L between two cocartesian fibrations is a left adjoint if +and only if it admits a relative right adjoint. Thus by (the (−)op of) [Lur17, Proposition 7.3.2.6], +14 + +L is a left adjoint if and only if it preserves locally cocartesian morphisms and is fiber-wise left +adjoint. By [Lur09, Proposition 2.4.2.8], locally cocartesian morphisms and cocartesian morphisms +coincide, concluding the proof. +Remark 4.9. There is a possible alternative route to Proposition 4.8. +Recall that [HHLN20a, +Theorem E] shows that Cocartlax +B +∼= Funlax(B, Cat), where the right hand side denotes the 2- +category of functors and lax natural transformations. Additionally, [Hau20, Theorem 4.6] shows that +for any 2-categories X, Y, the left adjoints in Funlax(X, Y) are those lax natural transformations +which are strong and point-wise left adjoint. Combined, this would imply Proposition 4.8. However, +the definition of Funlax in [HHLN20a] and [Hau20] rely on two different models of the Gray tensor +product, which to the best of our knowledge were not shown to be equivalent so far, and thus this +does not constitute a complete proof. +In the rest of the subsection we follow the notations of [HHLN20a, 3.4] for operads. Namely, an +operad O is a functor O → Fin∗ (satisfying certain properties), whereas in [Lur17], it is typically +denoted by O⊗. +Definition 4.10 ([HHLN20a, 3.4.1]). Let O be an operad. Let Monlax +O +⊂ Cocartlax +O +be the 1- +full 2-subcategory on the O-monoidal categories and lax O-monoidal functors (i.e. functors over O +preserving cocartesian morphisms lying over inert morphisms). +The following is a recast of [Lur17, Corollary 7.3.2.7] in an appropriate 2-category. +Corollary 4.11. The left adjoints in Monlax +O +are those lax O-monoidal functors that are strong +and fiber-wise left adjoint +(Monlax +O )L = MonL-fw +O +. +Proof. For the first direction, note that a left adjoint in Monlax +O +is in particular a left adjoint in +Cocartlax +O , thus it is strong O-monoidal (preserves all cocartesian morphisms) and fiber-wise left +adjoint. +For the second direction, let L: C → D over O in MonL-fw +O +. As MonL-fw +O +⊂ CocartL-fw +O += +(Cocartlax +O )L, there is a right adjoint R: D → C over O in Cocartlax +O +with some unit and counit. +Since Monlax +O +is a 1-full 2-subcategory of Cocartlax +O , all we need to show is that the right adjoint +is in Monlax +O , whence the unit and counit are also automatically there. Namely, we need to know +that R: D → C is lax O-monoidal, which follows from the construction, as explained in [Lur17, +Corollary 7.3.2.7]. +4.2 +Tensored Categories +As explained in [Hei20, 7.10], the category ωLModlax +V +is enhanced to a 2-category, and thus so is its +full subcategory LModlax +V . We repeat the construction in more detail, building on the definitions +above. See also [Bar22, Definition 2.14] for a closely related discussion. +We recall the operad Assoc classifying an associative algebra, and the operad LM classifying +an associative algebra and a left module over it. Consider the monoidal category V as an object in +Mon(Cat) ∼= MonAssoc ⊂ Monlax +Assoc. This leads us to the following: +Definition 4.12. We define the 2-category of V-tensored categories and lax V-linear functors to +be LModlax +V +:= {V} ×Monlax +Assoc Monlax +LM. +15 + +As a particular case of Corollary 4.11, we deduce the following, which is an “if and only if” +version of [Lur17, Example 7.3.2.8 and Remark 7.3.2.9] phrased in an appropriate 2-category. +Corollary 4.13. The left adjoints in LModlax +V +are those lax V-linear functors that are strong +V-linear and left adjoint (on the underlying category) +(LModlax +V )L = LModL-fw +V +. +Proof. The result follows from Corollary 4.11 by taking O = LM, and pulling back over V. In more +detail, consider the following diagram: +(LModlax +V )L +(Monlax +LM)L +{V}L +(Monlax +Assoc)L +{V} +Monlax +Assoc +≀ +The bottom square is a pullback square because the lower right morphism is an inclusion of a +2-subcategory, and the lower left morphism is the map from a point to itself. The upper square +is a pullback square, because (−)L commutes with limits, as it is given by hom from the walking +adjunction. Therefore, the outer square is also a pullback square. We thus finish by Corollary 4.11. +Remark 4.14. Definition 4.12 and Corollary 4.13 work for an arbitrary monoidal category V, without +needing to assume that it is presentably monoidal. +We now enhance PrL +V of Definition 2.3 to a 2-category. +Definition 4.15. We define the 2-category of presentably V-tensored categories to be the 1-full +2-subcategory PrL +V := LModV(PrL) ⊂ LModlax +V ( � +Cat) on the objects and morphisms of PrL +V. +Proposition 4.16. There is a full 2-subcategory inclusion +PrL +V ⊂ (LModcl,lax +V +( � +Cat))L. +Proof. First, we claim that there is a full 2-subcategory inclusion +PrL +V = LModV(PrL) ⊂ LModcl,L-fw +V +( � +Cat). +It is indeed an inclusion, since a presentably V-tensored category is automatically closed, and mor- +phism in both categories are the V-linear functors that are left adjoint on the underlying category. +Second, by Corollary 4.13, the target is (LModcl,lax +V +( � +Cat))L, which finishes the argument. +4.3 +Tensored Categories to Enriched Categories +We recall that CatV can be enhanced into a 2-category, as explained for example in [Hei20, 7.10]. +Indeed, by [GH15, Proposition 5.7.16], the functor Alg(PrL) → PrL given by V �→ CatV is lax +symmetric monoidal. Since V is a S-module in PrL, we get that CatV is a Cat-module in PrL as +well, which indeed models a 2-category (for example via χ). +16 + +Definition 4.17. We denote the 2-category of V-enriched categories by CatV. +Heine shows that χ of Theorem 2.2 enhances to a 2-equivalence. +Theorem 4.18 ([Hei20, Theorem 7.11]). The equivalence of Theorem 2.2 enhances to a 2-equivalence +χ: ωLModcl,lax +V +∼ +−→ CatV. +Using this we deduce the following: +Corollary 4.19. The functor from Corollary 2.4 enhances to a 2-fully faithful 2-functor +χ: PrL +V → ( � +CatV)L. +Proof. Consider the large version of Theorem 4.18, namely ωLModcl,lax +V +( � +Cat) +∼ +−→ � +CatV, and re- +strict it to the full 2-subcategory LModcl,lax +V +( � +Cat). Taking left adjoints and using Proposition 4.16, +we get the 2-fully faithful 2-functor PrL +V ⊂ (LModcl,lax +V +( � +Cat))L → ( � +CatV)L. +4.4 +Evaluation and Enriched Hom in Tensored Categories +In this subsection, we introduce homV in the presentably V-tensored context and show that it satis- +fies expected properties for adjunction in Proposition 4.21 (see also Remark 4.22 for the connection +to adjunction in V-enriched categories). With this definition in mind, we finish the proof of (1) of +Theorem 2.9 in Proposition 4.25. +Let C ∈ PrL +V. We have an equivalence of categories FunL +V(V, C) ∼ +−→ C given by evaluation at 1V. +Recall from Proposition 4.16 that there is a full 2-subcategory inclusion PrL +V ⊂ (LModcl,lax +V +( � +Cat))L. +Thus, passing to right adjoint, gives us a map FunL +V(V, C) → Funlax-V(C, V)op. Composing the two +and taking (−)op, we get Cop → Funlax-V(C, V). +Definition 4.20. We let homV(−, −): Cop × C → V to be the functor corresponding to the above +under the exponential adjunction. By construction, it is lax V-linear in the second argument. +Namely, this construction sends X to the lax V-linear functor homV(X, −): C → V, right adjoint +to the V-linear left adjoint − ⊗ X : V → C. +Proposition 4.21. Let L: C → D ∈ PrL +V be a V-linear left adjoint functor, and let R: D → C +denote its right adjoint. Then there is a natural isomorphism +homV(L(−), −) ∼= homV(−, R(−)) +of functors Cop × D → V lax V-linear in the second coordinate. +Proof. Indeed, consider the following diagram: +C +FunL +V(V, C) +Funlax-V(C, V) +D +FunL +V(V, D) +Funlax-V(D, V) +L +L◦− +−◦R +∼ +∼ +17 + +The left square commutes by passing to the right adjoints and noting that evaluation at 1V com- +mutes with post-composition. To see that the right square commutes, recall that the horizontal +morphisms were constructed by the full 2-subcategory inclusion PrL +V ⊂ (LModcl,lax +V +( � +Cat))L and +passing to the right adjoints, and the vertical morphisms are also adjoints in the same category, so +the commutativity is the fact that the right adjoint of a composition is the composition of the right +adjoints in reverse order. +Remark 4.22. We note that Proposition 4.21 is in the context of presentably V-tensored categories, +and not that of V-enriched categories. One of the main features of χ is that the V-enrichment +of χ(C) is given by homV(X, Y ), as is shown in [GH15, Corollary 7.4.9]. By Corollary 4.19, the +adjunction L ⊣ R produces an adjunction χ(L) ⊣ χ(R) in � +CatV. +One might expect that this +adjunction provides a natural isomorphism similar to the one of Proposition 4.21 for the V-enriched +hom’s of χ(C) and χ(D). However, we do not provide such a natural isomorphism, nor show its +compatibility with the one constructed above. +Let C0 ∈ CatV, X ∈ C0 and D ∈ PrL +V. +Recall from Definition 2.5 that FunV(C0, D) := +LModC0(Fun(C≃ +0 , D)). Consider i: pt → C≃ +0 choosing X, and the following diagram: +D +Fun(pt, D) +Fun(C≃ +0 , D) +LModC0(Fun(C≃ +0 , D)) +i! +i∗ +free +forget +Here the dashed arrows are the left adjoints of the solid arrows. Also recall from Definition 2.7 that +if we replace C0 by Cop +0 +and let D = V, the V-functor category is PV(C0). Moreover, in this case +the three categories in the diagram are presentably V-tensored and the two left adjoint functors are +V-linear, as in [Hin20, 6.2.3]. Via Proposition 4.16, the right adjoints are canonically lax V-linear. +Definition 4.23. Let C0 ∈ CatV, X ∈ C0 and D ∈ PrL +V be as above. We define the evaluation at +X functor evalX : FunV(C0, D) → D by evalX := i∗ ◦ forget. For the case evalX : PV(C0) → V, it is +canonically lax V-linear. +Proposition 4.24. Let C0 ∈ CatV and D ∈ PrL +V, and consider FunV(C0, D) ∈ PrL. Then evalX +commutes with (co)limits (i.e. (co)limits are computed level-wise), and are jointly conservative over +all X. +For PV(C0) ∈ PrL +V, the lax V-linear structure on evalX is strong (i.e., the V-action is +level-wise). +Proof. Recall that evalX = i∗ ◦ forget. The functor forget commutes with all (co)limits by [Lur17, +Corollary 4.2.3.3 and Corollary 4.2.3.5] as the forgetful from modules, while i∗ commutes with +(co)limits as they are computed level-wise in (unenriched) functor categories. +The forgetful from modules is always conservative by [Lur17, Corollary 4.2.3.2], and the evalu- +ation at X functors Fun(C≃ +0 , D) → D are jointly conservative. +For the case of PV(C0), the V-action is by construction given level-wise, as explained in [Hin20, +6.2.3]. +Finally, we are in position to prove our variant of Hinich’s enriched Yoneda lemma, appearing +as (1) of Theorem 2.9. +Proposition 4.25. Let C0 ∈ CatV, then homV(よV(X), −) ∼= evalX as lax V-linear functors, and +in particular homV(よV(X), −) is also strong V-linear. +18 + +Proof. We need to show that homV(よV(X), −) ∼= evalX as lax V-linear functors. By construction, +this is equivalent to showing that −⊗よV(X) ∼= free ◦i! as V-linear functors. Consider the following +diagram: +V +Fun(Cop,≃ +0 +, V) +LModCop +0 (Fun(Cop,≃ +0 +, V)) +S +Fun(Cop,≃ +0 +, S) +−⊗1V +hom(1V,−) +−⊗1V +hom(1V,−) +i! +i∗ +i! +i∗ +free +forget +Here the dashed arrows are the left adjoints of the solid arrows. Clearly, the solid square commutes, +and thus, the dashed square obtained by passing to left adjoints also commutes. +By [Hin20, 6.2.6], we have that よV(X) ∼= free((−⊗1V)◦よ(X)), where here よ(X) ∈ Fun(Cop,≃ +0 +, S) +is the image of X under the (unenriched) Yoneda embedding. Applying the naturality of the (un- +enriched) Yoneda embedding to i: pt → S, and noting that よ: pt → S sends よ(pt) = pt, we get +that よ(X) = よ(i(pt)) ∼= i!(よ(pt)) ∼= i!(pt). Using the commutativity of the dashed square above, +we conclude that +よV(X) ∼= free((− ⊗ 1V) ◦ よ(X)) ∼= free((− ⊗ 1V) ◦ i!(pt)) ∼= free(i!1V). +Since free and i! are V-linear functors, we finally get that +− ⊗ よV(X) ∼= − ⊗ free(i!1V) ∼= free ◦i!(− ⊗ 1V) ∼= free ◦i! +as V-linear functors, concluding the proof. +5 +Atomic Objects, Presheaves and Yoneda +In this section, building on the 2-categorical results of the previous section, we study internally left +adjoint functors and atomic objects, and connect them to the enriched Yoneda embedding. +We begin in Definition 5.1 by defining internally left adjoints between presentably V-tensored +categories. Namely, V-linear left adjoint functors, whose right adjoints are also left adjoint and their +canonical lax V-linear structure is strong. Following this, in Definition 5.4 we define a finiteness +condition called being atomic. We say that X in C ∈ PrL +V is atomic if − ⊗ X : V → C is internally +left adjoint, that is, if homV(X, −) commutes with colimits and is V-linear. For instance, in the case +V = Sp, this coincides with condition of being compact. From the definition, internally left adjoints +send atomic objects to atomic objects. The key technical result of this section is Proposition 5.16, +giving a converse result under the assumption that the source category is generated from the atomics +under weighted colimits. +Next, in Proposition 5.17 we show that よV(X) is atomic in PV(C0), and in Proposition 5.18 we +show that together they generate PV(C0) under weighted colimits. This allows us to use Proposi- +tion 5.16 to prove Theorem 5.20, the main result of this section. This result says that the partial +adjunction of Theorem 3.6 restricts to a (non-partial) adjunction between the enriched presheaves +functor and taking the atomics, with the unit being (the factorization through the atomics of) the +enriched Yoneda embedding. +19 + +5.1 +Internally Left Adjoints +Definition 5.1. A V-linear left adjoint functor is called internally left adjoint if it is left adjoint +in the 2-category PrL +V. We denote the wide 2-subcategory on the internally left adjoint functors by +PriL +V := (PrL +V)L. For C, D ∈ PriL +V , we denote the category of V-linear internally left adjoint functors +between them by FuniL +V (C, D) ∈ � +Cat and the corresponding homiL +V (C, D) ∈ �S. +Note that for any V-linear left adjoint functor L: C → D, Proposition 4.16 shows that it admits +a lax V-linear right adjoint R: D → C. +Proposition 5.2. A V-linear left adjoint functor L: C → D is internally left adjoint if and only if +the lax V-linear right adjoint R: D → C is strong and itself a left adjoint. +Proof. As above, R is a morphism in LModcl,lax +V +( � +Cat) between objects of the 2-subcategory PrL +V, +and the condition is that it is in fact a morphism in that 2-subcategory. By Proposition 4.16, PrL +V +is a full 2-subcategory of (LModcl,lax +V +( � +Cat))L, thus the condition is that R is a morphism in that +2-subcategory. The result then follows from Corollary 4.13. +Taking left adjoints in Corollary 4.19, we immediately get: +Corollary 5.3. χ restricts to a 2-fully faithful 2-functor +χ: PriL +V → ( � +CatV)LL. +5.2 +Atomic Objects +Definition 5.4. Let C ∈ PrL +V. We say that X ∈ C is atomic if the functor − ⊗ X : V → C is +internally left adjoint. We denote the full V-subcategory on the atomic objects by Cat ⊂ χ(C). +Example 5.5. The unit 1V ∈ V is always atomic, because − ⊗ 1V is the identity functor. +Example 5.6 ([BMS21, Proposition 2.6]). In the case V = Sp, atomic objects coincide with compact +objects. +Recall that for C ∈ PrL +V, there is an equivalence FunL +V(V, C) +∼ +−→ C given by evaluation at 1V, +with inverse sending X to − ⊗ X : V → C. Thus, immediately from the definition, we deduce the +following: +Proposition 5.7. Evaluation at 1V induces an equivalence FuniL +V (V, C) ∼ +−→ Cat. +Since the right adjoint of − ⊗ X : V → C is homV(X, −): C → V, the following follows immedi- +ately from Proposition 5.2. +Proposition 5.8. An object X ∈ C is atomic if and only if homV(X, −): C → V preserves colimits +and the lax V-linear structure is strong. +Proposition 5.9. The atomics are a small V-enriched category. +Proof. The argument is identical to [BMS21, Proposition 2.7]. We repeat the details for the conve- +nience of the reader. +Let κ be a regular cardinal such that the unit 1V is κ-compact, namely hom(1V, −): V → S +commutes with κ-filtered colimits. We show that all atomic objects are κ-compact. Let C ∈ PrL +V +20 + +and let X ∈ C be atomic. +By Proposition 5.8, homV(X, −): C → V commutes with κ-filtered +colimits. Since hom(X, −) ∼= hom(1V ⊗ X, −) ∼= hom(1V, homV(X, −)), we get that hom(X, −) +also commutes with κ-filtered colimits, i.e. X is κ-compact. We have shown that Cat ⊆ Cκ, the +latter being a small category, concluding the proof. +Proposition 5.10. Let L: C → D be an internally left adjoint, i.e. a morphism in PriL +V , then +it sends atomics to atomics. +Thus, the V-functor χ(L): χ(C) → χ(D) factors to a V-functor +L: Cat → Dat. +Proof. Let X ∈ Cat. Then, since L is V-linear, we have a natural isomorphism −⊗LX ∼= L(−⊗X) +of functors V → D. Since both − ⊗ X and L are internally left adjoints, so is − ⊗ LX, i.e. LX is +atomic as required. +Recall that Cat is a full V-subcategory of χ(C), thus Cat → χ(C) is a subobject (indeed, the +space of maps to Cat is exactly the subspace of maps to χ(C) that land in Cat). Furthermore, by +Proposition 5.10, the restriction of internally left adjoint functors to the atomics factor through +the atomics. Thus, by [Ram22, Proposition A.1], this assembles into an induced subfunctor, which +furthermore lands in CatV ⊂ � +CatV. +Definition 5.11. We denote the induced subfunctor of atomics by +(−)at : PriL +V → CatV, +equipped with a natural transformation (−)at ⇒ χ|PriL +V of functors PriL +V → � +CatV. +Definition 5.12. Let C ∈ PrL +V. We say that a collection of atomic objects B ⊆ Cat are atomic +generators, if C is generated from B under weighted colimits. If such B exists, we say that C is +molecular. +Remark 5.13. We note that this definition a priori differs from our definition of molecular in [BMS21] +(where we work over a mode). We expect that closure under weighted colimits coincides with closure +under colimits and the V-action, but we are unaware of a proof of this statement. Such a result +would make the connection transparent. +We now wish to prove a converse to Proposition 5.10 under the assumption that the source is +molecular. To that end, we begin with the following. +Definition 5.14. Let I ∈ CatV and C ∈ PrL +V, and fix f ∈ FunV(I, C). By (2) of Theorem 2.9, the +functor colim(−) +I +(f): PV(I) → C is colimit preserving, i.e. a left adjoint, and V-linear. We denote +the lax V-linear right adjoint by +homV(f(−), −): C → PV(I). +Lemma 5.15. There is a natural isomorphism of lax V-linear functors +evali ◦ homV(f(−), −) ∼= homV(f(i), −). +Proof. Observe that we have an equivalence of V-linear left adjoint functors +colim(−) +I +(f) ◦ (− ⊗ よV(i)) = colim(−⊗よV(i)) +I +(f) +(1) +∼= − ⊗ colimよV(i) +I +(f) +(2) +∼= − ⊗ f(i), +21 + +where (1) follows from (2) of Theorem 2.9, and (2) follows from (3) of Theorem 2.9. Passing to the +lax V-linear right adjoints, we get +homV(よV(i), homV(f(−), −)) ∼= homV(f(i), −). +We finish by recalling that homV(よV(i), −) is the evaluation at i by (1) of Theorem 2.9. +Proposition 5.16. Let L: C → D be in PrL +V. If C is molecular and L sends a collection of atomics +generators B ⊂ Cat to atomic objects of D, then L is internally left adjoint. +Proof. We adapt the proof of [BMS21, Proposition 2.18]. We wish to show that R: D → C, the +right adjoint of L, is itself a left adjoint and that the lax V-action is strong. Thus, it suffices to +show that for any v ∈ V and Y : I → D, the canonical map v ⊗ colimI RYi → R(v ⊗ colimI Yi) is an +isomorphism. By the (unenriched) Yoneda lemma in the category C, this is equivalent to checking +that for every X ∈ C the map +hom(X, v ⊗ colim +I +RYi) → hom(X, R(v ⊗ colim +I +Yi)) +(2) +is an isomorphism. We will in fact show the stronger statement that +homV(X, v ⊗ colim +I +RYi) → homV(X, R(v ⊗ colim +I +Yi)) +(3) +is an isomorphism, which implies the previous statement by taking hom(1V, −). Let A ⊂ C be the +collection of objects X for which it is an isomorphism, and we will show that A = C. +We first show that B ⊂ A. Let X ∈ B. By Proposition 4.21, the following diagram commutes, +and both vertical maps are isomorphisms: +v ⊗ colimI homV(X, RYi) +homV(X, v ⊗ colimI RYi) +homV(X, R(v ⊗ colimI Yi)) +v ⊗ colimI homV(LX, Yi) +homV(LX, v ⊗ colimI Yi) +≀ +≀ +The upper-left morphism is an isomorphism because X is atomic, and similarly the bottom mor- +phism is an isomorphism because LX is atomic since X ∈ B and L sends B to atomic objects. This +shows that the upper-right morphism is an isomorphism as well. +We now show that A is closed under weighted colimits. +Let J ∈ CatV, W ∈ PV(J), and +f ∈ FunV(J, C) a V-functor landing in A ⊂ C. We show that X := colimW +J (f) is in A. By definition, +homV(f(−), −): C → PV(J) is the lax V-linear right adjoint of colim(−) +J +(f): PV(J) → C. Thus, by +Proposition 4.21 we have a natural isomorphism of functors C → V +homV +C(X, −) = homV +C(colimW +J (f), −) ∼= homV +PV(J)(W, homV(f(−), −)). +Thus it suffices to check that the map +homV +C(f(−), v ⊗ colim +I +RYi) → homV +C(f(−), R(v ⊗ colim +I +Yi)) +∈ PV(J) +is an isomorphism. By Proposition 4.24, the evaluation at j ∈ J are jointly conservative, so it +suffices to check that the map is an isomorphism after evaluation at every j. By Lemma 5.15, this +means that we need to check that +homV(f(j), v ⊗ colim +I +RYi) → homV(f(j), R(v ⊗ colim +I +Yi)) +22 + +is an isomorphism, which holds since by assumption f lands in A. +Recall that B are atomic generators of C, and we have shown that B ⊂ A and that A is closed +under weighted colimits, thus A = C, as required. +5.3 +Atomics–Presheaves Adjunction +Proposition 5.17. Let C0 ∈ CatV, then the enriched Yoneda embedding lands in the atomic objects, +yielding a V-functor よV : C0 → PV(C0)at. +Proof. Recall from Proposition 5.8 that よV(X) is atomic if and only if homV(よV(X), −): PV(C0) → +V preserves colimits and the lax V-linear structure is strong. By (1) of Theorem 2.9, this V-functor +is the evaluation at X, concluding the proof by Proposition 4.24. +We thus get an induced natural transformation よV : id ⇒ PV(−)at of functors CatV → CatV. +Proposition 5.18. The category PV(C0) is molecular, with the image of the enriched Yoneda +embedding as atomic generators. +Proof. By Proposition 5.17, the image of よV are atomic in PV(C0). +The fact that PV(C0) is +generated from the image of enriched Yoneda embedding under weighted colimits is [Hin21, 6.3.1], +recalled as (3) of Theorem 2.9. +Proposition 5.19. The functor PV : CatV → PrL +V of Theorem 3.6 lands in PriL +V . +Proof. Let f : C0 → D0 be a V-functor, and we need to show that f! : PV(C0) → PV(D0) is internally +left adjoint. +Recall from Proposition 5.18 that PV(C0) is molecular with the image of よV +C0 as +atomic generators. As in Corollary 3.8, naturality of the enriched Yoneda embedding says that +よV +D0f ∼= f!よV +C0. +Thus, f! sends the image of よV +C0 to the image of よV +D0 which are atomic in +PV(D0). Thus, f! sends a collection of atomic generators to atomic objects, so it is internally left +adjoint by Proposition 5.16. +Theorem 5.20. The partial adjunctions of Theorem 3.6 restricts to an adjunction +PV : CatV ⇄ PriL +V :(−)at +with unit the enriched Yoneda embedding よV : id ⇒ PV(−)at. +Proof. We need to check that for any C0 ∈ CatV and D ∈ PriL +V , the map +homiL +V (PV(C0), D) +(−)at +−−−→ homV(PV(C0)at, Dat) +(よV)∗ +−−−−→ homV(C0, Dat) +(4) +is an equivalence. Recall that +homL +V(PV(C0), D) +χ−→ homV(χ(PV(C0)), χ(D)) +(よV)∗ +−−−−→ homV(C0, χ(D)) +(5) +is an equivalence. Furthermore, both the first and last spaces in (4) are a collection of connected +components of the first and last spaces in (5), showing that the composition in (4) is an inclusion +of connected components. +To finish the argument, we need to show that the composition in (4) hits every connected +component. +To that end, let f : C0 → Dat be a V-functor. +We can post-compose it with the +23 + +inclusion V-functor Dat → D, and using (5) we get ˜f : PV(C0) → D in PrL +V. It is left to show that ˜f +is internally left adjoint. Recall that the image of the enriched Yoneda embedding forms a collection +of atomic generators by Proposition 5.18. Furthermore, by construction, ˜f(よV(X)) = f(X) ∈ Dat +is atomic. Thus, we have shown that ˜f sends a collection of atomic generators to atomics, so it is +indeed internally left adjoint by Proposition 5.16. +We extend our notations from Definition 3.7: +Definition 5.21. For f : C0 → D0 a morphism in CatV, consider the internally left adjoint functor +f! : PV(C0) → PV(D0). It has a right adjoint in PrL +V, i.e. a V-linear left and right adjoint functor +denoted f ⊛ : PV(D0) → PV(C0). This functor thus has a further lax V-linear right adjoint denoted +f⊛ : PV(C0) → PV(D0). +Using Corollary 5.3 we get the composition +χ PV : CatV +PV +−−→ PriL +V +χ−→ (� +CatV)LL, +allowing to pass the (double) adjunction to � +CatV. +Corollary 5.22. For f : C0 → D0 a morphism in CatV, we get a double adjunction χ(f!) ⊣ χ(f ⊛) ⊣ +χ(f⊛) in � +CatV. +Corollary 5.23. Let f : C0 → D0 in CatV, then f ⊛(G)(X) ∼= G(f(X)). +Proof. By Corollary 3.8 we have f!よV ∼= よVf in FunV(C0, PV(D0)). We thus get +f ⊛(G)(X) ∼= homV(よV(X), f ⊛(G)) ∼= homV(f!(よV(X)), G) ∼= homV(よV(f(X)), G) ∼= G(f(X)), +where the first and last steps follow from the enriched Yoneda lemma of (1) of Theorem 2.9, the +second step is by Proposition 4.21, and the third step is by the isomorphism above. +6 +Heine’s Enriched Yoneda Embedding +Independently of Hinich’s enriched Yoneda embedding, in [Hei20], Heine defines an enriched Yoneda +embedding as well, which satisfies the exact same universal property appearing in Proposition 2.12. +Theorem 6.1 ([Hei20, Corollary 6.2]). Let C0 ∈ CatV. +There is a V-natural transformation +よV +Heine : C0 → χ(PV(C0)). For every D ∈ PrL +V, the composition +homL +V(PV(C0), D) +χ−→ homV(χ(PV(C0)), χ(D)) +(よV +Heine)∗ +−−−−−−→ homV(C0, χ(D)) +is an equivalence. +We also note these versions of the enriched Yoneda embedding agree point-wise. In particular, +Heine’s version also satisfies the enriched Yoneda lemma. +Proposition 6.2. For every X ∈ C0, there is an isomorphism よV +Heine(X) ∼= よV(X). In particular, +homV(よV +Heine(X), −): PV(C0) → V is given by evaluation at X. +24 + +Proof. Consider the composition +C≃ +0 +よ +−→ Fun(Cop,≃ +0 +, S) +1V⊗− +−−−−→ Fun(Cop,≃ +0 +, V) +free +−−→ LModCop +0 (Fun(Cop,≃ +0 +, V)) = PV(C0). +By [Hin20, 6.2.6] and [Hei20, Proposition 6.5], よV(X) and よV +Heine(X) (respectively) are the image +of X under this composition. +The second part then follows from (1) of Theorem 2.9. +Note that these two results do not show that Hinich’s and Heine’s enriched Yoneda embeddings +agree as functors, but rather only up to an automorphism of PV(C0) (which may not be the identity). +See Subsection 1.3.2 for further discussion. Nevertheless, we now explain that the main results of +this paper hold for Heine’s enriched Yoneda embedding as well. +The proofs in Section 3 and +Section 5 have relied on the the universal property of Proposition 2.12 and the enriched Yoneda +lemma of (1) of Theorem 2.9 for よV. These two results also hold for よV +Heine by the above. We +also used weighted colimits, and their relationship with the enriched Yoneda embedding appearing +in (2) and (3) of Theorem 2.9. However, this was only used in the proof of Proposition 5.18, whose +statement is about the image of the enriched Yoneda embedding which is the same for よV and +よV +Heine by Proposition 6.2, and in the proof of Lemma 5.15, whose statement does not involve +the enriched Yoneda embedding. Thus, we see that the results of Section 3 and Section 5 hold +for Heine’s enriched Yoneda embedding as well. Notably, the analogue of Theorem 5.20 gives an +adjunction +PV +Heine : CatV ⇄ PriL +V :(−)at +with unit Heine’s enriched Yoneda embedding よV +Heine : id ⇒ PV +Heine(−)at. +References +[Bar22] +Shaul Barkan. Explicit Square Zero Obstruction Theory. 2022. arXiv:2211.07034v1 +[math.AT]. +[BMS21] +Shay Ben Moshe and Tomer M. Schlank. Higher Semiadditive Algebraic K-Theory and +Redshift. 2021. arXiv:2111.10203v1 [math.KT]. +[GH15] +David Gepner and Rune Haugseng. +Enriched ∞-categories via non-symmetric ∞- +operads. Advances in Mathematics, 279:575–716, 2015. +[Hau20] +Rune Haugseng. On lax transformations, adjunctions, and monads in (∞, 2)-categories. +2020. arXiv:2002.01037v2 [math.CT]. +[Hei20] +Hadrian Heine. An equivalence between enriched ∞-categories and ∞-categories with +weak action. 2020. arXiv:2009.02428v1 [math.AT]. +[HHLN20a] Rune Haugseng, Fabian Hebestreit, Sil Linskens, and Joost Nuiten. Lax monoidal ad- +junctions, two-variable fibrations and the calculus of mates. 2020. arXiv:2011.08808v2 +[math.CT]. +[HHLN20b] Rune Haugseng, Fabian Hebestreit, Sil Linskens, and Joost Nuiten. +Two-variable +fibrations, factorisation systems and ∞-categories of spans. 2020. arXiv:2011.11042v2 +[math.CT]. +25 + +[Hin20] +Vladimir Hinich. Yoneda lemma for enriched ∞-categories. Advances in Mathematics, +367:107129, 2020. +[Hin21] +Vladimir Hinich. +Colimits in enriched ∞-categories and Day convolution. +2021. +arXiv:2101.09538v1 [math.CT]. +[Lur09] +Jacob Lurie. Higher Topos Theory (AM-170). Princeton University Press, 2009. +[Lur17] +Jacob Lurie. Higher Algebra. https://www.math.ias.edu/~lurie/papers/HA.pdf, +2017. +[Ram22] +Maxime Ramzi. +The Yoneda embedding is natural. +2022. +arXiv:2209.12575v2 +[math.CT]. +[RV22] +Emily Riehl and Dominic Verity. Elements of ∞-Category Theory. Cambridge Studies +in Advanced Mathematics. Cambridge University Press, 2022. +26 + diff --git a/QdAyT4oBgHgl3EQftvmX/content/tmp_files/load_file.txt b/QdAyT4oBgHgl3EQftvmX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e52aa2e6c6ccfbd10a7d6680bb2f01755fcd3193 --- /dev/null +++ b/QdAyT4oBgHgl3EQftvmX/content/tmp_files/load_file.txt @@ -0,0 +1,1520 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf,len=1519 +page_content='Naturality of the ∞-Categorical Enriched Yoneda Embedding Shay Ben Moshe Abstract We make Hinich’s ∞-categorical enriched Yoneda embedding natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To do so, we exhibit it as the unit of a partial adjunction between the functor taking enriched presheaves and Heine’s functor taking a tensored category to an enriched category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, we study a finiteness condition of objects in a tensored category called being atomic, and show that the partial adjunction restricts to a (non-partial) adjunction between taking enriched presheaves and taking atomic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 Overview .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 Further Questions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 6 2 Generalities on Enriched Categories 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 Enriched Categories and Tensored Categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': 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in O-Monoidal Categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 20 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 23 6 Heine’s Enriched Yoneda Embedding 24 References 25 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='00601v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='CT] 2 Jan 2023 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 Overview 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 Partial Adjunction The main goal of this paper is to make Hinich’s enriched Yoneda embedding in the ∞-categorical setting from [Hin20] into a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We work in the framework of enriched ∞- categories as developed in [Hin20, Hin21, GH15, Hei20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' These source differ in notation, and we introduce our notation along the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For brevity, henceforth we use the term (enriched) category to mean an (enriched) ∞-category, and a 2-category to mean an (∞, 2)-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Throughout this paper we fix a presentably monoidal category V ∈ Alg(PrL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For every V-enriched category C0 ∈ CatV and a presentably V-tensored category D ∈ LModV(PrL), Hinich defined an (unenriched) category of V-functors FunV(C0, D) ∈ PrL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Using this, he defined a presentably V-tensored category of enriched presheaves PV(C0) := FunVrev(Cop 0 , V) ∈ LModV(PrL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' One of the main results of [Hin20] is the construction of a V-enriched Yoneda embedding よV : C0 → PV(C0), namely, an object of FunV(C0, PV(C0)) ∈ PrL, satisfying the enriched Yoneda lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' However, this map is not shown to be natural in C0, which is the central question of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In a sequel paper [Hin21], Hinich shows that his enriched Yoneda embedding enjoys the following universal property: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 ([Hin21, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then, for every D ∈ LModV(PrL), composition with the enriched Yoneda embedding induces an equivalence (よV)∗ : FunL V(PV(C0), D) ∼ −→ FunV(C0, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This almost exhibits よV as a unit of an adjunction, but there are two problems: (1) FunV(C0, D) is not the hom in any category (note that C0 is V-enriched while D is V-tensored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (2) C0 is small while D is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To solve Problem (1), we use the fact that presentably V-tensored categories produce V-enriched categories, as was first proven in [GH15, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9], and made functorial in the work of [Hei20], a special case of which reads as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 ([Hei20, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' There is an functor χ: LModV(PrL) → � CatV witnessing the source as a (non-full non-wide) subcategory of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We shall also use [Hei20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3], showing that for every C0 ∈ CatV and D ∈ LModV(PrL) there is an equivalence FunV(C0, D)≃ ∼= homV(C0, χ(D)), where the right hand side denotes the space of morphisms in � CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Combining these result, in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12 we conclude that there is an isomorphism of spaces homL V(PV(C0), D) ∼ −→ homV(C0, χ(D)), 2 where the left hand side denotes the underlying space of FunL V(PV(C0), D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To deal with Problem (2), we recall that one can define adjoints partially, namely an adjoint and a (co)unit map defined only on a full subcategory (or, more generally, relative to a functor), see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, we recall the folklore result that (partial) adjoints can be constructed point-wise, as we show in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' With these results in mind, we deduce our first main result: Theorem A (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The functor χ: LModV(PrL) → � CatV has a partial left adjoint PV : CatV → LModV(PrL) with partial unit agreeing with the enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In particular, for every f : C0 → D0 in CatV, we get an induced V-linear left adjoint f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' : PV(C0) → PV(D0), and an isomorphism f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='よV ∼= よVf, as explained in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 Atomics–Presheaves Adjunction In a somewhat different direction, we study atomic objects, a finiteness condition on objects of presentably V-tensored categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We show that this condition is closely related to the enriched Yoneda embedding, and, in particular, gives another solution for Problem (2), leading to a (non- partial) adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C ∈ LModV(PrL), and let X ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The functor − ⊗ X : V → C is a V-linear left adjoint functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We say that X is atomic if the right adjoint homV(X, −) is itself a left adjoint and the canonical lax V-linear structure on it is strong (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' These objects form a small full V-subcategory Cat ⊂ χ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We show that the construction C �→ Cat is functorial in internally left adjoint functors, that is, V-linear left adjoint functors L: C → D ∈ LModV(PrL) whose right adjoint is itself a left adjoint and the canonical lax V-linear structure on it is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Namely, we obtain a functor (−)at : LModV(PrL)iL → CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' As a key example, in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='17 we show that for any C0 ∈ CatV and X ∈ C0, the image よV(X) ∈ PV(C0) is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To see this, recall that by the enriched Yoneda lemma homV(よV(X), −) is given by evaluation at X, which preserves (co)limits and is V-linear, as we show in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Therefore, we get a factorization of the enriched Yoneda embedding through the atomics よV : C0 → PV(C0)at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, we show that the functoriality of PV obtained in Theorem A sends V-functors to internally left adjoints, namely, restricts to a functor PV : CatV → LModV(PrL)iL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Our second main result is that the partial adjunction of Theorem A restricts accordingly: Theorem B (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' There is an adjunction PV : CatV ⇄ LModV(PrL)iL :(−)at with unit agreeing with the enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In particular, for every f : C0 → D0 in CatV, the V-linear left adjoint f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' : PV(C0) → PV(D0), is internally left adjoint, and thus admits a right adjoint f ⊛ : PV(D0) → PV(C0) which is itself a V-linear left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Using this, in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='23 we show that f ⊛(G)(X) ∼= G(f(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We warn the reader that this does not imply that f ⊛ is the functor given by pre-composition, see Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 2-Categorical Structures In our discussion of atomic objects and their relation to the enriched Yoneda embedding, we use certain 2-categorical aspects of the theory of enriched categories and tensored categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Notably, we have considered the condition for a V-linear left adjoint functor to be internally left adjoint, which we show has a particularly simple 2-categorical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12 we recall that the category of V-tensored categories and lax V-linear functors enhances to a 2-category LModlax V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We show that a 1-morphism in that 2-category is a left adjoint, if and only if it is strong V-linear and left adjoint (on the underlying category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This result is a special case of a more general result we deduce from Lurie’s work on relative adjunctions and [HHLN20a], concerning the 2-category Monlax O of O-monoidal categories and lax O-monoidal functors for some operad O, which may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Corollary C (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The left adjoints in Monlax O are the lax O-monoidal functors that are strong and fiber-wise left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The category LModV(PrL) also enhances into a 2-category, which by the above result is a full 2- subcategory LModV(PrL) ⊂ LModlax V ( � Cat)L of the left adjoints in the 2-category LModlax V ( � Cat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, using the above result again, we see that internally left adjoints are precisely the left adjoint morphisms in the 2-category LModV(PrL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' As explained in [Hei20, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10], by [GH15, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16], the category of V-enriched cate- gories also enhances to a 2-category CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, by [Hei20, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11], the functor χ of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 enhances to a 2-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Using these results we also deduce the following result which may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Here a 2-functor is called 2-fully faithful if it induces an isomorphism on hom categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' if it exhibits the source as a full 2-subcategory of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Corollary D (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='19, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The functor χ enhances to a 2-fully faithful 2-functor χ: LModV(PrL) → ( � CatV)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Taking left adjoints again we get the 2-fully faithful 2-functor χ: LModV(PrL)iL → ( � CatV)LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In particular, for f : C0 → D0 in CatV, since f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' : PV(C0) → PV(D0) is internally left adjoint, we get a corresponding double adjunction χ(f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=') ⊣ χ(f ⊛) ⊣ χ(f⊛) in � CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 Relation to Previous Work This paper shares many of the ideas on atomic objects developed in our previous paper with Tomer Schlank, most notably [BMS21, Theorem D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To avoid the usage of enriched categories, in our previous paper we work over a mode M, that is, an idempotent algebra in PrL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The key feature of modes is that LModM(PrL) is a full subcategory of PrL, namely, a left adjoint functor is automatically (and uniquely) M-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This allowed us, for example, to simplify the definition of X being atomic to having homM(X, −): C → M commute with colimits (and thus automatically M-linear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Particularly, in the previous paper we have ignored the M-enriched structure on the category of atomic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In addition, we worked with unenriched presheaves and the unenriched Yoneda embedding, further tensored with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In particular, [BMS21, Theorem D] is a weaker statement than the main results of the present paper in these respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 4 On the other hand, in the unenriched context, the two different functorialities of presheaves and naturalities of the Yoneda embedding were shown coincide, contrary to the enriched case of the present paper (see the discussion in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In addition, our goal in the previous paper was different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The unenriched presheaves functor is endowed with a symmetric monoidal structure, and through the adjunction, this makes the atomics functor lax symmetric monoidal, and the Yoneda a symmetric monoidal natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In the present paper we do not deal with the multiplicative structure (though see the discussion in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 Further Questions We now list several further questions left open, which we expect have a positive answer, but we do not know how to approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 Naturality for Pre-composition The first question is closely related to [Ram22] and [HHLN20b, Corollary F], which deal with the case V = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that using the universal property of PV(C0), we deduced Theorem A, assembling PV into a functor, and the enriched Yoneda embedding into a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In addition, we saw that f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' has a right adjoint f ⊛, and we showed that f ⊛(G)(X) ∼= G(f(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The construction of PV(C0) as enriched functors from Cop 0 to V shows that it admits an- other functoriality in C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' More specifically, [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4] implies that it assembles into a functor PV : CatV → (PrR)op, sending f to f ∗ : PV(D0) → PV(C0) given by pre-composition, admitting a left adjoint f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=', commonly spelled “f lower what”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Note that this functoriality does not a priori give the V-tensored structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Also note that f ⊛(G)(X) ∼= G(f(X)) ∼= f ∗(G)(X), however it is not clear that this holds naturally in X, G or C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, Hinich’s construction of the enriched Yoneda embedding, as described for example in [Hin21, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2], seems to interact with the f ∗, and thus f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=', functoriality, but we don’t know how to extract naturality from his results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Can the f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' functoriality be extended to V-modules?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Can the enriched Yoneda embedding be made natural for the f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' functoriality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Do these agree with the f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' functoriality and naturality of the enriched Yoneda embedding?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 Heine’s and Hinich’s Enriched Yoneda Embeddings In [Hei20], Heine also defines an enriched Yoneda embedding よV Heine : C0 → χ(PV(C0)) by different means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' As we explain in Section 6, the main results of this paper hold for this version as well, producing an adjunction PV Heine : CatV ⇄ LModV(PrL)iL :(−)at with unit agreeing with Heine’s enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The uniqueness of adjoints implies that there is a natural isomorphism ψ: PV Heine ∼ −→ PV, together with an isomorphism ψよV Heine ∼= よV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' On the other hand, by the construction of the adjunction, for every C0 ∈ CatV there is an isomorphism PV Heine(C0) ∼= PV(C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We thus get an automorphism PV(C0) ψC0 −−→ PV Heine(C0) ∼= PV(C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' It is not clear to us that this automorphism is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Showing this is equivalent to showing that these two versions of the enriched Yoneda embedding coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 5 Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Do the enriched Yoneda embeddings constructed by Hinich and Heine coincide?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Namely, is the above automorphism the identity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 � Cat-Enrichment Recall that χ enhances to a 2-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, the universal property of the enriched Yoneda embedding of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 is originally stated for functor categories, though one of the sides is not constructed as the � Cat-enriched hom in a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This leads to the following question: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Can the (partial) adjunctions be made � Cat-enriched?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4 Multiplicative Structure For simplicity, assume that V is presentably symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' [Hei20, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16] enhances χ into a symmetric monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Via the adjunction, it may be possible to endow PV with an oplax symmetric monoidal structure, either by making the subfunctor (−)at lax symmetric monoidal and using the main result of [HHLN20a], or by proving a version of it for partial adjunctions and directly applying to PV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Analogously to the case of V = S, one would expect the resulting oplax symmetric monoidal structure on PV to be strong, and thus make the adjunction symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Assuming this, for any operad O, we get that if C0 is O-monoidal, then PV(C0) and the enriched Yoneda embedding are endowed with an O-monoidal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Separately, in [Hin21, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2], Hinich studies the multiplicative structure of enriched presheaves and the enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In particular, under the above assumptions, he endows PV(C0) and the enriched Yoneda embedding with an O-monoidal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' He further proves an O-monoidal version of the universal property of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Can the adjunction be made symmetric monoidal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Does the induced O-monoidal structure on PV(C0) and the enriched Yoneda embedding agree with those constructed by Hinich?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4 Acknowledgments We would like to thank Lior Yanovski for numerous useful conversations about atomic objects, enriched categories and weighted colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We also thank Shai Keidar, Shaul Ragimov, Maxime Ramzi and Noam Zimhoni for comments on earlier drafts of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Particularly, we thank Maxime for pointing several subtle missing components in earlier drafts, and suggesting corrections for some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 2 Generalities on Enriched Categories In this section we review some generalities on enriched categories, their relationship to tensored categories, and the enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We shall not delve into the details of the con- structions, as most of them will not play a role in the present paper, but rather only the formal properties of the resulting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 Enriched Categories and Tensored Categories Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We denote the (large) category of V-enriched categories, defined in [Hin20, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2], by CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For C0, D0 ∈ CatV, we denote the space of V-functors between them by homV(C0, D0) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Similarly, we let � CatV be the (huge) category of large V-enriched categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' V-enriched categories are closely related to categories tensored over V, as was first proven in [GH15, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9], and made functorial in the work of [Hei20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Indeed, Heine constructs the category ωLModlax V (denoted ωLModV there) of weakly V-tensored categories and lax V-linear functors, and a full subcategory ωLModcl,lax V thereof on the closed weakly V-tensored categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 ([Hei20, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' There is an equivalence χ: ωLModcl,lax V ∼ −→ CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Considering the large version of this equivalence, one can restrict the source to the subcategory with objects presentable categories with V-action commuting with colimits (which are automatically closed) and morphisms the (strong) V-linear left adjoint functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We define the category of presentably V-tensored categories to be PrL V := LModV(PrL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For C, D ∈ PrL V, we denote the space of V-linear left adjoint functors between them by homL V(C, D) ∈ �S, which is the space of objects of the category FunL V(C, D) ∈ � Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4 ([Hei20, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' There is a functor χ: PrL V → � CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' witnessing the source as a (non-full non-wide) subcategory of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For C ∈ PrL V, this constructs χ(C) ∈ � CatV, both of which have the same underlying category and thus space of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 Enriched Yoneda Lemma and Weighted Colimits In [Hin20, Hin21] Hinich constructs enriched presheaves and the enriched Yoneda embedding, which we recall in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We begin with Hinich’s model for the category of V-functors from a V-enriched category to a V-tensored category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In Hinich’s model, a V-enriched category C0 ∈ CatV is an algebra in some operad constructed from the space of objects C≃ 0 (see [Hin20, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For a presentably V-tensored category D ∈ PrL V, Hinich endows the (unenriched) category of functors Fun(C≃ 0 , D) with a left module structure over this operad (see [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In particular, one can take left C0-modules inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5 ([Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV and D ∈ PrL V, then the category of V-functors from C0 to D is defined to be FunV(C0, D) := LModC0(Fun(C≃ 0 , D)) ∈ PrL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We note that FunV(C0, D) is indeed presentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To see this, note that Fun(C≃ 0 , D) is an (unen- riched) presheaf category, and thus is presentable by [Lur09, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then, the module category FunV(C0, D) is also presentable by [Lur17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 7 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We warn the reader that in [Hei20], FunV(C0, D) is denoted by FunV(C0, χ(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7 ([Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The V-tensored category of V-enriched presheaves is defined to be PV(C0) := FunVrev(Cop 0 , V) ∈ PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' V is a V-V-bimodule in PrL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' One of the V-module structures is used to define the presentable category of Vrev-enriched functors, and the other is used to endow the resulting category with a V-module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We now record the main results about Hinich’s enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We remark that in [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7], Hinich proves his version of the enriched Yoneda lemma, very closely related to (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' However, the form described below is somewhat different, and relies on a definition of homV and the evaluation at X that were not introduced thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We postpone these definitions to Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='20 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='23, and the proof to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='25, but include the statement here for completeness of the exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Note that (1) (as well as (2)) will not be used before Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9 ([Hin20, Hin21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then, there is a V-enriched Yoneda embedding V-functor よV : C0 → PV(C0), that is, an object of FunV(C0, PV(C0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For every D ∈ PrL V, there is a weighted colimit functor colim(−) C0 (−): PV(C0) × FunV(C0, D) → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' These satisfy the following properties: (1) For every X ∈ C0, the functor homV(よV(X), −): PV(C0) → V agrees with evaluation at X as V-linear functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (2) colim(−) C0 (−) commutes with colimits in both arguments separately, and commutes with the V-action in the first argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (3) There is an equivalence (よV)∗ : FunL V(PV(C0), D) ⇄ FunV(C0, D) :colim(−) C0 (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' よV is constructed in [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (1) is deferred to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='25, though see [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' colim(−) C0 (−) is constructed in [Hin21, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2], where (2) is explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (3) is [Hin21, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In [Hei20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3, see also the discussion in the beginning of Section 5], Heine shows that FunV(C0, D) is closely related to the hom in V-enriched categories via χ (see also [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We recall a special case of Heine’s results as follows: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10 ([Hei20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV and D ∈ PrL V, then there is an equiva- lence FunV(C0, D)≃ ∼= homV(C0, χ(D)) 8 natural in C0 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Namely, there is a natural isomorphism between the functors (CatV)op × PrL V i×χ −−→ (� CatV)op × � CatV homV −−−−→ �S and (CatV)op × PrL V FunV(−,−) −−−−−−−→ Cat (−)≃ −−−→ S ⊂ �S We use this to transform the enriched Yoneda embedding to a morphism in � CatV, and deduce a universal property similar to (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9 where both the source and the target are the hom in some category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We denote by the same notation the enriched Yoneda embedding V-functor よV : C0 → χ(PV(C0)), corresponding to the enriched Yoneda embedding under the equivalence FunV(C0, PV(C0))≃ ∼= homV(C0, χ(PV(C0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV and D ∈ PrL V, then the composition homL V(PV(C0), D) χ−→ homV(χ(PV(C0)), χ(D)) (よV)∗ −−−−→ homV(C0, χ(D)) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Naturality in the PrL V coordinate of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10 shows that for C0 ∈ CatV and D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' E ∈ PrL V we get a commutative square: homL V(E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' D) homV(χ(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ(D)) hom(FunV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' E)≃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' FunV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' D)≃) hom(homV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ(E)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' homV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ(D))) χ ∼ Using the exponential adjunction we get a commutative square: homL V(E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' D) × FunV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' E)≃ homV(χ(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ(D)) × homV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ(E)) FunV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' D)≃ homV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ(D)) χ ∼ Taking E = PV(C0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' and picking the point よV ∈ FunV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' PV(C0))≃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' we get a commutative square: homL V(PV(C0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' D) homV(χ(PV(C0)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ(D)) FunV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' D)≃ homV(C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ(D)) χ (よV)∗ (よV)∗ ∼ The left morphism is an equivalence by applying (−)≃ to (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, the left-bottom composition is an equivalence, and by the commutativity of the diagram, so is the upper-right composition, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 9 3 Partial Adjunction In this section we prove Theorem A, namely the naturality of the enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To do so, we first define (unenriched) partial adjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then, we show the folklore result that (partial) adjoints can be constructed point-wise, from which the naturality of the enriched Yoneda embedding follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let L: C0 → D0 be a functor, and i: E0 → D0 another functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The data of a partial right adjoint (of L relative to i) is a functor R: E0 → C0 and a natural transformation ε: LR ⇒ i called the partial counit, such that for every X ∈ C0 and Y ∈ E0 the composition homC0(X, RY ) L−→ homD0(LX, LRY ) εY ◦− −−−→ homD0(LX, iY ) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' A partial left adjoint with a partial unit is defined dually, by taking (−)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' What we call a partial adjunction is typically called a relative adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' However, in Section 4 we use the distinct concept of relative adjunctions developed by Lurie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To avoid confusion, we call what is typically called a relative adjunction a partial adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In this paper we only use the notion of a partial adjunction when i: E0 → D0 is an inclusion of a full subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In this case, one can think of R as an adjoint defined only partially on D0, justifying the name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Unlike in a standard adjunction, a partial adjunction is not a symmetric concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Particularly, note that there is no partial unit (in fact, L and R can not be composed in the other direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let L: C0 → D0 be a functor, and i: E0 → D0 another functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Assume that for every Y ∈ E0 we are given an object RY ∈ C0 and a morphism εY : LRY → iY , such that for every X ∈ C0 the composition homC0(X, RY ) L−→ homD0(LX, LRY ) εY ◦− −−−→ homD0(LX, iY ) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then, R and ε assemble into the data of a partial right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the functor ˜R: D0 → P(C0) given by the composition D0 よD0 −−−→ P(D0) L∗ −−→ P(C0), which sends Z ∈ D0 to homD0(L(−), Z): Cop 0 → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let Y ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the natural transformation homC0(−, RY ) L−→ homD0(L(−), LRY ) εY ◦− −−−→ homD0(L(−), iY ) = ˜R(iY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (1) By assumption, it is an isomorphism at every X ∈ C0, and thus it is a natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Namely, the presheaf ˜R(iY ) ∈ P(C0) is representable by RY ∈ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In other words, the composition ˜Ri: E0 → P(C0) lands in the essential image of the (unenriched) Yoneda embedding よC0 : C0 → P(C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' As the Yoneda embedding is fully faithful, we get an induced functor R: E0 → C0, together with a natural isomorphism よC0R ∼= ˜Ri = L∗よD0i of functors E0 → P(C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 10 By construction, R agrees with RY at every Y ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, the natural isomorphism at Y is given by (1), constructed via εY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We now extract the partial counit ε: LR ⇒ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the following composition: よD0LR (1) ∼= (LR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='よE0 (2) ⇒ (LR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='R∗よC0R (3) ∼= (LR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='R∗L∗よD0i ∼= (LR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (LR)∗よD0i (4) ⇒ よD0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (1) is naturality of the (unenriched) Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (2) is the fact that R is a functor, giving homE0(−, −) ⇒ homC0(R(−), R(−)), which by exponential adjunction is the same as よE0 ⇒ R∗よC0R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (3) is the isomorphism よC0R ∼= L∗よD0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (4) is the counit of the adjunction (LR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' ⊣ (LR)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, we have constructed a natural transformation よD0LR ⇒ よD0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Since よD0 is fully faithful, this induces a natural transformation ε: LR ⇒ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that the isomorphism used at step (3) is given at Y ∈ E0 by (1), which shows that ε indeed agrees with εY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The functor χ: PrL V → � CatV has a partial left adjoint PV : CatV → PrL V with partial unit よV : id|CatV → χ PV, agreeing with enriched presheaves and enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This follows immediately from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12 and (the dual statement to) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For f : C0 → D0 a morphism in CatV, we denote the induced morphism in PrL V by f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' : PV(C0) → PV(D0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' An instance of the naturality of the enriched Yoneda embedding is the following: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let f : C0 → D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then, よV D0f ∼= χ(f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' )よV C0 in homV(C0, χ(PV(D0))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Similarly, よV D0f ∼= f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='よV C0 in FunV(C0, PV(D0)) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The first part is an immediate application of the adjunction of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For the sec- ond part, recall from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10 that there is a natural isomorphism homV(−, χ(−)) ∼= FunV(−, −)≃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11 we have defined よV C0 ∈ homV(C0, χ(PV(C0))) to be the map corresponding to よV C0 ∈ FunV(C0, PV(C0)) via this isomorphism, and similarly for D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By applying the naturality of the isomorphism in the target to the morphism f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=', we get that χ(f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' )よV C0 ∈ homV(C0, χ(PV(D0))) corresponds to f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='よV C0 ∈ FunV(C0, PV(D0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Similarly, by ap- plying the naturality of the isomorphism in the source to the morphism f, we get that よV D0f ∈ homV(C0, χ(PV(D0))) corresponds to よV D0f ∈ FunV(C0, PV(D0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, under the natural isomor- phism, the isomorphism よV D0f ∼= χ(f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' )よV C0 corresponds to an isomorphism よV D0f ∼= f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='よV C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 4 2-Categorical Structures In this section we study certain 2-categorical aspects of enriched categories and tensored categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The first main result of this section is Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11, showing that a lax O-monoidal functor between 11 O-monoidal categories is a left adjoint in the 2-category Monlax O , if and only if it is strong O-monoidal and fiber-wise left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This is proven by applying Lurie’s [Lur17, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 Relative Adjunctions] to the 2-categories constructed in [HHLN20a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' From this, we deduce Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='13, the corresponding result for lax V-linear functors between V-tensored categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then, after recalling that in [Hei20, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11] Heine shows that χ is a 2-functor, we deduce Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='19, our second main result, which says that χ enhances to a 2-fully faithful 2-functor χ: PrL V → ( � CatV)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Finally, in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='25 we finish the proof of (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 Adjunctions in O-Monoidal Categories Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 ([HHLN20a, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let B be a a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Denote by Cocartlax B the 2-category of cocartesian fibrations over B and functors over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We shall not recall the precise definition of this 2-category (for which we refer the reader to [HHLN20a, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7]), rather, let us recall its objects, 1-morphisms and 2-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Cocartlax B has objects: cocartesian fibrations q: C → B, 1-morphisms: functors over B, namely, a 1-morphism from q: C → B to p: D → B is a functor F : C → D and a natural isomorphism pF ∼= q, 2-morphisms: natural transformations over B, namely, a 2-morphism from F to G is a natural transformation α: F ⇒ G and an identification of the natural transformation q ∼= pF pα ==⇒ pG ∼= q with idq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' exhibiting the following square as commutative: pF pG q q pα ≀ ≀ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By construction, the underlying category of the 2-category Cocartlax B is the full subcategory of Cat/B on the cocartesian fibrations, which explains the objects and the 1-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For the 2- morphisms, we recall that as explained in [HHLN20a, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9], a 2-morphism is commutative diagram of the form: C × [1] D B q p ¯α Under the exponential adjunction, ¯α: C × [1] → D corresponds to the natural transformation α: F ⇒ G, and the isomorphism p¯α ∼= q corresponds to the identification pα ∼= idq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' A 2-morphism in Cocartlax B given by α: F ⇒ G together with an identification pα ∼= idq is invertible if and only if α is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Clearly, if the 2-morphism is invertible then in particular α is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For the other direction, assume that α has an inverse α−1 : G ⇒ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We shall enhance it to an inverse 2-morphism in Cocartlax B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the following diagram: pF pG pF q q q ≀ ≀ ≀ pα pα−1 The identification pα ∼= idq is precisely an (invertible) 3-morphism in Cat making the left square commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The outer square commutes as pα−1pα ∼= idpF ∼= idq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, by composing the outer square with the inverse of the left square, we get commutativity data for the right square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This makes α−1 into a 2-morphism in Cocartlax B , which by construction is the required inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Our next goal is to understand left adjoints in the 2-category Cocartlax B , achieved in Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To that end, we shall employ Lurie’s [Lur17, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 Relative Adjunctions].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The definitions and results in that section are phrased for cartesian fibrations and right adjoints, which by taking (−)op correspond to our case of cocartesian fibrations and left adjoints, as we present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' First, we recall that in a 2-category, we say that a 1-morphism L: X → Y is a left adjoint if there exists a 1-morphism R: Y → X and two 2-morphisms u: idX ⇒ RL and c: LR ⇒ idY satisfying the zigzag identities, namely (R uR ==⇒ RLR Rc ==⇒ R) ∼= idR, (L Lu ==⇒ LRL cL ==⇒ L) ∼= idL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We also recall that to check that L is a left adjoint, we can check a weaker condition: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4 ([RV22, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In any 2-category, a 1-morphism L: X → Y is a left adjoint if and only if there exist a 1-morphism R: Y → X and 2-morphisms u: idX ⇒ RL and c: LR ⇒ idY such that the zigzag morphisms R uR ==⇒ RLR Rc ==⇒ R, L Lu ==⇒ LRL cL ==⇒ L are invertible (in which case there exists a possibly different 2-morphism ˜c: LR ⇒ idY for which the zigzag identities hold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The proof of [RV22, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11] works in an arbitrary 2-category, although stated in the context of functors between (∞-)categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5 ([Lur17, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C D B q p L be a commutative diagram of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We say that L admits a relative right adjoint if L has a (non-relative) right adjoint R: D → C, and the counit map c: LR ⇒ idD satisfies the condition that pc is equivalent to idp (and in particular qR ∼= p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 13 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The definition of [Lur17, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2] is via the two equivalent conditions of [Lur17, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' First, we note that these are phrased for admitting a relative left adjoint, but the theory is symmetric via taking (−)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Second, Lurie assumes that p and q are categorical fibrations, but every functor is equivalent to a categorical fibration, so this assumption can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Finally, and most importantly, we note that our definition clearly implies condition (1) and is implied by condition (2) in [Lur17, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1], and thus is also equivalent to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' A 1-morphism L: C → D over B in Cocartlax B is a left adjoint if and only if L admits a relative right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' If L is a left adjoint in Cocartlax B then it clearly admits a relative right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For the other direction, assume that L admits a relative right adjoint, and we shall show that it is a left adjoint in Cocartlax B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let R: D → C, u: idC ⇒ RL and c: LR ⇒ idD be the (non-relative) adjunction data, satisfying that pc ∼= idp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The identification pc ∼= idp makes c into a 2-morphism in Cocartlax B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This also identifies qR ∼= p, making R into a 1-morphism in Cocartlax B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In the rest of the proof we make u into a 2-morphism as well, and show that L is a left adjoint using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We begin by making u into a 2-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the following diagram: pL pLRL pL q q q pLu pcL ≀ ≀ ≀ The upper composition is p(−) applied the zigzag morphism, which is equivalent to idpL, thus making the outer square commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The assumption that pc is equivalent to idp makes the right square commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, by composing the outer square with the inverse of the right square, we get that the left square commutes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In other words, we obtained an identification of pLu with idq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The identification pL ∼= q thus shows that we got an identification of qu with idq, making u into a 2-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4, to see that L is a left adjoint in Cocartlax B it suffices to check that the zigzag morphisms corresponding to these 2-morphisms are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3, a 2-morphism in Cocartlax B is invertible if and only if the underlying natural transformations are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Indeed, the underlying natural transformations are simply R uR ==⇒ RLR Rc ==⇒ R, L Lu ==⇒ LRL cL ==⇒ L which are invertible (in fact, equivalent to the identities) by the assumption that u and c are a unit and a counit for a (non-relative) adjunction L ⊣ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The following is a recast of [Lur17, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6] in an appropriate 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The left adjoints in Cocartlax B are those functors over B that preserve cocarte- sian morphisms and are fiber-wise left adjoint (Cocartlax B )L = CocartL-fw B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7, a 1-morphism L between two cocartesian fibrations is a left adjoint if and only if it admits a relative right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus by (the (−)op of) [Lur17, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6], 14 L is a left adjoint if and only if it preserves locally cocartesian morphisms and is fiber-wise left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By [Lur09, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8], locally cocartesian morphisms and cocartesian morphisms coincide, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' There is a possible alternative route to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that [HHLN20a, Theorem E] shows that Cocartlax B ∼= Funlax(B, Cat), where the right hand side denotes the 2- category of functors and lax natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Additionally, [Hau20, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6] shows that for any 2-categories X, Y, the left adjoints in Funlax(X, Y) are those lax natural transformations which are strong and point-wise left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Combined, this would imply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' However, the definition of Funlax in [HHLN20a] and [Hau20] rely on two different models of the Gray tensor product, which to the best of our knowledge were not shown to be equivalent so far, and thus this does not constitute a complete proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In the rest of the subsection we follow the notations of [HHLN20a, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4] for operads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Namely, an operad O is a functor O → Fin∗ (satisfying certain properties), whereas in [Lur17], it is typically denoted by O⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10 ([HHLN20a, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let O be an operad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let Monlax O ⊂ Cocartlax O be the 1- full 2-subcategory on the O-monoidal categories and lax O-monoidal functors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' functors over O preserving cocartesian morphisms lying over inert morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The following is a recast of [Lur17, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7] in an appropriate 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The left adjoints in Monlax O are those lax O-monoidal functors that are strong and fiber-wise left adjoint (Monlax O )L = MonL-fw O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For the first direction, note that a left adjoint in Monlax O is in particular a left adjoint in Cocartlax O , thus it is strong O-monoidal (preserves all cocartesian morphisms) and fiber-wise left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For the second direction, let L: C → D over O in MonL-fw O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' As MonL-fw O ⊂ CocartL-fw O = (Cocartlax O )L, there is a right adjoint R: D → C over O in Cocartlax O with some unit and counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Since Monlax O is a 1-full 2-subcategory of Cocartlax O , all we need to show is that the right adjoint is in Monlax O , whence the unit and counit are also automatically there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Namely, we need to know that R: D → C is lax O-monoidal, which follows from the construction, as explained in [Lur17, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 Tensored Categories As explained in [Hei20, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10], the category ωLModlax V is enhanced to a 2-category, and thus so is its full subcategory LModlax V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We repeat the construction in more detail, building on the definitions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' See also [Bar22, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='14] for a closely related discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We recall the operad Assoc classifying an associative algebra, and the operad LM classifying an associative algebra and a left module over it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the monoidal category V as an object in Mon(Cat) ∼= MonAssoc ⊂ Monlax Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This leads us to the following: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We define the 2-category of V-tensored categories and lax V-linear functors to be LModlax V := {V} ×Monlax Assoc Monlax LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 15 As a particular case of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11, we deduce the following, which is an “if and only if” version of [Lur17, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8 and Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9] phrased in an appropriate 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The left adjoints in LModlax V are those lax V-linear functors that are strong V-linear and left adjoint (on the underlying category) (LModlax V )L = LModL-fw V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The result follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11 by taking O = LM, and pulling back over V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In more detail, consider the following diagram: (LModlax V )L (Monlax LM)L {V}L (Monlax Assoc)L {V} Monlax Assoc ≀ The bottom square is a pullback square because the lower right morphism is an inclusion of a 2-subcategory, and the lower left morphism is the map from a point to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The upper square is a pullback square, because (−)L commutes with limits, as it is given by hom from the walking adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Therefore, the outer square is also a pullback square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We thus finish by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='13 work for an arbitrary monoidal category V, without needing to assume that it is presentably monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We now enhance PrL V of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 to a 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We define the 2-category of presentably V-tensored categories to be the 1-full 2-subcategory PrL V := LModV(PrL) ⊂ LModlax V ( � Cat) on the objects and morphisms of PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' There is a full 2-subcategory inclusion PrL V ⊂ (LModcl,lax V ( � Cat))L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' First, we claim that there is a full 2-subcategory inclusion PrL V = LModV(PrL) ⊂ LModcl,L-fw V ( � Cat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' It is indeed an inclusion, since a presentably V-tensored category is automatically closed, and mor- phism in both categories are the V-linear functors that are left adjoint on the underlying category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Second, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='13, the target is (LModcl,lax V ( � Cat))L, which finishes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 Tensored Categories to Enriched Categories We recall that CatV can be enhanced into a 2-category, as explained for example in [Hei20, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Indeed, by [GH15, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16], the functor Alg(PrL) → PrL given by V �→ CatV is lax symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Since V is a S-module in PrL, we get that CatV is a Cat-module in PrL as well, which indeed models a 2-category (for example via χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 16 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We denote the 2-category of V-enriched categories by CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Heine shows that χ of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 enhances to a 2-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='18 ([Hei20, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The equivalence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 enhances to a 2-equivalence χ: ωLModcl,lax V ∼ −→ CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Using this we deduce the following: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The functor from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4 enhances to a 2-fully faithful 2-functor χ: PrL V → ( � CatV)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the large version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='18, namely ωLModcl,lax V ( � Cat) ∼ −→ � CatV, and re- strict it to the full 2-subcategory LModcl,lax V ( � Cat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Taking left adjoints and using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16, we get the 2-fully faithful 2-functor PrL V ⊂ (LModcl,lax V ( � Cat))L → ( � CatV)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4 Evaluation and Enriched Hom in Tensored Categories In this subsection, we introduce homV in the presentably V-tensored context and show that it satis- fies expected properties for adjunction in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='21 (see also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='22 for the connection to adjunction in V-enriched categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' With this definition in mind, we finish the proof of (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9 in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C ∈ PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We have an equivalence of categories FunL V(V, C) ∼ −→ C given by evaluation at 1V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16 that there is a full 2-subcategory inclusion PrL V ⊂ (LModcl,lax V ( � Cat))L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, passing to right adjoint, gives us a map FunL V(V, C) → Funlax-V(C, V)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Composing the two and taking (−)op, we get Cop → Funlax-V(C, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We let homV(−, −): Cop × C → V to be the functor corresponding to the above under the exponential adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By construction, it is lax V-linear in the second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Namely, this construction sends X to the lax V-linear functor homV(X, −): C → V, right adjoint to the V-linear left adjoint − ⊗ X : V → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let L: C → D ∈ PrL V be a V-linear left adjoint functor, and let R: D → C denote its right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then there is a natural isomorphism homV(L(−), −) ∼= homV(−, R(−)) of functors Cop × D → V lax V-linear in the second coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Indeed, consider the following diagram: C FunL V(V, C) Funlax-V(C, V) D FunL V(V, D) Funlax-V(D, V) L L◦− −◦R ∼ ∼ 17 The left square commutes by passing to the right adjoints and noting that evaluation at 1V com- mutes with post-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To see that the right square commutes, recall that the horizontal morphisms were constructed by the full 2-subcategory inclusion PrL V ⊂ (LModcl,lax V ( � Cat))L and passing to the right adjoints, and the vertical morphisms are also adjoints in the same category, so the commutativity is the fact that the right adjoint of a composition is the composition of the right adjoints in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We note that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='21 is in the context of presentably V-tensored categories, and not that of V-enriched categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' One of the main features of χ is that the V-enrichment of χ(C) is given by homV(X, Y ), as is shown in [GH15, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='19, the adjunction L ⊣ R produces an adjunction χ(L) ⊣ χ(R) in � CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' One might expect that this adjunction provides a natural isomorphism similar to the one of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='21 for the V-enriched hom’s of χ(C) and χ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' However, we do not provide such a natural isomorphism, nor show its compatibility with the one constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV, X ∈ C0 and D ∈ PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5 that FunV(C0, D) := LModC0(Fun(C≃ 0 , D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider i: pt → C≃ 0 choosing X, and the following diagram: D Fun(pt, D) Fun(C≃ 0 , D) LModC0(Fun(C≃ 0 , D)) i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' i∗ free forget Here the dashed arrows are the left adjoints of the solid arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Also recall from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7 that if we replace C0 by Cop 0 and let D = V, the V-functor category is PV(C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Moreover, in this case the three categories in the diagram are presentably V-tensored and the two left adjoint functors are V-linear, as in [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Via Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16, the right adjoints are canonically lax V-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV, X ∈ C0 and D ∈ PrL V be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We define the evaluation at X functor evalX : FunV(C0, D) → D by evalX := i∗ ◦ forget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For the case evalX : PV(C0) → V, it is canonically lax V-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV and D ∈ PrL V, and consider FunV(C0, D) ∈ PrL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then evalX commutes with (co)limits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (co)limits are computed level-wise), and are jointly conservative over all X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For PV(C0) ∈ PrL V, the lax V-linear structure on evalX is strong (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=', the V-action is level-wise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that evalX = i∗ ◦ forget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The functor forget commutes with all (co)limits by [Lur17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5] as the forgetful from modules, while i∗ commutes with (co)limits as they are computed level-wise in (unenriched) functor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The forgetful from modules is always conservative by [Lur17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2], and the evalu- ation at X functors Fun(C≃ 0 , D) → D are jointly conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For the case of PV(C0), the V-action is by construction given level-wise, as explained in [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Finally, we are in position to prove our variant of Hinich’s enriched Yoneda lemma, appearing as (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV, then homV(よV(X), −) ∼= evalX as lax V-linear functors, and in particular homV(よV(X), −) is also strong V-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We need to show that homV(よV(X), −) ∼= evalX as lax V-linear functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By construction, this is equivalent to showing that −⊗よV(X) ∼= free ◦i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' as V-linear functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the following diagram: V Fun(Cop,≃ 0 , V) LModCop 0 (Fun(Cop,≃ 0 , V)) S Fun(Cop,≃ 0 , S) −⊗1V hom(1V,−) −⊗1V hom(1V,−) i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' i∗ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' i∗ free forget Here the dashed arrows are the left adjoints of the solid arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Clearly, the solid square commutes, and thus, the dashed square obtained by passing to left adjoints also commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6], we have that よV(X) ∼= free((−⊗1V)◦よ(X)), where here よ(X) ∈ Fun(Cop,≃ 0 , S) is the image of X under the (unenriched) Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Applying the naturality of the (un- enriched) Yoneda embedding to i: pt → S, and noting that よ: pt → S sends よ(pt) = pt, we get that よ(X) = よ(i(pt)) ∼= i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (よ(pt)) ∼= i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Using the commutativity of the dashed square above, we conclude that よV(X) ∼= free((− ⊗ 1V) ◦ よ(X)) ∼= free((− ⊗ 1V) ◦ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (pt)) ∼= free(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Since free and i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' are V-linear functors, we finally get that − ⊗ よV(X) ∼= − ⊗ free(i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1V) ∼= free ◦i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (− ⊗ 1V) ∼= free ◦i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' as V-linear functors, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 5 Atomic Objects, Presheaves and Yoneda In this section, building on the 2-categorical results of the previous section, we study internally left adjoint functors and atomic objects, and connect them to the enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We begin in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 by defining internally left adjoints between presentably V-tensored categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Namely, V-linear left adjoint functors, whose right adjoints are also left adjoint and their canonical lax V-linear structure is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Following this, in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4 we define a finiteness condition called being atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We say that X in C ∈ PrL V is atomic if − ⊗ X : V → C is internally left adjoint, that is, if homV(X, −) commutes with colimits and is V-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For instance, in the case V = Sp, this coincides with condition of being compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' From the definition, internally left adjoints send atomic objects to atomic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The key technical result of this section is Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16, giving a converse result under the assumption that the source category is generated from the atomics under weighted colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Next, in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='17 we show that よV(X) is atomic in PV(C0), and in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='18 we show that together they generate PV(C0) under weighted colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This allows us to use Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16 to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='20, the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This result says that the partial adjunction of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6 restricts to a (non-partial) adjunction between the enriched presheaves functor and taking the atomics, with the unit being (the factorization through the atomics of) the enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 Internally Left Adjoints Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' A V-linear left adjoint functor is called internally left adjoint if it is left adjoint in the 2-category PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We denote the wide 2-subcategory on the internally left adjoint functors by PriL V := (PrL V)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For C, D ∈ PriL V , we denote the category of V-linear internally left adjoint functors between them by FuniL V (C, D) ∈ � Cat and the corresponding homiL V (C, D) ∈ �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Note that for any V-linear left adjoint functor L: C → D, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16 shows that it admits a lax V-linear right adjoint R: D → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' A V-linear left adjoint functor L: C → D is internally left adjoint if and only if the lax V-linear right adjoint R: D → C is strong and itself a left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' As above, R is a morphism in LModcl,lax V ( � Cat) between objects of the 2-subcategory PrL V, and the condition is that it is in fact a morphism in that 2-subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16, PrL V is a full 2-subcategory of (LModcl,lax V ( � Cat))L, thus the condition is that R is a morphism in that 2-subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The result then follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Taking left adjoints in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='19, we immediately get: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' χ restricts to a 2-fully faithful 2-functor χ: PriL V → ( � CatV)LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 Atomic Objects Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C ∈ PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We say that X ∈ C is atomic if the functor − ⊗ X : V → C is internally left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We denote the full V-subcategory on the atomic objects by Cat ⊂ χ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The unit 1V ∈ V is always atomic, because − ⊗ 1V is the identity functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6 ([BMS21, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In the case V = Sp, atomic objects coincide with compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that for C ∈ PrL V, there is an equivalence FunL V(V, C) ∼ −→ C given by evaluation at 1V, with inverse sending X to − ⊗ X : V → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, immediately from the definition, we deduce the following: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Evaluation at 1V induces an equivalence FuniL V (V, C) ∼ −→ Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Since the right adjoint of − ⊗ X : V → C is homV(X, −): C → V, the following follows immedi- ately from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' An object X ∈ C is atomic if and only if homV(X, −): C → V preserves colimits and the lax V-linear structure is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The atomics are a small V-enriched category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The argument is identical to [BMS21, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We repeat the details for the conve- nience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let κ be a regular cardinal such that the unit 1V is κ-compact, namely hom(1V, −): V → S commutes with κ-filtered colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We show that all atomic objects are κ-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C ∈ PrL V 20 and let X ∈ C be atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8, homV(X, −): C → V commutes with κ-filtered colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Since hom(X, −) ∼= hom(1V ⊗ X, −) ∼= hom(1V, homV(X, −)), we get that hom(X, −) also commutes with κ-filtered colimits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' X is κ-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We have shown that Cat ⊆ Cκ, the latter being a small category, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let L: C → D be an internally left adjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' a morphism in PriL V , then it sends atomics to atomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, the V-functor χ(L): χ(C) → χ(D) factors to a V-functor L: Cat → Dat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let X ∈ Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Then, since L is V-linear, we have a natural isomorphism −⊗LX ∼= L(−⊗X) of functors V → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Since both − ⊗ X and L are internally left adjoints, so is − ⊗ LX, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' LX is atomic as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that Cat is a full V-subcategory of χ(C), thus Cat → χ(C) is a subobject (indeed, the space of maps to Cat is exactly the subspace of maps to χ(C) that land in Cat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10, the restriction of internally left adjoint functors to the atomics factor through the atomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, by [Ram22, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1], this assembles into an induced subfunctor, which furthermore lands in CatV ⊂ � CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We denote the induced subfunctor of atomics by (−)at : PriL V → CatV, equipped with a natural transformation (−)at ⇒ χ|PriL V of functors PriL V → � CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C ∈ PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We say that a collection of atomic objects B ⊆ Cat are atomic generators, if C is generated from B under weighted colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' If such B exists, we say that C is molecular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We note that this definition a priori differs from our definition of molecular in [BMS21] (where we work over a mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We expect that closure under weighted colimits coincides with closure under colimits and the V-action, but we are unaware of a proof of this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Such a result would make the connection transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We now wish to prove a converse to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='10 under the assumption that the source is molecular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To that end, we begin with the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let I ∈ CatV and C ∈ PrL V, and fix f ∈ FunV(I, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9, the functor colim(−) I (f): PV(I) → C is colimit preserving, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' a left adjoint, and V-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We denote the lax V-linear right adjoint by homV(f(−), −): C → PV(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' There is a natural isomorphism of lax V-linear functors evali ◦ homV(f(−), −) ∼= homV(f(i), −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Observe that we have an equivalence of V-linear left adjoint functors colim(−) I (f) ◦ (− ⊗ よV(i)) = colim(−⊗よV(i)) I (f) (1) ∼= − ⊗ colimよV(i) I (f) (2) ∼= − ⊗ f(i), 21 where (1) follows from (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9, and (2) follows from (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Passing to the lax V-linear right adjoints, we get homV(よV(i), homV(f(−), −)) ∼= homV(f(i), −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We finish by recalling that homV(よV(i), −) is the evaluation at i by (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let L: C → D be in PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' If C is molecular and L sends a collection of atomics generators B ⊂ Cat to atomic objects of D, then L is internally left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We adapt the proof of [BMS21, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We wish to show that R: D → C, the right adjoint of L, is itself a left adjoint and that the lax V-action is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, it suffices to show that for any v ∈ V and Y : I → D, the canonical map v ⊗ colimI RYi → R(v ⊗ colimI Yi) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By the (unenriched) Yoneda lemma in the category C, this is equivalent to checking that for every X ∈ C the map hom(X, v ⊗ colim I RYi) → hom(X, R(v ⊗ colim I Yi)) (2) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We will in fact show the stronger statement that homV(X, v ⊗ colim I RYi) → homV(X, R(v ⊗ colim I Yi)) (3) is an isomorphism, which implies the previous statement by taking hom(1V, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let A ⊂ C be the collection of objects X for which it is an isomorphism, and we will show that A = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We first show that B ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let X ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='21, the following diagram commutes, and both vertical maps are isomorphisms: v ⊗ colimI homV(X, RYi) homV(X, v ⊗ colimI RYi) homV(X, R(v ⊗ colimI Yi)) v ⊗ colimI homV(LX, Yi) homV(LX, v ⊗ colimI Yi) ≀ ≀ The upper-left morphism is an isomorphism because X is atomic, and similarly the bottom mor- phism is an isomorphism because LX is atomic since X ∈ B and L sends B to atomic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This shows that the upper-right morphism is an isomorphism as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We now show that A is closed under weighted colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let J ∈ CatV, W ∈ PV(J), and f ∈ FunV(J, C) a V-functor landing in A ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We show that X := colimW J (f) is in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By definition, homV(f(−), −): C → PV(J) is the lax V-linear right adjoint of colim(−) J (f): PV(J) → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='21 we have a natural isomorphism of functors C → V homV C(X, −) = homV C(colimW J (f), −) ∼= homV PV(J)(W, homV(f(−), −)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus it suffices to check that the map homV C(f(−), v ⊗ colim I RYi) → homV C(f(−), R(v ⊗ colim I Yi)) ∈ PV(J) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='24, the evaluation at j ∈ J are jointly conservative, so it suffices to check that the map is an isomorphism after evaluation at every j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='15, this means that we need to check that homV(f(j), v ⊗ colim I RYi) → homV(f(j), R(v ⊗ colim I Yi)) 22 is an isomorphism, which holds since by assumption f lands in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that B are atomic generators of C, and we have shown that B ⊂ A and that A is closed under weighted colimits, thus A = C, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 Atomics–Presheaves Adjunction Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV, then the enriched Yoneda embedding lands in the atomic objects, yielding a V-functor よV : C0 → PV(C0)at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8 that よV(X) is atomic if and only if homV(よV(X), −): PV(C0) → V preserves colimits and the lax V-linear structure is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9, this V-functor is the evaluation at X, concluding the proof by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We thus get an induced natural transformation よV : id ⇒ PV(−)at of functors CatV → CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The category PV(C0) is molecular, with the image of the enriched Yoneda embedding as atomic generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='17, the image of よV are atomic in PV(C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The fact that PV(C0) is generated from the image of enriched Yoneda embedding under weighted colimits is [Hin21, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1], recalled as (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The functor PV : CatV → PrL V of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6 lands in PriL V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let f : C0 → D0 be a V-functor, and we need to show that f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' : PV(C0) → PV(D0) is internally left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='18 that PV(C0) is molecular with the image of よV C0 as atomic generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' As in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8, naturality of the enriched Yoneda embedding says that よV D0f ∼= f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='よV C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' sends the image of よV C0 to the image of よV D0 which are atomic in PV(D0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' sends a collection of atomic generators to atomic objects, so it is internally left adjoint by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The partial adjunctions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6 restricts to an adjunction PV : CatV ⇄ PriL V :(−)at with unit the enriched Yoneda embedding よV : id ⇒ PV(−)at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We need to check that for any C0 ∈ CatV and D ∈ PriL V , the map homiL V (PV(C0), D) (−)at −−−→ homV(PV(C0)at, Dat) (よV)∗ −−−−→ homV(C0, Dat) (4) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that homL V(PV(C0), D) χ−→ homV(χ(PV(C0)), χ(D)) (よV)∗ −−−−→ homV(C0, χ(D)) (5) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, both the first and last spaces in (4) are a collection of connected components of the first and last spaces in (5), showing that the composition in (4) is an inclusion of connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To finish the argument, we need to show that the composition in (4) hits every connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' To that end, let f : C0 → Dat be a V-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We can post-compose it with the 23 inclusion V-functor Dat → D, and using (5) we get ˜f : PV(C0) → D in PrL V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' It is left to show that ˜f is internally left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Recall that the image of the enriched Yoneda embedding forms a collection of atomic generators by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Furthermore, by construction, ˜f(よV(X)) = f(X) ∈ Dat is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, we have shown that ˜f sends a collection of atomic generators to atomics, so it is indeed internally left adjoint by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We extend our notations from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='7: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For f : C0 → D0 a morphism in CatV, consider the internally left adjoint functor f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' : PV(C0) → PV(D0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' It has a right adjoint in PrL V, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' a V-linear left and right adjoint functor denoted f ⊛ : PV(D0) → PV(C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' This functor thus has a further lax V-linear right adjoint denoted f⊛ : PV(C0) → PV(D0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Using Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3 we get the composition χ PV : CatV PV −−→ PriL V χ−→ (� CatV)LL, allowing to pass the (double) adjunction to � CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For f : C0 → D0 a morphism in CatV, we get a double adjunction χ(f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=') ⊣ χ(f ⊛) ⊣ χ(f⊛) in � CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let f : C0 → D0 in CatV, then f ⊛(G)(X) ∼= G(f(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='8 we have f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='よV ∼= よVf in FunV(C0, PV(D0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We thus get f ⊛(G)(X) ∼= homV(よV(X), f ⊛(G)) ∼= homV(f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' (よV(X)), G) ∼= homV(よV(f(X)), G) ∼= G(f(X)), where the first and last steps follow from the enriched Yoneda lemma of (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9, the second step is by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='21, and the third step is by the isomorphism above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 6 Heine’s Enriched Yoneda Embedding Independently of Hinich’s enriched Yoneda embedding, in [Hei20], Heine defines an enriched Yoneda embedding as well, which satisfies the exact same universal property appearing in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='1 ([Hei20, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Let C0 ∈ CatV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' There is a V-natural transformation よV Heine : C0 → χ(PV(C0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For every D ∈ PrL V, the composition homL V(PV(C0), D) χ−→ homV(χ(PV(C0)), χ(D)) (よV Heine)∗ −−−−−−→ homV(C0, χ(D)) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We also note these versions of the enriched Yoneda embedding agree point-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In particular, Heine’s version also satisfies the enriched Yoneda lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' For every X ∈ C0, there is an isomorphism よV Heine(X) ∼= よV(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' In particular, homV(よV Heine(X), −): PV(C0) → V is given by evaluation at X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 24 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Consider the composition C≃ 0 よ −→ Fun(Cop,≃ 0 , S) 1V⊗− −−−−→ Fun(Cop,≃ 0 , V) free −−→ LModCop 0 (Fun(Cop,≃ 0 , V)) = PV(C0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' By [Hin20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='6] and [Hei20, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='5], よV(X) and よV Heine(X) (respectively) are the image of X under this composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The second part then follows from (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Note that these two results do not show that Hinich’s and Heine’s enriched Yoneda embeddings agree as functors, but rather only up to an automorphism of PV(C0) (which may not be the identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' See Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2 for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Nevertheless, we now explain that the main results of this paper hold for Heine’s enriched Yoneda embedding as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' The proofs in Section 3 and Section 5 have relied on the the universal property of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='12 and the enriched Yoneda lemma of (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9 for よV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' These two results also hold for よV Heine by the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' We also used weighted colimits, and their relationship with the enriched Yoneda embedding appearing in (2) and (3) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' However, this was only used in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='18, whose statement is about the image of the enriched Yoneda embedding which is the same for よV and よV Heine by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='2, and in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='15, whose statement does not involve the enriched Yoneda embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Thus, we see that the results of Section 3 and Section 5 hold for Heine’s enriched Yoneda embedding as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Notably, the analogue of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content='20 gives an adjunction PV Heine : CatV ⇄ PriL V :(−)at with unit Heine’s enriched Yoneda embedding よV Heine : id ⇒ PV Heine(−)at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' References [Bar22] Shaul Barkan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' Explicit Square Zero Obstruction Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAyT4oBgHgl3EQftvmX/content/2301.00601v1.pdf'} +page_content=' 2022.' metadata={'source': 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a/QdAzT4oBgHgl3EQfIvvq/content/tmp_files/2301.01069v1.pdf.txt b/QdAzT4oBgHgl3EQfIvvq/content/tmp_files/2301.01069v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..35ee32596955f4cc5ef889126dbed706c18aebb0 --- /dev/null +++ b/QdAzT4oBgHgl3EQfIvvq/content/tmp_files/2301.01069v1.pdf.txt @@ -0,0 +1,817 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 +1 +Saliency-Aware Spatio-Temporal Artifact Detection +for Compressed Video Quality Assessment +Liqun Lin, Yang Zheng, Weiling Chen, Chengdong Lan, Tiesong Zhao +Abstract—Compressed videos often exhibit visually annoying +artifacts, known as Perceivable Encoding Artifacts (PEAs), which +dramatically degrade video visual quality. Subjective and ob- +jective measures capable of identifying and quantifying various +types of PEAs are critical in improving visual quality. In this +paper, we investigate the influence of four spatial PEAs (i.e. +blurring, blocking, bleeding, and ringing) and two temporal PEAs +(i.e. flickering and floating) on video quality. For spatial artifacts, +we propose a visual saliency model with a low computational +cost and higher consistency with human visual perception. In +terms of temporal artifacts, self-attention based TimeSFormer is +improved to detect temporal artifacts. Based on the six types of +PEAs, a quality metric called Saliency-Aware Spatio-Temporal +Artifacts Measurement (SSTAM) is proposed. Experimental re- +sults demonstrate that the proposed method outperforms state- +of-the-art metrics. We believe that SSTAM will be beneficial for +optimizing video coding techniques. +Index Terms—Video quality assessment, saliency detection, +Perceivable Encoding Artifacts (PEAs), compression artifact. +I. INTRODUCTION +W +ITH the growing demand of users, High-Definition +(HD)/Ultra-HD (UHD) video is being used extensively. +Due to the large amount of data, HD/UHD videos are encoded +to meet the limited transmission bandwidth and storage space. +The encoded videos have visually annoying Perceivable En- +coding Artifacts (PEAs) which lead to visual quality degra- +dation [1]. Thus, Video Quality Assessment (VQA) based on +PEAs is necessary for guiding the optimization of video coding +to reduce distortion and improving the visual experience of +users. +VQA can be divided into subjective and objective meth- +ods. For subjective VQA, Mean Opinion Score (MOS) and +Different Mean Opinion Score (DMOS) suggested by the +International Telecommunications Union (ITU) [2] are used to +reflect video subjective quality. Subjective quality assessment +is accurate and reliable, but it is time-consuming and labor- +intensive. +There are three types of objective VQA methods: Full +Reference (FR), Reduced Reference (RR), and No Reference +This work was mainly supported by the Natural Science Foundation of +China under Grant 62171134 and Grant 61901119. It was also supported by +the Natural Science Foundation of Fujian Province under Grant 2022J02015 +and Grant 2022J05117. (Corresponding author: C. Lan.) +L. Lin and T. Zhao are with the Fujian Key Lab for Intelligent Processing +and Wireless Transmission of Media Information, College of Physics and +Information Engineering, Fuzhou University, Fuzhou, Fujian 350116, China, +and also with Fujian Science & Technology Innovation Laboratory for Opto- +electronic Information of China (e-mails: {lin liqun, t.zhao}@fzu.edu.cn). +Y. Zheng, W. Chen, C. Lan are with the Fujian Key Lab for Intelligent +Processing and Wireless Transmission of Media Information, College of +Physics and Information Engineering, Fuzhou University, Fuzhou, Fujian +350116, China (e-mails: {211127117, weiling.chen, lancd}@fzu.edu.cn). +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 1. Examples of PEAs. (a) Blocking: +Blocking: +Blocking: Popular encoders are block-based, +and all compression processes take place within non-overlapping blocks. This +usually results in spurious discontinuities across block boundaries. The visual +appearance of blocking may vary depending on the area of visual discontinuity +[3]. (b) Blurring: +Blurring: +Blurring: Video signal reconstruction results in a potentially significant +loss of high frequency energy, which can lead to visual blurring. Perceptually, +blurring usually manifests itself as a loss of spatial detail or sharpness of edge +or texture areas in the image [4]. (c) Color bleeding: +Color bleeding: +Color bleeding: It shows when the edges +of one color in the video unintentionally overflow or overlap into another color +[5]. (d) Ringing: +Ringing: +Ringing: It appears in the form of “halos”, “rings” or “shadows” near +sharp edges [6]. (e) Flickering: +Flickering: +Flickering: It refers to frequent changes in brightness or +chromaticity along the time dimension. It is very eye-catching and annoying +to the viewer. (f) Floating: +Floating: +Floating: It is the illusory movement of certain areas while +the surrounding areas remain stationary. Visually, the areas appear to float just +above the surrounding background. +(NR) [7]. The FR-VQA methods measure video quality by +calculating the difference between the original videos and the +compressed videos. Some FR methods like Peak Signal to +Noise Ratio (PSNR) as well as Structural SIMilarity (SSIM) +[8] are widely used. The RR methods evaluate the quality of +the compressed video by utilizing a few features of the original +video. RR-VQA metrics like Spatial-Temporal RR Entropy +Differences (STRRED) [9] and Spatial Effective Entropy Dif- +ference Quality Assessment (SpEED-QA) [10] also exhibited +positive performance. However, these methods call for original +video data that is typically absent from VQA, reducing the +generalizability of models. +The most popular quality assessment scheme is NR-VQA +because original videos are out of reach for end users. Some +NR-VQAs such as Video Intrinsic Integrity and Distortion +Evaluation Oracle (VIIDEO) [11] and Two-Level Video Qual- +ity Model (TLVQM) [12] improved the performance of video +quality evaluation from different perspectives. Recently, more +attention has been paid to the influence of compression ar- +tifacts on the compressed video quality. For instance, Bovik +[13] took into account video coding artifacts without specific +classification. In [6], an NR-VQA method was developed by +exploiting ringing and blocking artifacts. More than two types +of spatial PEAs were considered in TLVQM [12] and Saliency- +arXiv:2301.01069v1 [eess.IV] 3 Jan 2023 + +MEG7Visual +nfoiow-upselaten +EC 15938-6:2001/PDAM4[N11611 +IEC15938-6:20D1/FPDAW6 +Video Sienature Tocls Iheto-1 +IEC15938-7:2001/PDAMGON11615JOURNAL OF LATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 +2 +Fig. 2. +The overall framework of SSTAM. In spatial PEA detection, each video frame sequentially passes through the spatial PEA module to detect the +intensities of spatial PEAs Is +v. During the detection of temporal PEAs, z(l−1) in Transformer encoder block l ∈ [1, L] passes through the temporal attention +module, the spatial attention module, and the MLP in sequence. The output of temporal PEA module is It +v. +Aware Artifact Measurement (SAAM) [14]. The effects of +blurring, blocking, and flickering artifacts on video quality are +mentioned in [15], but no specific detection is performed. +As mentioned above, several VQA methods have taken +into account compression artifacts, but most of them involve +only a few types of compression artifacts. Moreover, these +methods were very computationally intensive [14]. To address +these issues, we propose a method named Saliency-Aware +Spatio-Temporal Artifacts Measurement (SSTAM). Here, we +consider six typical PEAs which have a great impact on +compressed video quality [16], as illustrated in Fig. 1. The +main contributions are summarized as follows. +(1)An NR-VQA method is proposed based on PEA de- +tection. The PEA detection module accurately recognizes six +typical types of PEAs. Based on the PEA detection module, +the PEA intensities are obtained to predict video quality. +(2)A saliency model that leverages visual saliency to achieve +highly consistent with human visual perception is put forward. +It makes our method more relevant to quality evaluation of +compressed videos, while excluding regions that have very lit- +tle impact on the video visual quality to reduce computational +consumption. +(3)A comprehensive validation on four publicly available +databases is conducted to verify our proposed SSTAM. The +results reveal the promising performance of SSTAM in video +quality evaluation. +II. PEA-BASED VIDEO QUALITY INDEX +This section systematically introduces the framework of +SSTAM, whose overall structure is depicted in Fig. 2. SSTAM +consists of a spatial PEA detection model, a temporal PEA +detection model and a video quality prediction model. +A. Spatial PEA Detection Model +The spatial PEA detection model can be divided into visual +saliency model and spatial PEA detection. First, the visual +saliency model extracts the saliency areas. This model can +exclude areas that have little impact on the visual experience, +making it better suited to human visual perception while +reducing computational complexity. Then, spatial PEAs within +the saliency regions are obtained by the spatial PEA detection +part. +Visual Saliency Model: +Visual Saliency Model: +Visual Saliency Model: An essential component of the Hu- +man Visual System (HVS) and a determinant of the perceived +quality of videos is visual saliency. Saliency object detection +identifies the most visually distinctive objects in an image +or video. Focusing on saliency regions can also significantly +reduce the computational burden on the network. Existing +saliency models are focused on the extraction of a specific +class of targets [17]–[20]. For videos, changes in target mo- +tion and appearance contrast are more attention-grabbing, so +extracted saliency targets should be more oriented to such +regions [17]. Given the characteristics of compression artifacts, +we focus more on channel attention, which is also known +as feature attention. Channel linkage is used to reflect the +importance of each feature. In addition, spatial attention is +employed to complement channel attention. The channel and +spatial features are extracted in this work using Convolutional +Block Attention Module (CBAM) [21]. +However, it is important to combine the extracted channel +and spatial features with semantic information to enable the +network to better identify saliency regions. Though high-level +semantic features are helpful for locating saliency objects, +semantic features at lower levels can also provide essential +local details. The low-level semantic information is often +diluted in the top-down process. The absence of local in- +formation often disturbs the localization of saliency objects. +Drawing inspiration from U-Net [22] and U2-Net [23], we +extract multi-level features and fuse them together for better +detection. For the information at each level to be used properly, +Global Guiding Flows (GGFs) [24] is utilized to determine the +weights of the global guidance information for each part of the +top-down path. It can integrate the information generated by +the Pyramid Pooling Module (PPM) [24] with the feature maps +of the top-down pathway output. Inspired by this, we present +an improved network U2-Convolutional Pyramid Network +(U2-CPNet), incorporating the attention mechanism and global +guidance. +In addition, the loss function is also improved to make the + +Spatial PEA Detection +DenseNet-PR +>U>甲 +>Di +Intensity +Saliency +calculation +Binary +72×72 +W +map S; +imageBi +patches Pij +个 +Video Quality Prediction +Channel +Spatial +PPM +Bagging+SVR +>Q +Atten. +Atten. +GGFs +U2-CPNet +w = s(q,k) +Time&Space Attention +It +self attention +(1-1) +z' = S(w,v) +(0) +个 +Z(p,t) +weight w +scaled dot-product +output z' +SM +Linear +MLP +Embedding +M +Layer +Lx +>≥() +p +Norm. +Temporal PEA DetectionJOURNAL OF LATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 +3 +model more target to identify the saliency regions. During +the training process, the Binary Cross Entropy (BCE) loss +is pixel-level, which does not consider the labels of adjacent +regions and gives equal weights to the pixels in saliency +regions and the background. This loss function aims to achieve +convergence on all pixels, and does not fit well with the +purpose of extracting saliency regions. Loss function should +pay more attention to the correlation between adjacent pixels +and the global aspect during training. In this paper, the BCE +function is used to allow the network to fully acquire pixel- +level data. The global background and the local information +among neighboring pixels are addressed using intersection +over union (IoU) and structural similarity (SSIM), respectively. +IoU enables the network to attend more to the global context +of the saliency map and concentrate on the foreground targets +[25]. The expression of BCE and IoU loss function is shown +in Eq. 1 and Eq. 2. +Lbce = −(1−G(a,b)) log (1 − S(a,b))−G(a,b) log S(a,b), (1) +Liou = +H +� +a=1 +W +� +b=1 +S(a,b)G(a,b) +H +� +a=1 +W +� +b=1 +[S(a,b) + G(a,b) − S(a,b)G(a,b)] +, +(2) +where a and b represent the vertical and horizontal coordinates +of the pixel in each video frame. H and W indicate the height +and width of the video frame. G(a,b) ∈ {0, 1} denotes the +real label of pixel (a, b) and S(a,b) ∈ {0, 1} is the predicted +probability of the saliency object. +SSIM deals with the structural relationships among adjacent +pixels in the image content. We mix it into the training loss to +highlight the structural information of the saliency objects. It +ensures that there is enough space for feature exploitation. +While the probability is reduced to zero in the detected +regions of saliency, the backdrop is perceived visually as being +considerably cleaner. The expression of the SSIM loss function +is shown in Eq. 3. +Lssim = 1 − +(2µGµP + C2 +1)(2σGP + C2 +2) +(µ2 +G + µ2 +P + C2 +1)(σ2 +G + σ2 +P + C2 +2), +(3) +where the subscript G denotes the binary ground truth mask, +and the subscript P is the predicted probabilistic map mask. +The µG and µP refer to the mean value, σG and σP represent +the standard deviation of both, and σGP indicates as their +covariance. C1 and C2 are 0.01 and 0.03 [26], respectively. +They are employed to avoid the division by zero error. +The strategy for mixing losses during model training is +shown in Eq. 4, where L is the loss of the training process, +and the weight of each component is 1. +L = Lbce + Liou + Lssim. +(4) +Spatial PEA Detection: +Spatial PEA Detection: +Spatial PEA Detection: As for spatial PEA recognition, +DenseNet for PEA Recognition (DenseNet- PR) [16] can +alleviate the problem of gradient disappearance and improve +feature transmission while greatly reducing the number of pa- +rameters. Video frames are pre-processed to fit the DenseNet- +PR input: after obtaining the saliency map Si extracted by +U2-CPNet, we binarize it into Bi, then map the Bi back +to the video frame and partition the saliency regions in the +video frame into 72 × 72 patches. Four types of spatial PEAs +recognition models are trained to detect the PEAs in patches. +The probability list of each 72 × 72 patch, namely Iij, is +obtained. From which the intensity of each spatial PEA in +the whole video sequence is finally calculated through the +following equation. +Is +v = +1 +NfNp +Nf +� +i=1 +Np +� +j=1 +Iij, +(5) +where Np refers to the total number of patches in the saliency +regions of each frame. Nf denotes the total number of video +frames. Is +v represents the intensity value of spatial PEAs in +each video. Iij, as mentioned above, denotes the probability +of certain PEA of each 72 × 72 patch. i is the i-th frame of +the video and j is the j-th patch of the video frame. +B. Temporal PEA Detection Model +Video-specific tasks consider both spatial and temporal in- +formation. Compared to other video understanding tasks, tem- +poral PEAs detection is more dependent on spatio-temporal +features which are more challenging for the network to cap- +ture. Meanwhile, there is a great lack of research on the +analysis and detection of temporal PEAs. Therefore, it is +imperative to exploit temporal PEA detection. Self-attention +mechanism can capture both local and global dependencies +over a long range by directly comparing all spatio-temporal +features. Hence, self-attention based TimeSFormer [27] is +improved and applied to the detection of temporal PEAs. Since +the spatial features of temporal PEA are relatively insignifi- +cant, we focus more on temporal features when training the +network. +Training data of flickering and floating is derived from +the database PEA265 [16]. Temporal PEA detection model +is shown in Fig. 2. The intensity of the temporal PEA for +each video is finally computed as follows. +It +v = SoftMax(MLP(LN( +Nc +� +j=1 +Nt +� +i=1 +TSFj(Fi)))), +(6) +where we divide the consecutive Nt frames into a clip and +Nc is the total number of clips for each video. TSF() is the +processing of TimeSFormer and LN() refers to Layer Norm. +It +v is the intensity of temporal PEA in each video. +C. Video Quality Prediction +Predicting video quality requires effectively mapping the +correlation between the spatial and temporal artifacts detected +in the previous sections and the subjective quality scores. +We use the Bootstrap Aggregating (Bagging) [28] method in +integrated learning. Multiple base learners using different parts +of the training set are combined to achieve better generaliza- +tion performance than a single learner. Assuming a complete +dataset, D, we match the MOS|DMOS values to the intensities +of the six PEAs of each video in D. The matching process is +shown in the following equation. +D = {(IV1, MOS1|DMOS1), . . . , (IVk, MOSm|DMOSm)}, +(7) + +JOURNAL OF LATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 +4 +where MOSm|DMOSm denotes the subjective quality score +of the m-th compressed video, IVk is the summary of the +intensities of all spatial and temporal PEAs. +For training, we divide the dataset into a training set and a +testing set in a ratio of 8:2, followed by dividing the training +set into ten sub-training sets evenly. Support Vector Regression +(SVR) models are trained using these ten sub-training sets. The +performance of these base learners is evaluated by the Pearson +Linear Correlation Coefficient (PLCC) between the predicted +and true quality scores. The top three learners in PLCC scores +among the ten base learners are chosen to predict the video +quality Qv. The expression for Qv is shown below. +Qv = +n +� +i=1 +ωiyi(x), +(8) +n +� +i=1 +ωi = 1 +ωi ∈ [0, 1), +(9) +where x is testing set. yi(x) represents the prediction output of +the i-th base learner. ωi is the weight of the i-th base learner. +n denotes the number of base learners, which value is 10. +III. EXPERIMENTS AND DISCUSSIONS +Experiments and evaluations are carried out on four com- +monly used and publicly accessible Video Quality Databases +(VQDs), including LIVE [29], CSIQ [30], IVP [31], and +FERIT-RTRK [32], to verify the generalizability of SSTAM. +LIVE contains 40 compressed videos with the resolution of +768 × 432. CSIQ consists of 36 compressed videos at a +resolution of 832 × 480. IVP and FERIT-RTRK contain 40 +and 30 compressed videos with the resolution of 1920×1080, +respectively. In this work, we matched the six compression +artifacts intensities of compressed videos from these four +databases with their MOS|DMOS to form four complete +datasets for evaluation. We compared SSTAM with several +classical and popular quality assessment methods to verify its +performance. These methods are PSNR, SSIM [8], MS-SSIM +[33], STRRED [9], SpEED-QA [10], VIIDEO [11], TLVQM +[12], SAAM [14], BRISQUE [34] and NIQE [35]. PLCC and +Spearman Rank Correlation Coefficient (SRCC) values are +employed to benchmark the performance of these methods. +Performance Comparison: +Performance Comparison: +Performance Comparison: The results can be seen in Tables +I and II. The bolded result corresponds to the best performing +method in each database. It is clear that our model is highly +competitive on these four databases. Weighing the quantity +of compressed videos in each database gets the overall per- +formance. As can be seen from the last column, the total +performance of SSTAM is better than the other methods. +The high improvement of SAAM and SSTAM compared to +other methods is because they fully consider the influence +of PEA on compressed video quality. This experiment also +fully demonstrates that there is a strong correlation between +the PEAs and the subjective quality scores of compressed +videos. We find that methods containing a saliency model +predict video quality more effectively in FERIT-RTRK. Since +the saliency objects in FERIT-RTRK are relatively few and +large, the network can extract the saliency regions of the video +TABLE I +PERFORMANCE COMPARISON IN TERMS OF PLCC. +Methods +LIVE +CSIQ +IVP +FERIT-RTRK +Overall +PSNR +0.5735 +0.8220 +0.7998 +0.7756 +0.7383 +SSIM +0.6072 +0.8454 +0.8197 +0.6870 +0.7406 +MS-SSIM +0.6855 +0.8782 +0.8282 +0.8724 +0.8105 +STRRED +0.8392 +0.8772 +0.5947 +0.8425 +0.7823 +SpEED-QA +0.7933 +0.8554 +0.6822 +0.6978 +0.7586 +BRISQUE +0.2154 +0.5526 +0.2956 +0.7653 +0.4335 +NIQE +0.3311 +0.5350 +0.3955 +0.5817 +0.4505 +VIIDEO +0.6829 +0.7211 +0.4358 +0.3933 +0.5651 +TLVQM +0.7511 +0.7740 +- +- +0.7619 +SAAM +0.9023 +0.9244 +0.8717 +0.9499 +0.9091 +SSTAM +0.9490 +0.9490 +0.9490 0.9523 +0.9523 +0.9523 0.9478 +0.9478 +0.9478 +0.9769 +0.9769 +0.9769 +0.9552 +0.9552 +0.9552 +TABLE II +PERFORMANCE COMPARISON IN TERMS OF SRCC. +Methods +LIVE +CSIQ +IVP +FERIT-RTRK +Overall +PSNR +0.4146 +0.8028 +0.8154 +0.7685 +0.6928 +SSIM +0.5677 +0.8440 +0.8049 +0.7236 +0.7328 +MS-SSIM +0.6773 +0.9465 +0.7917 +0.8508 +0.8107 +STRRED +0.8358 0.9770 +0.9770 +0.9770 +0.8595 +0.8310 +0.8761 +SpEED-QA +0.7895 +0.9639 +0.8812 +0.7945 +0.8587 +BRISQUE +0.2638 +0.5655 +0.1051 +0.7574 +0.3961 +NIQE +0.1769 +0.5012 +0.2351 +0.4855 +0.3362 +VIIDEO +0.6593 +0.7153 +0.1621 +0.3177 +0.4667 +TLVQM +0.7338 +0.7956 +- +- +0.7631 +SAAM +0.8691 +0.8810 +0.8413 +0.9429 +0.8796 +SSTAM +0.9201 +0.9201 +0.9201 +0.9177 0.9048 +0.9048 +0.9048 +0.9823 +0.9823 +0.9823 +0.9281 +0.9281 +0.9281 +TABLE III +ABLATION EXPERIMENTS. +Indicators +SAAM +SAAM2.0 +SSTAM +PLCC +0.9091 +0.9475 +0.9552 +SRCC +0.8796 +0.9109 +0.9281 +more precisely. Accordingly, a better extraction of saliency +regions contributes significantly to the performance of quality +assessment. +Ablation Experiments: +Ablation Experiments: +Ablation Experiments: To further verify the improvement +of our proposed strategies for video quality assessment, we +compared the effects of saliency model and temporal PEAs +separately. Visual saliency model called Attentive CNN-LSTM +Network (ACLNet) [36] was applied in SAAM. To evaluate +the performance, we substitute ACLNet in SAAM with U2- +CPNet. After changing the saliency network, we refer to the +method as SAAM2.0. As shown in Table III, our saliency +model performs better in recognizing saliency regions and +enhances the performance of the network. The comparisons +of SAAM2.0 and SSTAM reveal that temporal PEAs have an +effect. +IV. CONCLUSIONS +This paper proposes a compressed video quality index based +on saliency-aware spatio-temporal artifact detection. 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Xie, et al, “Revisiting video saliency prediction +in the deep learning era,” in IEEE Transactions on Pattern Analysis and +Machine Intelligence, vol. 43, no. 1, pp. 220–237, 2019. + diff --git a/QdAzT4oBgHgl3EQfIvvq/content/tmp_files/load_file.txt b/QdAzT4oBgHgl3EQfIvvq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28bbafbfa686626afc4f5eff54a6a0fe02e4d436 --- /dev/null +++ b/QdAzT4oBgHgl3EQfIvvq/content/tmp_files/load_file.txt @@ -0,0 +1,627 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf,len=626 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 18, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 9, SEPTEMBER 2020 1 Saliency-Aware Spatio-Temporal Artifact Detection for Compressed Video Quality Assessment Liqun Lin, Yang Zheng, Weiling Chen, Chengdong Lan, Tiesong Zhao Abstract—Compressed videos often exhibit visually annoying artifacts, known as Perceivable Encoding Artifacts (PEAs), which dramatically degrade video visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Subjective and ob- jective measures capable of identifying and quantifying various types of PEAs are critical in improving visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' In this paper, we investigate the influence of four spatial PEAs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' blurring, blocking, bleeding, and ringing) and two temporal PEAs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' flickering and floating) on video quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' For spatial artifacts, we propose a visual saliency model with a low computational cost and higher consistency with human visual perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' In terms of temporal artifacts, self-attention based TimeSFormer is improved to detect temporal artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Based on the six types of PEAs, a quality metric called Saliency-Aware Spatio-Temporal Artifacts Measurement (SSTAM) is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Experimental re- sults demonstrate that the proposed method outperforms state- of-the-art metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' We believe that SSTAM will be beneficial for optimizing video coding techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Index Terms—Video quality assessment, saliency detection, Perceivable Encoding Artifacts (PEAs), compression artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' INTRODUCTION W ITH the growing demand of users, High-Definition (HD)/Ultra-HD (UHD) video is being used extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Due to the large amount of data, HD/UHD videos are encoded to meet the limited transmission bandwidth and storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The encoded videos have visually annoying Perceivable En- coding Artifacts (PEAs) which lead to visual quality degra- dation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Thus, Video Quality Assessment (VQA) based on PEAs is necessary for guiding the optimization of video coding to reduce distortion and improving the visual experience of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' VQA can be divided into subjective and objective meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' For subjective VQA, Mean Opinion Score (MOS) and Different Mean Opinion Score (DMOS) suggested by the International Telecommunications Union (ITU) [2] are used to reflect video subjective quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Subjective quality assessment is accurate and reliable, but it is time-consuming and labor- intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' There are three types of objective VQA methods: Full Reference (FR), Reduced Reference (RR), and No Reference This work was mainly supported by the Natural Science Foundation of China under Grant 62171134 and Grant 61901119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' It was also supported by the Natural Science Foundation of Fujian Province under Grant 2022J02015 and Grant 2022J05117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (Corresponding author: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Lan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=') L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Lin and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Zhao are with the Fujian Key Lab for Intelligent Processing and Wireless Transmission of Media Information, College of Physics and Information Engineering, Fuzhou University, Fuzhou, Fujian 350116, China, and also with Fujian Science & Technology Innovation Laboratory for Opto- electronic Information of China (e-mails: {lin liqun, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='zhao}@fzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Zheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Lan are with the Fujian Key Lab for Intelligent Processing and Wireless Transmission of Media Information, College of Physics and Information Engineering, Fuzhou University, Fuzhou, Fujian 350116, China (e-mails: {211127117, weiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='chen, lancd}@fzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Examples of PEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (a) Blocking: Blocking: Blocking: Popular encoders are block-based, and all compression processes take place within non-overlapping blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' This usually results in spurious discontinuities across block boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The visual appearance of blocking may vary depending on the area of visual discontinuity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (b) Blurring: Blurring: Blurring: Video signal reconstruction results in a potentially significant loss of high frequency energy, which can lead to visual blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Perceptually, blurring usually manifests itself as a loss of spatial detail or sharpness of edge or texture areas in the image [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (c) Color bleeding: Color bleeding: Color bleeding: It shows when the edges of one color in the video unintentionally overflow or overlap into another color [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (d) Ringing: Ringing: Ringing: It appears in the form of “halos”, “rings” or “shadows” near sharp edges [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (e) Flickering: Flickering: Flickering: It refers to frequent changes in brightness or chromaticity along the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' It is very eye-catching and annoying to the viewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (f) Floating: Floating: Floating: It is the illusory movement of certain areas while the surrounding areas remain stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Visually, the areas appear to float just above the surrounding background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (NR) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The FR-VQA methods measure video quality by calculating the difference between the original videos and the compressed videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Some FR methods like Peak Signal to Noise Ratio (PSNR) as well as Structural SIMilarity (SSIM) [8] are widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The RR methods evaluate the quality of the compressed video by utilizing a few features of the original video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' RR-VQA metrics like Spatial-Temporal RR Entropy Differences (STRRED) [9] and Spatial Effective Entropy Dif- ference Quality Assessment (SpEED-QA) [10] also exhibited positive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' However, these methods call for original video data that is typically absent from VQA, reducing the generalizability of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The most popular quality assessment scheme is NR-VQA because original videos are out of reach for end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Some NR-VQAs such as Video Intrinsic Integrity and Distortion Evaluation Oracle (VIIDEO) [11] and Two-Level Video Qual- ity Model (TLVQM) [12] improved the performance of video quality evaluation from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Recently, more attention has been paid to the influence of compression ar- tifacts on the compressed video quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' For instance, Bovik [13] took into account video coding artifacts without specific classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' In [6], an NR-VQA method was developed by exploiting ringing and blocking artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' More than two types of spatial PEAs were considered in TLVQM [12] and Saliency- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='01069v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='IV] 3 Jan 2023 MEG7Visual nfoiow-upselaten EC 15938-6:2001/PDAM4[N11611 IEC15938-6:20D1/FPDAW6 Video Sienature Tocls Iheto-1 IEC15938-7:2001/PDAMGON11615JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 18, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 9, SEPTEMBER 2020 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The overall framework of SSTAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' In spatial PEA detection, each video frame sequentially passes through the spatial PEA module to detect the intensities of spatial PEAs Is v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' During the detection of temporal PEAs, z(l−1) in Transformer encoder block l ∈ [1, L] passes through the temporal attention module, the spatial attention module, and the MLP in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The output of temporal PEA module is It v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Aware Artifact Measurement (SAAM) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The effects of blurring, blocking, and flickering artifacts on video quality are mentioned in [15], but no specific detection is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' As mentioned above, several VQA methods have taken into account compression artifacts, but most of them involve only a few types of compression artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Moreover, these methods were very computationally intensive [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' To address these issues, we propose a method named Saliency-Aware Spatio-Temporal Artifacts Measurement (SSTAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Here, we consider six typical PEAs which have a great impact on compressed video quality [16], as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The main contributions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (1)An NR-VQA method is proposed based on PEA de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The PEA detection module accurately recognizes six typical types of PEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Based on the PEA detection module, the PEA intensities are obtained to predict video quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (2)A saliency model that leverages visual saliency to achieve highly consistent with human visual perception is put forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' It makes our method more relevant to quality evaluation of compressed videos, while excluding regions that have very lit- tle impact on the video visual quality to reduce computational consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (3)A comprehensive validation on four publicly available databases is conducted to verify our proposed SSTAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The results reveal the promising performance of SSTAM in video quality evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' PEA-BASED VIDEO QUALITY INDEX This section systematically introduces the framework of SSTAM, whose overall structure is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' SSTAM consists of a spatial PEA detection model, a temporal PEA detection model and a video quality prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Spatial PEA Detection Model The spatial PEA detection model can be divided into visual saliency model and spatial PEA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' First, the visual saliency model extracts the saliency areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' This model can exclude areas that have little impact on the visual experience, making it better suited to human visual perception while reducing computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Then, spatial PEAs within the saliency regions are obtained by the spatial PEA detection part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Visual Saliency Model: Visual Saliency Model: Visual Saliency Model: An essential component of the Hu- man Visual System (HVS) and a determinant of the perceived quality of videos is visual saliency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Saliency object detection identifies the most visually distinctive objects in an image or video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Focusing on saliency regions can also significantly reduce the computational burden on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Existing saliency models are focused on the extraction of a specific class of targets [17]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' For videos, changes in target mo- tion and appearance contrast are more attention-grabbing, so extracted saliency targets should be more oriented to such regions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Given the characteristics of compression artifacts, we focus more on channel attention, which is also known as feature attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Channel linkage is used to reflect the importance of each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' In addition, spatial attention is employed to complement channel attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The channel and spatial features are extracted in this work using Convolutional Block Attention Module (CBAM) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' However, it is important to combine the extracted channel and spatial features with semantic information to enable the network to better identify saliency regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Though high-level semantic features are helpful for locating saliency objects, semantic features at lower levels can also provide essential local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The low-level semantic information is often diluted in the top-down process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The absence of local in- formation often disturbs the localization of saliency objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Drawing inspiration from U-Net [22] and U2-Net [23], we extract multi-level features and fuse them together for better detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' For the information at each level to be used properly, Global Guiding Flows (GGFs) [24] is utilized to determine the weights of the global guidance information for each part of the top-down path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' It can integrate the information generated by the Pyramid Pooling Module (PPM) [24] with the feature maps of the top-down pathway output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Inspired by this, we present an improved network U2-Convolutional Pyramid Network (U2-CPNet), incorporating the attention mechanism and global guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' In addition, the loss function is also improved to make the Spatial PEA Detection DenseNet-PR >U>甲 >Di Intensity Saliency calculation Binary 72×72 W map S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' imageBi patches Pij 个 Video Quality Prediction Channel Spatial PPM Bagging+SVR >Q Atten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Atten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=" GGFs U2-CPNet w = s(q,k) Time&Space Attention It self attention (1-1) z' = S(w,v) (0) 个 Z(p,t) weight w scaled dot-product output z' SM Linear MLP Embedding M Layer Lx >≥() p Norm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Temporal PEA DetectionJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 18, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 9, SEPTEMBER 2020 3 model more target to identify the saliency regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' During the training process, the Binary Cross Entropy (BCE) loss is pixel-level, which does not consider the labels of adjacent regions and gives equal weights to the pixels in saliency regions and the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' This loss function aims to achieve convergence on all pixels, and does not fit well with the purpose of extracting saliency regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Loss function should pay more attention to the correlation between adjacent pixels and the global aspect during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' In this paper, the BCE function is used to allow the network to fully acquire pixel- level data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The global background and the local information among neighboring pixels are addressed using intersection over union (IoU) and structural similarity (SSIM), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' IoU enables the network to attend more to the global context of the saliency map and concentrate on the foreground targets [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The expression of BCE and IoU loss function is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Lbce = −(1−G(a,b)) log (1 − S(a,b))−G(a,b) log S(a,b), (1) Liou = H � a=1 W � b=1 S(a,b)G(a,b) H � a=1 W � b=1 [S(a,b) + G(a,b) − S(a,b)G(a,b)] , (2) where a and b represent the vertical and horizontal coordinates of the pixel in each video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' H and W indicate the height and width of the video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' G(a,b) ∈ {0, 1} denotes the real label of pixel (a, b) and S(a,b) ∈ {0, 1} is the predicted probability of the saliency object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' SSIM deals with the structural relationships among adjacent pixels in the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' We mix it into the training loss to highlight the structural information of the saliency objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' It ensures that there is enough space for feature exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' While the probability is reduced to zero in the detected regions of saliency, the backdrop is perceived visually as being considerably cleaner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The expression of the SSIM loss function is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Lssim = 1 − (2µGµP + C2 1)(2σGP + C2 2) (µ2 G + µ2 P + C2 1)(σ2 G + σ2 P + C2 2), (3) where the subscript G denotes the binary ground truth mask, and the subscript P is the predicted probabilistic map mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The µG and µP refer to the mean value, σG and σP represent the standard deviation of both, and σGP indicates as their covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' C1 and C2 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='03 [26], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' They are employed to avoid the division by zero error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The strategy for mixing losses during model training is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 4, where L is the loss of the training process, and the weight of each component is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' L = Lbce + Liou + Lssim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' (4) Spatial PEA Detection: Spatial PEA Detection: Spatial PEA Detection: As for spatial PEA recognition, DenseNet for PEA Recognition (DenseNet- PR) [16] can alleviate the problem of gradient disappearance and improve feature transmission while greatly reducing the number of pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Video frames are pre-processed to fit the DenseNet- PR input: after obtaining the saliency map Si extracted by U2-CPNet, we binarize it into Bi, then map the Bi back to the video frame and partition the saliency regions in the video frame into 72 × 72 patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Four types of spatial PEAs recognition models are trained to detect the PEAs in patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The probability list of each 72 × 72 patch, namely Iij, is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' From which the intensity of each spatial PEA in the whole video sequence is finally calculated through the following equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Is v = 1 NfNp Nf � i=1 Np � j=1 Iij, (5) where Np refers to the total number of patches in the saliency regions of each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Nf denotes the total number of video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Is v represents the intensity value of spatial PEAs in each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Iij, as mentioned above, denotes the probability of certain PEA of each 72 × 72 patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' i is the i-th frame of the video and j is the j-th patch of the video frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Temporal PEA Detection Model Video-specific tasks consider both spatial and temporal in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Compared to other video understanding tasks, tem- poral PEAs detection is more dependent on spatio-temporal features which are more challenging for the network to cap- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Meanwhile, there is a great lack of research on the analysis and detection of temporal PEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Therefore, it is imperative to exploit temporal PEA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Self-attention mechanism can capture both local and global dependencies over a long range by directly comparing all spatio-temporal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Hence, self-attention based TimeSFormer [27] is improved and applied to the detection of temporal PEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Since the spatial features of temporal PEA are relatively insignifi- cant, we focus more on temporal features when training the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Training data of flickering and floating is derived from the database PEA265 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Temporal PEA detection model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The intensity of the temporal PEA for each video is finally computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' It v = SoftMax(MLP(LN( Nc � j=1 Nt � i=1 TSFj(Fi)))), (6) where we divide the consecutive Nt frames into a clip and Nc is the total number of clips for each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' TSF() is the processing of TimeSFormer and LN() refers to Layer Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' It v is the intensity of temporal PEA in each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Video Quality Prediction Predicting video quality requires effectively mapping the correlation between the spatial and temporal artifacts detected in the previous sections and the subjective quality scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' We use the Bootstrap Aggregating (Bagging) [28] method in integrated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Multiple base learners using different parts of the training set are combined to achieve better generaliza- tion performance than a single learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Assuming a complete dataset, D, we match the MOS|DMOS values to the intensities of the six PEAs of each video in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The matching process is shown in the following equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' D = {(IV1, MOS1|DMOS1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' , (IVk, MOSm|DMOSm)}, (7) JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 18, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 9, SEPTEMBER 2020 4 where MOSm|DMOSm denotes the subjective quality score of the m-th compressed video, IVk is the summary of the intensities of all spatial and temporal PEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' For training, we divide the dataset into a training set and a testing set in a ratio of 8:2, followed by dividing the training set into ten sub-training sets evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Support Vector Regression (SVR) models are trained using these ten sub-training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The performance of these base learners is evaluated by the Pearson Linear Correlation Coefficient (PLCC) between the predicted and true quality scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The top three learners in PLCC scores among the ten base learners are chosen to predict the video quality Qv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The expression for Qv is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Qv = n � i=1 ωiyi(x), (8) n � i=1 ωi = 1 ωi ∈ [0, 1), (9) where x is testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' yi(x) represents the prediction output of the i-th base learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' ωi is the weight of the i-th base learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' n denotes the number of base learners, which value is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' EXPERIMENTS AND DISCUSSIONS Experiments and evaluations are carried out on four com- monly used and publicly accessible Video Quality Databases (VQDs), including LIVE [29], CSIQ [30], IVP [31], and FERIT-RTRK [32], to verify the generalizability of SSTAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' LIVE contains 40 compressed videos with the resolution of 768 × 432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' CSIQ consists of 36 compressed videos at a resolution of 832 × 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' IVP and FERIT-RTRK contain 40 and 30 compressed videos with the resolution of 1920×1080, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' In this work, we matched the six compression artifacts intensities of compressed videos from these four databases with their MOS|DMOS to form four complete datasets for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' We compared SSTAM with several classical and popular quality assessment methods to verify its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' These methods are PSNR, SSIM [8], MS-SSIM [33], STRRED [9], SpEED-QA [10], VIIDEO [11], TLVQM [12], SAAM [14], BRISQUE [34] and NIQE [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' PLCC and Spearman Rank Correlation Coefficient (SRCC) values are employed to benchmark the performance of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Performance Comparison: Performance Comparison: Performance Comparison: The results can be seen in Tables I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The bolded result corresponds to the best performing method in each database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' It is clear that our model is highly competitive on these four databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Weighing the quantity of compressed videos in each database gets the overall per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' As can be seen from the last column, the total performance of SSTAM is better than the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The high improvement of SAAM and SSTAM compared to other methods is because they fully consider the influence of PEA on compressed video quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' This experiment also fully demonstrates that there is a strong correlation between the PEAs and the subjective quality scores of compressed videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' We find that methods containing a saliency model predict video quality more effectively in FERIT-RTRK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Since the saliency objects in FERIT-RTRK are relatively few and large, the network can extract the saliency regions of the video TABLE I PERFORMANCE COMPARISON IN TERMS OF PLCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Methods LIVE CSIQ IVP FERIT-RTRK Overall PSNR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='5735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='8220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='7998 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9281 TABLE III ABLATION EXPERIMENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Indicators SAAM SAAM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='0 SSTAM PLCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9552 SRCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='8796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='9281 more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Accordingly, a better extraction of saliency regions contributes significantly to the performance of quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Ablation Experiments: Ablation Experiments: Ablation Experiments: To further verify the improvement of our proposed strategies for video quality assessment, we compared the effects of saliency model and temporal PEAs separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Visual saliency model called Attentive CNN-LSTM Network (ACLNet) [36] was applied in SAAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' To evaluate the performance, we substitute ACLNet in SAAM with U2- CPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' After changing the saliency network, we refer to the method as SAAM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' As shown in Table III, our saliency model performs better in recognizing saliency regions and enhances the performance of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The comparisons of SAAM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content='0 and SSTAM reveal that temporal PEAs have an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' CONCLUSIONS This paper proposes a compressed video quality index based on saliency-aware spatio-temporal artifact detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' The ex- perimental results demonstrate the improved performance and generalizability of the proposed SSTAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' We believe it will be essential in compressed video quality assessment and video coding optimization in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 18, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' 9, SEPTEMBER 2020 5 REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Shi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdAzT4oBgHgl3EQfIvvq/content/2301.01069v1.pdf'} +page_content=' Teunissen, and I.' metadata={'source': 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b/V9AyT4oBgHgl3EQfhvhg/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e3b34d1a0c764fb60affef2ab67a9ce21522ae83f45728f22f1811a95eb9b77d +size 144342 diff --git a/WdE5T4oBgHgl3EQfCA5G/content/tmp_files/2301.05391v1.pdf.txt b/WdE5T4oBgHgl3EQfCA5G/content/tmp_files/2301.05391v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e1fbf26c13a8b17ad941cc6aae1d297249bd32b --- /dev/null +++ b/WdE5T4oBgHgl3EQfCA5G/content/tmp_files/2301.05391v1.pdf.txt @@ -0,0 +1,812 @@ +Hierarchical Deep Q-Learning Based Handover in +Wireless Networks with Dual Connectivity +Pedro Enrique Iturria-Rivera1, Student Member, IEEE, Medhat Elsayed2, Majid Bavand2, Raimundas Gaigalas2, +Steve Furr2 and Melike Erol-Kantarci1, Senior Member, IEEE +1School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada +2Ericsson Inc., Ottawa, Canada +Emails:{pitur008, melike.erolkantarci}@uottawa.ca, {medhat.elsayed, majid.bavand, raimundas.gaigalas, steve.furr}@ericsson.com +Abstract—5G New Radio proposes the usage of frequencies +above 10 GHz to speed up LTE’s existent maximum data rates. +However, the effective size of 5G antennas and consequently its +repercussions in the signal degradation in urban scenarios makes +it a challenge to maintain stable coverage and connectivity. In +order to obtain the best from both technologies, recent dual +connectivity solutions have proved their capabilities to improve +performance when compared with coexistent standalone 5G and +4G technologies. Reinforcement learning (RL) has shown its +huge potential in wireless scenarios where parameter learning is +required given the dynamic nature of such context. In this paper, +we propose two reinforcement learning algorithms: a single agent +RL algorithm named Clipped Double Q-Learning (CDQL) and a +hierarchical Deep Q-Learning (HiDQL) to improve Multiple Ra- +dio Access Technology (multi-RAT) dual-connectivity handover. +We compare our proposal with two baselines: a fixed parameter +and a dynamic parameter solution. Simulation results reveal +significant improvements in terms of latency with a gain of +47.6% and 26.1% for Digital-Analog beamforming (BF), 17.1% +and 21.6% for Hybrid-Analog BF, and 24.7% and 39% for +Analog-Analog BF when comparing the RL-schemes HiDQL and +CDQL with the with the existent solutions, HiDQL presented +a slower convergence time, however obtained a more optimal +solution than CDQL. Additionally, we foresee the advantages +of utilizing context-information as geo-location of the UEs to +reduce the beam exploration sector, and thus improving further +multi-RAT handover latency results. +Index Terms—5G, dual-connectivity (DC), hierarchical deep +Q-learning, clipped double deep Q-learning, context-awareness, +handover. +I. INTRODUCTION +With the deployment of 5th Generation New Radio (5G) +and the existing 4th Generation LTE (4G) technologies, +mobile users with LTE or 5G capabilities must be able to +seamlessly adapt to the dual existent infrastructure. Back in +2013, the 3rd Generation Partnership Project (3GPP) pro- +posed dual connectivity (DC) architectures in [1] that allowed +master eNodeBs (eNBs) and secondary eNodeBs to share +partially their IP layer in a dual connection manner with +the purpose of maximizing network performance. A more +recent technical report [2] proposed a generalization of multi- +radio dual connectivity architectures to support LTE and 5G +users. Dual connection architectures take advantage of the +technologies involved to improve key performance indicators +(KPIs) of interest. Concurrently, the stringent requirements of +diverse services in terms of latency and throughput, such as +Ultra-Reliable and Low Latency Communications (URLLC) +and enhanced Mobile Broadband (eMMB) have demanded a +gNB 1 +Satellite +gNB 2 +gNB 3 +eNB +GPS +Fig. 1: Overview of the scenario: a UE uses GPS context information and dual +connectivity to improve handover latency in an urban scenario. +more optimized parameter tuning in any wireless mechanism +utilized. +Reinforcement Learning (RL) techniques have been widely +recognized for its effectiveness in the autonomous learning +context. More specifically, RL has sparked the wireless net- +work community’s attention [3] since optimized parameter +learning has become a challenge in such dynamic environ- +ments. +Typically, an RL agent optimizes its action based on a +customized reward or objective function. The design of a +reward function will intend to maximize or minimize certain +metric of interest. In addition, a reward function could also be +used in scenarios where our agent’s goal is known. However, +in goal-directed problems as in the majority of RL problems, +sparse rewards are a significant challenge that affects the +learning of robust value functions and thus, optimal action +selection. Hierarchical reinforcement learning (hRL) [4] helps +to solve the aforementioned problem by splitting the value +function into two levels: a meta-controller and a controller. +In this paper, we address the handover problem in an +LTE-NR network with dual connectivity using a hierarchical +and a non-hierarchical architecture. Additionally, we con- +sider the context information to further improve the DC +algorithm performance. To do so, we use a novel hierar- +chical Deep Q-learning algorithm named hierarchical Deep +Q-Learning (HiDQL) and a non-hierarchical RL approached +named Clipped Double Q-Learning (CDQL) to improve han- +dover latency in a DC architecture. HiDQL presents an +arXiv:2301.05391v1 [cs.NI] 13 Jan 2023 + +QuinceMediagNB 1 +Satellite +gNB 2 +gNB 3 +eNB +GPS +Environment +Meta-Controller +Controller +Q-Value Net +Q-Value Net +hDQN Agent +Replay +buffer +Replay +buffer +State (SINR RT) +Updated +in c steps +Action +(Outage, TTT) +Updated +periodically +State (SINR RT) +Goal Action for +next c steps +([s,g],a,r,[s_,g],done) +(s0,g,R,s,done) +Intrinsic reward (r) +R +g +a +s +s +Fig. 2: Hierarchical Deep Q-Learning (HiDQL) applied in a Multiple Radio Access Technology (multi-RAT) LTE-NR scenario. The HiDQL agent located in the coordinator will +optimize TTT and SINR outage metrics to reduce multi-RAT handover latency. +architecture comprised by a meta-controller and a controller. +Two parameters are adjusted in this problem: Threshold to +Time (TTT) and signal-to-interference-plus noise ratio (SINR) +outage. TTT is defined as the time to hold the condition +of triggering the handover algorithm named Secondary Cell +Handover (SCH), discussed in section III. On the other hand, +SINR outage is defined as the SINR threshold in which we +consider a cell being in outage from the User Equipment +(UE). The controller will propose actions in terms of TTT +and SINR outage aided by the proposed goals by the meta- +controller in order to minimize handover latency. We compare +our results with an algorithm presented in [5] named Clipped +Double Q-Learning (CDQL) that proved its efficiency in load +balancing problems. Our results show an improvement in +terms of latency with a gain of 47.6% and 26.1% for Digital- +Analog beamforming (BF), 17.1% and 21.6% for Hybrid- +Analog BF, and 24.7% and 39% for Analog-Analog BF +when comparing the RL-schemes with the baseline’s best +results. Additionally, we observed faster convergence when +utilizing the CDQL algorithm, meanwhile the HiDQL showed +improved results in terms of handover latency. Finally, we ob- +tained an improvement regarding handover latency under line +of sight assumptions when considering context information in +comparison with no context available. +The rest of this paper is organized as follows. Section +II presents the existing works related to dual connectivity +and hierarchical algorithms applications in wireless networks. +System model is demonstrated in Section III. Section IV +provides a description of the Markov Decision process of the +RL algorithms and baselines utilized. Section V depicts the +performance evaluation and comparison with the baseline so- +lutions. Section VI introduces context information in the DC +architecture and its performance evaluation. Finally, section +VII concludes the paper. +II. RELATED WORK +To the best of our knowledge, there is no previous work +where hRL has been utilized to optimize handover key +parameters such as TTT and SINR cell outage in a 5G +dual connectivity scenario. The aforementioned parameters +are defined in detail in section V. However, dual connectivity +(DC) has been studied thoroughly in the literature. These +studies are summarized below. +In [6], the authors give an overview of the 4G LTE-NR +DC based on the new specifications given by 3GPP. A study +case showed that DC is capable of providing coverage and +capacity improvements when compared to standalone LTE- +NR deployed individually. More recently, in [7] the authors +perform an analysis of the dual connectivity proposal in the +3GPP release 15 [2]. In addition to the existent specification, +the authors remarked the possibilities to include a Secondary +Cell controller to manage the multi-radio DC signaling. +Furthermore, in [8], the authors present a survey on recent +developments of 4G/5G DC, as well 4G/5G internetworking +performance and future research challenges and open issues. +Besides, the body of works in DC, the concept of hRL +has not been explored much in the the wireless community. +In [9], the authors study outage avoidance in a two-hop +cooperative relay network by utilizing hRL in optimizing +relay selection and both source and relays transmission power. +In [10], the authors propose a hierarchical deep Q-network to +perform dynamic multi-channel spectrum sensing in cognitive +networks. In this work the actions corresponds to the selection +of channel and the reward corresponds to the channel status +(busy/idle). In the following sections, we will introduce the +dual connectivity architecture utilized in this work and explain +how RL emerges as effective solution in the optimization +of the handover in DC scenarios. Additionally, we forsee +the advantage of embedding context-awareness in the DC +architecture. +III. SYSTEM MODEL +In this work, we use a DC architecture presented in [11] +inspired by a 3GPP’s previous DC proposal [1]. In [11], the +authors leverage the usage of a DC framework to improve, +among others, handover latency by using an algorithm named +Secondary Cell Handover (SCH). SCH enables fast switching + +Quince Mediabetween the LTE and 5G RATs and is managed by a defined +entity named coordinator. In this architecture the eNB takes +the role of coordinator and controls the multi-RAT switch. +Two main parameters are controlled by the coordinator: TTT +and SINR cell outage. As part of the SCH algorithm, the +coordinator will trigger the SCH when any of the RATs +involved report a better SINR and neither of the RATs are in +outage. Additionally, it will check the condition to trigger the +SCH algorithm for TTT seconds to avoid ping pong scenarios. +This algorithm resembles the classical LTE and 5G handover +with the exception that no initial access is necessary and no +interaction with the MME is required thanks to the nature +of the DC architecture. As mentioned, the handover will be +triggered based on SINR and specifically using a table named +the Complete Report Table (CRT) comprised by the UE’s +SINR report tables from each gNodeB (gNB). Each gNB will +perform a sweep over a number of predefined directions and +will sense the Sounding Reference Signals from the UE’s to +obtain SINR measurements. The collected data will be sent +via X2 interface to the coordinator. The delay incurred to +perform the measurements sweeps is directly related with the +beamforming (BF) technology used by the UEs and gNBs. +The aforementioned delay, D, is calculated as: +D = NgNBNUETper +L +, +(1) +where NgNB and NUE correspond to the number of required +sweep directions needed for measurement collection by the +gNBs and UEs, respectively. Tper is the SRS periodicity +and L corresponds to the BF capabilities that will present +different values if the transceiver is fully digital or analog. +The relationship between D and L can be calculated using +typical values as depicted in Table I (Modified from [11]). +TABLE I: Relationship between L and D +gNB-UE BF +L (gNB/UE) +D*(ms) +Analog-Analog +1/1 +25.6 +Hybrid-Analog +2/1 +16.8 +Digital-Analog +NgNB/1 +1.6 +* D in Eq. 1 is calculated with Tper = 200µs, +NgNB = 16 and NUE = 8 +In this paper, we consider an LTE-NR network with dual +connectivity and consisting of MT gNBs. Additionally, an +eNB will act as a coordinator and thus, serving the MT gNBs. +A mobile user is dual connected to one gNB and one eNB +at the same time. We consider an urban scenario with two +buildings as shown in Fig. 1. The channel considered for +the eNB is the 3GPP Channel Model, meanwhile the 3GPP +building channel type and losses based on the 3GPP UMi +Street Canyon propagation model is used for the gNBs. +IV. HIERARCHICAL DEEP Q-LEARNING (HIDQL) +In this work, we use a state-of-the-art hierarchical Deep +Q-learning (HiDQL) algorithm. This algorithm is presented +in [4], where the goal-directed behavior is studied under +some specific sparse reward problems such as the Montezuma +Revenge and a complex discrete stochastic decision process. +As shown in figure 2, a hierarchical agent located in +the coordinator observes the user’s SINR report table and +receives the extrinsic reward based on the feedback from the +environment in terms of handover latency. This agent will +adjust key parameters as TTT and SINR outage in order +to maximize the agent’s extrinsic reward. In the following +subsection, we will present an overview of HiDQL and then +formally define our solution. +Algorithm 1 Hierarchical Deep Q-Learning (HiDQL) +Initialize cmax, κ, experience replay buffers {Bmc, Bc}, policy networks {µmc, +µc}, for the meta-controller and controller, respectively. +Function Done(c, cmax, Sκ, κ): +if (c = cmax or Sκ < κ) then +return True +else +return False +end +End Function +for environment step t ← 1 to T do +c ← 0 +Init environment and initial state +Execute goal from meta-controller: g = µmc(s) +while cmax not reached do +R ← 0; s0 ← s; Sκ = false +while not Done do +Execute action from controller: a = µc(s, g) +Get next state s′ and similarity condition Sκ ; +Calculate intrinsic reward r and receive extrinsic reward rg; +Store ([s, g], a, r, [s′, g], Done) in replay buffer Bc; +Update µc; +c := c + 1; R := R + rg; s := s′ +end +Store (s0, g, R, s, Done) in replay buffer Bc; +Update µmc; +if not Done then +Execute goal from meta-controller: g = µmc(s) +end +end +end +A. Meta-controller and controller action space selection +In the proposed hierarchical approach described in Al- +gorithm 1, the meta-controller proposes goals or high level +actions when the maximum c steps, cmax is achieved or +when the similarity between the goal and low-level action, Sκ +falls under a predefined threshold index κ as defined in the +function Done. Meanwhile, the controller’s low level actions +are executed c inner steps. For both levels, the actions are +defined in the same fashion as: +A(t) = +�aout(t), attt(t)� +, +(2) +where aout and attt are the SINR outage and TTT values, +respectively. Such values are discretized into Ko and KT T T +levels according the maximum and minimum defined values. +The possible values for aout and attt are defined as: +Aout = {Omin, Omin + Omax − Omin +Ko − 1 +, ..., Omax}, +(3) +Attt = {TTTmin,TTTmin+ +TTTmax − TTTmin +KT T T − 1 +, · · · , TTTmax}, +(4) + +where Omin, Omax and TTTmin, TTTmax are predefined +maximum and minimum values of SINR outage and TTT +values, respectively. Finally, the size of the meta-controller +and controller action spaces become S = |Aout| ∗ |Attt|. +B. State space selection +The state space is comprised by the Complete Report Table +(CRT), which contains the relationship in terms of SINR of +each user and the perceived gNB SINR. The SINR between +an ith gNB and nth UE is calculated as follows: +SINRi,n = +PT X +P Li,n Gi,n +� +k̸=i +PT X +P Lk,n Gk,n + Wtot × N0 +, +(5) +where Gi,n and PLi,n are the beamforming gain and the +pathlosss obtained between ith gNB and nth UE. PT X is the +transmit power and Wtot × N0 is the thermal noise power. +Thus, the state space is defined as follows: +S(t) = +�C(t)� +, +(6) +where C(t) corresponds to the CRT with a size of |C(t)| = +MT . +C. Intrinsic and extrinsic reward +The intrinsic reward function is calculated by the controller +by taking into account the goal selected by the meta-controller +in c inner steps. Such a reward is defined as: +r(t) = ri(t) + rl2(t), +(7) +where ri corresponds to the immediate reward obtained from +the environment and rl2 corresponds to reward based on +the l2-norm between the controller’s action and the meta- +controller’s goal. Such rewards are defined as follows: +ri(t) = +� +−f−1(−Du +avg) +if Du +avg < DBF +tol , +0 +otherwise +(8) +where f−1(·) : [−e−1, 0) → [−1, −∞) is the lower branch of +the Lambert function y = f−1(yey) [12]. DBF +tol is the max- +imum tolerable delay corresponding to the value of the best +latency results per BF technology in the non-RL approaches. +Du +avg is the average latency of the UE. Additionally, we scale +linearly Du +avg → [e−1, 0]. +rl2(t) = +� +1 +if Sκ < κ, +−1 +otherwise, +(9) +where Sκ = |a−g|2 and κ is the similarity threshold between +the action and goal. Finally, we can express the extrinsic +reward as: +R = +c +� +j=t−c +r(j), +(10) +where the extrinsic reward is defined as the sum of the +intrinsic reward during the c inner steps. +TABLE II: Network settings +Parameter +Value +MT +3 +Number of UEs +1 +gNB center frequency +28 GHz +gNB system bandwidth +200 MHz +gNB numerology +2 +gNB Pathloss Model +3GPP Umi-Street Canyon +gNB Channel condition model +Buildings +gNB antenna height +10 m +gNB number of antennas +64 +eNB Center Frequency +2 GHz +eNB System Bandwidth +100 MHz +eNB Pathloss Model +3GPP +Max Tx power +30 dBm +UE number of antennas +16 +UE antenna height +1.6 m +UE speed +13 m/s +UE Tx power +10 dBm +Traffic Model +On-Off UDP application +Packet payload size = 512 Bytes +Packet window size = 256 +Interval = 20 µs +D. Clipped Double Q-Learning: RL scheme +In addition to HiDQL, we utilized an RL solution named +Clipped Double Q-Leaning (CDQL) [5]. For the CDQL +algorithm, the state and action space correspond to the ones +used in the lower level by the HiDQL algorithm in (6) and +(2), respectively, meanwhile the reward corresponds to the +intrinsic reward defined in (8). +E. Baselines: Fixed TTT and Dynamic TTT +For the present work, as baselines, we take a fixed and a +dynamic solution into consideration. The fixed and dynamic +solutions were proposed in [11], where the fixed solution has +a fixed value of TTT and in the dynamic solution TTT is +calculated as follows: +fT T T (∆) = TTTmax− +∆ − ∆min +∆max − ∆min +(TTTmax−TTTmin), +(11) +where ∆ corresponds to the difference between the serving +cell and the best gNB in terms of SINR. +V. PERFORMANCE EVALUATION +A. Simulation Setting +We implement our proposed solution using the discrete +network simulator ns-3 and its module mmWave [13]. Addi- +tionally, ns3-ai module [14] is used to interface the simulation +environment to our RL algorithm written in Python 3.9. +Simulation settings and RL parameters are depicted in Table +II and Table III, respectively. +B. Simulation Results +We present the performance results of our proposed scheme +in terms of convergence and handover latency. Figure 3 (a), +(b) depicts the convergence behavior for the CDQL and + +0 +20 +40 +60 +80 +100 +Episode +0 +20 +40 +60 +80 +100 +Reward +CDQL - Convergence +Digital-Analog +Hybrid-Analog +Analog-Analog +(a) +0 +50 +100 +150 +200 +Episode +−5 +0 +5 +Reward +HiDQL - Convergence +Digital-Analog +Hybrid-Analog +Analog-Analog +(b) +Digital-Analog +Hybrid-Analog +Analog-Analog +BF Architecture (gNB-UE) +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +Latency(s) +Latency performance +HiDQL +CDQL +Dynamic TTT +Fixed TTT +(c) +Fig. 3: (a) CDQL convergence performance. (b) HiDQL convergence performance. contain. (c) Latency performance of the baselines Dynamic TTT and Fixed TTT and the +proposed RL schemes CDQL and HiDQL. +TABLE III: Learning parameters +Parameter +Value +Handover algorithm +Threshold to time: TTTmax = 150 ms; TTTmin = 25 ms +SINR Outage: Omax = −3 dBm; Omin = −8 dBm +DBF +tol = min(DBF +f +, DBF +d +) * +Gym environment step time +Event-based +Batch size +32 +Number of hidden layers (Nh) : 2 +Number of neurons/layer (nl) : 64 +HiDQL +κ = 0.25 +Optimizer : Meta-Controller: Adam (1e-4), +Controller: Adam (1e-4) +cmax +50 steps +CDQL +Update target model type : Polyak averaging +γ = 0.95, ϵ = 1.0, ϵmin = 0.001, ϵdecay = 0.995 +Optimizer : Adam (1e-4) +*DBF +f +and DBF +d +correspond to the latency performance per BF technology of the fixed TTT and dynamic TTT +solutions, respectively. +HiDQL algorithms for different BF mechanisms, respectively. +It can be observed that CDQL presents a faster conver- +gence when compared with HiDQL. However, converged +reward values are not achieved equally among the tested +BF architectures. HiDQL presents a slower convergence time +than CDQL however with a similar maximum convergence +reward value among BF methods. The intuition behind the +previous results correspond to the inner nature of hierarchical +learning. HiDQL performs an iterative search through the +meta-controller’s proposal of goals to the controller and thus, +obtaining optimum action selection in challenging scenarios +as described in [4]. The difference between reward scales +between HiDQL and CDQL corresponds to how the reward +is defined in both algorithms, thus a comparison in terms of +the value of the reward is not relevant. +TABLE IV: RL vs non-RL latency comparison +Latency improvement (%) +Digital-Analog BF +Hybrid-Analog BF +Analog-Analog BF +RL Scheme +F-TTT +D-TTT +F-TTT +D-TTT +F-TTT +D-TTT +HiDQL +76.1 +47.6 +83.5 +21.6 +87.0 +39.0 +CDQL +66.2 +26.1 +82.5 +17.1 +83.8 +24.7 +In Fig. 3 (c), we present the latency performance of our +techniques and the baselines: fixed TTT (F-TTT), dynamic +TTT (D-TTT). From the two baselines, the best performance +in terms of latency is achieved by the dynamic TTT parameter +selection. On the other hand, we obtained an average 9% +improvement when comparing HiDQL and CDQL in terms +of latency results. However, based on more detailed look into +results, we further focus in Table IV on the latency results +improvements over the previous solutions F-TTT and D-TTT. +As shown in Table IV, when compared with the dynamic +TTT results, a gain of 47.6% and 26.1%, for Digital-Analog +BF, 17.1% and 21.6%, for Hybrid-Analog BF, and 24.7% +and 39% for Analog-Analog BF were obtained for HiDQL +and CDQL, respectively. +TABLE V: CI vs. non-CI latency comparison +Latency improvement (%) +Digital-Analog BF +Hybrid-Analog BF +Analog-Analog BF +CI +Non-CI +F-TTT +74.9 +2.57 +28.5 +D-TTT +40.1 +8.7 +29.7 +VI. CONTEXT INFORMATION AWARE DC HANDOVER +In addition to the SCH algorithm used as part of the +proposed DC architecture, we foresee the possible advantages +of using context information to improve handover time. In +[15], the authors give an overview of the typical initial access +(IA) cell search algorithms where contrary to LTE, 5G NR +provide mechanisms to determine suitable initial directions +of transmission to avoid high isotropic losses. Among the IA +algorithms, context information (CI)-based algorithms allow +users to be aware of the location of the surrounded 5G NR +antennas through an LTE link and GPS. In this work, we +leverage GPS information as context and reduce the sector in +which a gNB should sense the space towards an UE. Figure +4 depicts the beforementioned mechanism, where given the +user position P1 the gNB is able to calculate a reduced sector +to obtain the periodic SRS signals from the surrounded UEs. +We also assume that each UE has line of sight both with +the coordinator and the gNBs. The proposed mechanism is +described in Algorithm 2. As shown in Table V, context +awareness directly improves the DC handover mechanism +with a considerable latency improvement. +VII. CONCLUSIONS +In this paper, we presented two Reinforcement Learning +(RL)-based dual connectivity (DC) solutions that further + +90o +0o +180o +270o +P1(x1,y1) +gNB (X, Y) +δ +2 +1 +3 +4 +gNB +UE +ψ +ψ +P1’(x1,Y) +a) +b) +Fig. 4: a) ±ψ indicates the sweep angle to obtain SINR measurements. b) δ corresponds +to the relative angle used to derive the sweep sector angle. +Algorithm 2 CI-aware sector sweep selection algorithm +Result: Sector’s angle to perform measurements sweeps. +Given P1(x1, y1) and (X, Y ), the UE’s and gNB geolocations, respectively. +Additionally, δ += +arcsin +⃗ +d2 +⃗ +d1 +and ΘgNB += +2π +N ; where +⃗d2 += +−−−→ +P1P ′ +1 and +⃗d1 = −−−−−→ +P1gNB +# Step 1: Detect UE’s relative quadrant q +∈ +{1, 2, 3, 4}. +Calculate angle φ respect gNB quadrant one. +if y1 > Y and x1 > X then +# UE in 1st quadrant +φ = δ +else if y1 > Y and x1 < X then +# UE in 2nd quadrant φ = π +2 + δ +else if y1 < Y and x1 < X then +# UE in 3rd quadrant φ = π + δ +else +# UE in 4th quadrant φ = 3π +2 + δ +end +# Step 2: Calculate sector angle (SA) +SA ∈ [φ − 2Θ, φ + 2Θ] +Digital-Analog +Hybrid-Analog +Analog-Analog +BF Architecture (gNB-UE) +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +Latency(s) +Latency performance +Dynamic TTT (C) +Dynamic TTT +Fixed TTT (C) +Fixed TTT +Fig. 5: Context information (C) latency performance results when utilizing Dynamic +TTT and Fixed TTT baselines. +improve latency metric in inter-RAT handover. Additionally, +we propose leveraging an initial access cell search algorithm +that utilizes context information to reduce the latency in gNB- +UE’s measurement collection. We compared two RL solutions +named Hierarchical Deep Q-Learning (HiDQL) and Clipped +Double Q-Learning (CDQL) with two previously proposed +baselines, fixed Threshold to Time (TTT) and dynamic TTT, +obtaining a significant gain in both cases in terms of latency +with a 47.6% and 26.1%, for Digital-Analog BF, 17.1% +and 21.6%, for Hybrid-Analog BF and 24.7% and 39% for +Analog-Analog BF when compared with the baselines’ best +results. Finally, we presented a context-aware mechanism +under line of sight assumptions and obtained a significant +improvement in terms of latency when compared with the +case of no context. As future work we forsee the integration +of the presented context-aware mechanism and reinforcement +learning to further improve the DC handover mechanism. +VIII. ACKNOWLEDGMENT +This research is supported by Mitacs and Ericsson. +REFERENCES +[1] 3GPP TR 36.842, “Study on small cell enhancements for E-UTRA and +E-UTRAN, v12.0.0 ,” Tech. Rep., 2013. +[2] 3GPP TS 37.340, “Universal Mobile Telecommunications System +(UMTS), LTE, 5G, NR, Multi-connectivity, Overall description, Stage- +2,” Tech. Rep., 2019. +[3] M. Elsayed and M. Erol-Kantarci, “AI-Enabled Future Wireless Net- +works: Challenges, Opportunities, and Open Issues,” IEEE Veh. Tech- +nol. Mag., 2019. +[4] T. D. Kulkarni, K. R. Narasimhan, A. Saeedi, and J. B. Tenenbaum, “Hi- +erarchical deep reinforcement learning: Integrating temporal abstraction +and intrinsic motivation,” in Advances in Neural Information Processing +Systems (NeurIPS), 2016. +[5] P. E. Iturria-Rivera and M. Erol-Kantarci, “QoS-Aware Load Balancing +in Wireless Networks using Clipped Double Q-Learning,” in 2021 IEEE +18th International Conference on Mobile Ad Hoc and Smart Systems +(MASS), 2021. +[6] O. N. C. 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Mag., vol. 54, no. 11, +pp. 40–47, 2016. + diff --git a/WdE5T4oBgHgl3EQfCA5G/content/tmp_files/load_file.txt b/WdE5T4oBgHgl3EQfCA5G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..85568ac9f52479932232d14301907c25792a0767 --- /dev/null +++ b/WdE5T4oBgHgl3EQfCA5G/content/tmp_files/load_file.txt @@ -0,0 +1,388 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf,len=387 +page_content='Hierarchical Deep Q-Learning Based Handover in Wireless Networks with Dual Connectivity Pedro Enrique Iturria-Rivera1, Student Member, IEEE, Medhat Elsayed2, Majid Bavand2, Raimundas Gaigalas2, Steve Furr2 and Melike Erol-Kantarci1, Senior Member, IEEE 1School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada 2Ericsson Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=', Ottawa, Canada Emails:{pitur008, melike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='erolkantarci}@uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='ca, {medhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='elsayed, majid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='bavand, raimundas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='gaigalas, steve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='furr}@ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='com Abstract—5G New Radio proposes the usage of frequencies above 10 GHz to speed up LTE’s existent maximum data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' However, the effective size of 5G antennas and consequently its repercussions in the signal degradation in urban scenarios makes it a challenge to maintain stable coverage and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In order to obtain the best from both technologies, recent dual connectivity solutions have proved their capabilities to improve performance when compared with coexistent standalone 5G and 4G technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Reinforcement learning (RL) has shown its huge potential in wireless scenarios where parameter learning is required given the dynamic nature of such context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In this paper, we propose two reinforcement learning algorithms: a single agent RL algorithm named Clipped Double Q-Learning (CDQL) and a hierarchical Deep Q-Learning (HiDQL) to improve Multiple Ra- dio Access Technology (multi-RAT) dual-connectivity handover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' We compare our proposal with two baselines: a fixed parameter and a dynamic parameter solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Simulation results reveal significant improvements in terms of latency with a gain of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6% and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1% for Digital-Analog beamforming (BF), 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1% and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6% for Hybrid-Analog BF, and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='7% and 39% for Analog-Analog BF when comparing the RL-schemes HiDQL and CDQL with the with the existent solutions, HiDQL presented a slower convergence time, however obtained a more optimal solution than CDQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, we foresee the advantages of utilizing context-information as geo-location of the UEs to reduce the beam exploration sector, and thus improving further multi-RAT handover latency results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Index Terms—5G, dual-connectivity (DC), hierarchical deep Q-learning, clipped double deep Q-learning, context-awareness, handover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' INTRODUCTION With the deployment of 5th Generation New Radio (5G) and the existing 4th Generation LTE (4G) technologies, mobile users with LTE or 5G capabilities must be able to seamlessly adapt to the dual existent infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Back in 2013, the 3rd Generation Partnership Project (3GPP) pro- posed dual connectivity (DC) architectures in [1] that allowed master eNodeBs (eNBs) and secondary eNodeBs to share partially their IP layer in a dual connection manner with the purpose of maximizing network performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' A more recent technical report [2] proposed a generalization of multi- radio dual connectivity architectures to support LTE and 5G users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Dual connection architectures take advantage of the technologies involved to improve key performance indicators (KPIs) of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Concurrently, the stringent requirements of diverse services in terms of latency and throughput, such as Ultra-Reliable and Low Latency Communications (URLLC) and enhanced Mobile Broadband (eMMB) have demanded a gNB 1 Satellite gNB 2 gNB 3 eNB GPS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' 1: Overview of the scenario: a UE uses GPS context information and dual connectivity to improve handover latency in an urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' more optimized parameter tuning in any wireless mechanism utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Reinforcement Learning (RL) techniques have been widely recognized for its effectiveness in the autonomous learning context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' More specifically, RL has sparked the wireless net- work community’s attention [3] since optimized parameter learning has become a challenge in such dynamic environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Typically, an RL agent optimizes its action based on a customized reward or objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The design of a reward function will intend to maximize or minimize certain metric of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In addition, a reward function could also be used in scenarios where our agent’s goal is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' However, in goal-directed problems as in the majority of RL problems, sparse rewards are a significant challenge that affects the learning of robust value functions and thus, optimal action selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Hierarchical reinforcement learning (hRL) [4] helps to solve the aforementioned problem by splitting the value function into two levels: a meta-controller and a controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In this paper, we address the handover problem in an LTE-NR network with dual connectivity using a hierarchical and a non-hierarchical architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, we con- sider the context information to further improve the DC algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' To do so, we use a novel hierar- chical Deep Q-learning algorithm named hierarchical Deep Q-Learning (HiDQL) and a non-hierarchical RL approached named Clipped Double Q-Learning (CDQL) to improve han- dover latency in a DC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' HiDQL presents an arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='05391v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='NI] 13 Jan 2023 QuinceMediagNB 1 Satellite gNB 2 gNB 3 eNB GPS Environment Meta-Controller Controller Q-Value Net Q-Value Net hDQN Agent Replay buffer Replay buffer State (SINR RT) Updated in c steps Action (Outage, TTT) Updated periodically State (SINR RT) Goal Action for next c steps ([s,g],a,r,[s_,g],done) (s0,g,R,s,done) Intrinsic reward (r) R g a s s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' 2: Hierarchical Deep Q-Learning (HiDQL) applied in a Multiple Radio Access Technology (multi-RAT) LTE-NR scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The HiDQL agent located in the coordinator will optimize TTT and SINR outage metrics to reduce multi-RAT handover latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' architecture comprised by a meta-controller and a controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Two parameters are adjusted in this problem: Threshold to Time (TTT) and signal-to-interference-plus noise ratio (SINR) outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' TTT is defined as the time to hold the condition of triggering the handover algorithm named Secondary Cell Handover (SCH), discussed in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' On the other hand, SINR outage is defined as the SINR threshold in which we consider a cell being in outage from the User Equipment (UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The controller will propose actions in terms of TTT and SINR outage aided by the proposed goals by the meta- controller in order to minimize handover latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' We compare our results with an algorithm presented in [5] named Clipped Double Q-Learning (CDQL) that proved its efficiency in load balancing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Our results show an improvement in terms of latency with a gain of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6% and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1% for Digital- Analog beamforming (BF), 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1% and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6% for Hybrid- Analog BF, and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='7% and 39% for Analog-Analog BF when comparing the RL-schemes with the baseline’s best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, we observed faster convergence when utilizing the CDQL algorithm, meanwhile the HiDQL showed improved results in terms of handover latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Finally, we ob- tained an improvement regarding handover latency under line of sight assumptions when considering context information in comparison with no context available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Section II presents the existing works related to dual connectivity and hierarchical algorithms applications in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' System model is demonstrated in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Section IV provides a description of the Markov Decision process of the RL algorithms and baselines utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Section V depicts the performance evaluation and comparison with the baseline so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Section VI introduces context information in the DC architecture and its performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Finally, section VII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' RELATED WORK To the best of our knowledge, there is no previous work where hRL has been utilized to optimize handover key parameters such as TTT and SINR cell outage in a 5G dual connectivity scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The aforementioned parameters are defined in detail in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' However, dual connectivity (DC) has been studied thoroughly in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' These studies are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In [6], the authors give an overview of the 4G LTE-NR DC based on the new specifications given by 3GPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' A study case showed that DC is capable of providing coverage and capacity improvements when compared to standalone LTE- NR deployed individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' More recently, in [7] the authors perform an analysis of the dual connectivity proposal in the 3GPP release 15 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In addition to the existent specification, the authors remarked the possibilities to include a Secondary Cell controller to manage the multi-radio DC signaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Furthermore, in [8], the authors present a survey on recent developments of 4G/5G DC, as well 4G/5G internetworking performance and future research challenges and open issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Besides, the body of works in DC, the concept of hRL has not been explored much in the the wireless community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In [9], the authors study outage avoidance in a two-hop cooperative relay network by utilizing hRL in optimizing relay selection and both source and relays transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In [10], the authors propose a hierarchical deep Q-network to perform dynamic multi-channel spectrum sensing in cognitive networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In this work the actions corresponds to the selection of channel and the reward corresponds to the channel status (busy/idle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In the following sections, we will introduce the dual connectivity architecture utilized in this work and explain how RL emerges as effective solution in the optimization of the handover in DC scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, we forsee the advantage of embedding context-awareness in the DC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' SYSTEM MODEL In this work, we use a DC architecture presented in [11] inspired by a 3GPP’s previous DC proposal [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In [11], the authors leverage the usage of a DC framework to improve, among others, handover latency by using an algorithm named Secondary Cell Handover (SCH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' SCH enables fast switching Quince Mediabetween the LTE and 5G RATs and is managed by a defined entity named coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In this architecture the eNB takes the role of coordinator and controls the multi-RAT switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Two main parameters are controlled by the coordinator: TTT and SINR cell outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' As part of the SCH algorithm, the coordinator will trigger the SCH when any of the RATs involved report a better SINR and neither of the RATs are in outage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, it will check the condition to trigger the SCH algorithm for TTT seconds to avoid ping pong scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' This algorithm resembles the classical LTE and 5G handover with the exception that no initial access is necessary and no interaction with the MME is required thanks to the nature of the DC architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' As mentioned, the handover will be triggered based on SINR and specifically using a table named the Complete Report Table (CRT) comprised by the UE’s SINR report tables from each gNodeB (gNB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Each gNB will perform a sweep over a number of predefined directions and will sense the Sounding Reference Signals from the UE’s to obtain SINR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The collected data will be sent via X2 interface to the coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The delay incurred to perform the measurements sweeps is directly related with the beamforming (BF) technology used by the UEs and gNBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The aforementioned delay, D, is calculated as: D = NgNBNUETper L , (1) where NgNB and NUE correspond to the number of required sweep directions needed for measurement collection by the gNBs and UEs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Tper is the SRS periodicity and L corresponds to the BF capabilities that will present different values if the transceiver is fully digital or analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The relationship between D and L can be calculated using typical values as depicted in Table I (Modified from [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' TABLE I: Relationship between L and D gNB-UE BF L (gNB/UE) D*(ms) Analog-Analog 1/1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6 Hybrid-Analog 2/1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='8 Digital-Analog NgNB/1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6 D in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' 1 is calculated with Tper = 200µs, NgNB = 16 and NUE = 8 In this paper, we consider an LTE-NR network with dual connectivity and consisting of MT gNBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, an eNB will act as a coordinator and thus, serving the MT gNBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' A mobile user is dual connected to one gNB and one eNB at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' We consider an urban scenario with two buildings as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The channel considered for the eNB is the 3GPP Channel Model, meanwhile the 3GPP building channel type and losses based on the 3GPP UMi Street Canyon propagation model is used for the gNBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' HIERARCHICAL DEEP Q-LEARNING (HIDQL) In this work, we use a state-of-the-art hierarchical Deep Q-learning (HiDQL) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' This algorithm is presented in [4], where the goal-directed behavior is studied under some specific sparse reward problems such as the Montezuma Revenge and a complex discrete stochastic decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' As shown in figure 2, a hierarchical agent located in the coordinator observes the user’s SINR report table and receives the extrinsic reward based on the feedback from the environment in terms of handover latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' This agent will adjust key parameters as TTT and SINR outage in order to maximize the agent’s extrinsic reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In the following subsection, we will present an overview of HiDQL and then formally define our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Algorithm 1 Hierarchical Deep Q-Learning (HiDQL) Initialize cmax, κ, experience replay buffers {Bmc, Bc}, policy networks {µmc, µc}, for the meta-controller and controller, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Function Done(c, cmax, Sκ, κ): if (c = cmax or Sκ < κ) then return True else return False end End Function for environment step t ← 1 to T do c ← 0 Init environment and initial state Execute goal from meta-controller: g = µmc(s) while cmax not reached do R ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' s0 ← s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Sκ = false while not Done do Execute action from controller: a = µc(s, g) Get next state s′ and similarity condition Sκ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Calculate intrinsic reward r and receive extrinsic reward rg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Store ([s, g], a, r, [s′, g], Done) in replay buffer Bc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Update µc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' c := c + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' R := R + rg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' s := s′ end Store (s0, g, R, s, Done) in replay buffer Bc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Update µmc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' if not Done then Execute goal from meta-controller: g = µmc(s) end end end A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Meta-controller and controller action space selection In the proposed hierarchical approach described in Al- gorithm 1, the meta-controller proposes goals or high level actions when the maximum c steps, cmax is achieved or when the similarity between the goal and low-level action, Sκ falls under a predefined threshold index κ as defined in the function Done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Meanwhile, the controller’s low level actions are executed c inner steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' For both levels, the actions are defined in the same fashion as: A(t) = �aout(t), attt(t)� , (2) where aout and attt are the SINR outage and TTT values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Such values are discretized into Ko and KT T T levels according the maximum and minimum defined values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The possible values for aout and attt are defined as: Aout = {Omin, Omin + Omax − Omin Ko − 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=', Omax}, (3) Attt = {TTTmin,TTTmin+ TTTmax − TTTmin KT T T − 1 , · · · , TTTmax}, (4) where Omin, Omax and TTTmin, TTTmax are predefined maximum and minimum values of SINR outage and TTT values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Finally, the size of the meta-controller and controller action spaces become S = |Aout| ∗ |Attt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' State space selection The state space is comprised by the Complete Report Table (CRT), which contains the relationship in terms of SINR of each user and the perceived gNB SINR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The SINR between an ith gNB and nth UE is calculated as follows: SINRi,n = PT X P Li,n Gi,n � k̸=i PT X P Lk,n Gk,n + Wtot × N0 , (5) where Gi,n and PLi,n are the beamforming gain and the pathlosss obtained between ith gNB and nth UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' PT X is the transmit power and Wtot × N0 is the thermal noise power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Thus, the state space is defined as follows: S(t) = �C(t)� , (6) where C(t) corresponds to the CRT with a size of |C(t)| = MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Intrinsic and extrinsic reward The intrinsic reward function is calculated by the controller by taking into account the goal selected by the meta-controller in c inner steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Such a reward is defined as: r(t) = ri(t) + rl2(t), (7) where ri corresponds to the immediate reward obtained from the environment and rl2 corresponds to reward based on the l2-norm between the controller’s action and the meta- controller’s goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Such rewards are defined as follows: ri(t) = � −f−1(−Du avg) if Du avg < DBF tol , 0 otherwise (8) where f−1(·) : [−e−1, 0) → [−1, −∞) is the lower branch of the Lambert function y = f−1(yey) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' DBF tol is the max- imum tolerable delay corresponding to the value of the best latency results per BF technology in the non-RL approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Du avg is the average latency of the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, we scale linearly Du avg → [e−1, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' rl2(t) = � 1 if Sκ < κ, −1 otherwise, (9) where Sκ = |a−g|2 and κ is the similarity threshold between the action and goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Finally, we can express the extrinsic reward as: R = c � j=t−c r(j), (10) where the extrinsic reward is defined as the sum of the intrinsic reward during the c inner steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' TABLE II: Network settings Parameter Value MT 3 Number of UEs 1 gNB center frequency 28 GHz gNB system bandwidth 200 MHz gNB numerology 2 gNB Pathloss Model 3GPP Umi-Street Canyon gNB Channel condition model Buildings gNB antenna height 10 m gNB number of antennas 64 eNB Center Frequency 2 GHz eNB System Bandwidth 100 MHz eNB Pathloss Model 3GPP Max Tx power 30 dBm UE number of antennas 16 UE antenna height 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6 m UE speed 13 m/s UE Tx power 10 dBm Traffic Model On-Off UDP application Packet payload size = 512 Bytes Packet window size = 256 Interval = 20 µs D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Clipped Double Q-Learning: RL scheme In addition to HiDQL, we utilized an RL solution named Clipped Double Q-Leaning (CDQL) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' For the CDQL algorithm, the state and action space correspond to the ones used in the lower level by the HiDQL algorithm in (6) and (2), respectively, meanwhile the reward corresponds to the intrinsic reward defined in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Baselines: Fixed TTT and Dynamic TTT For the present work, as baselines, we take a fixed and a dynamic solution into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The fixed and dynamic solutions were proposed in [11], where the fixed solution has a fixed value of TTT and in the dynamic solution TTT is calculated as follows: fT T T (∆) = TTTmax− ∆ − ∆min ∆max − ∆min (TTTmax−TTTmin), (11) where ∆ corresponds to the difference between the serving cell and the best gNB in terms of SINR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' PERFORMANCE EVALUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Simulation Setting We implement our proposed solution using the discrete network simulator ns-3 and its module mmWave [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Addi- tionally, ns3-ai module [14] is used to interface the simulation environment to our RL algorithm written in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Simulation settings and RL parameters are depicted in Table II and Table III, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Simulation Results We present the performance results of our proposed scheme in terms of convergence and handover latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Figure 3 (a), (b) depicts the convergence behavior for the CDQL and 0 20 40 60 80 100 Episode 0 20 40 60 80 100 Reward CDQL - Convergence Digital-Analog Hybrid-Analog Analog-Analog (a) 0 50 100 150 200 Episode −5 0 5 Reward HiDQL - Convergence Digital-Analog Hybrid-Analog Analog-Analog (b) Digital-Analog Hybrid-Analog Analog-Analog BF Architecture (gNB-UE) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='010 Latency(s) Latency performance HiDQL CDQL Dynamic TTT Fixed TTT (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' 3: (a) CDQL convergence performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' (b) HiDQL convergence performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' (c) Latency performance of the baselines Dynamic TTT and Fixed TTT and the proposed RL schemes CDQL and HiDQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' TABLE III: Learning parameters Parameter Value Handover algorithm Threshold to time: TTTmax = 150 ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' TTTmin = 25 ms SINR Outage: Omax = −3 dBm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Omin = −8 dBm DBF tol = min(DBF f , DBF d ) * Gym environment step time Event-based Batch size 32 Number of hidden layers (Nh) : 2 Number of neurons/layer (nl) : 64 HiDQL κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='25 Optimizer : Meta-Controller: Adam (1e-4), Controller: Adam (1e-4) cmax 50 steps CDQL Update target model type : Polyak averaging γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='95, ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='0, ϵmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='001, ϵdecay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='995 Optimizer : Adam (1e-4) DBF f and DBF d correspond to the latency performance per BF technology of the fixed TTT and dynamic TTT solutions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' HiDQL algorithms for different BF mechanisms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' It can be observed that CDQL presents a faster conver- gence when compared with HiDQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' However, converged reward values are not achieved equally among the tested BF architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' HiDQL presents a slower convergence time than CDQL however with a similar maximum convergence reward value among BF methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The intuition behind the previous results correspond to the inner nature of hierarchical learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' HiDQL performs an iterative search through the meta-controller’s proposal of goals to the controller and thus, obtaining optimum action selection in challenging scenarios as described in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The difference between reward scales between HiDQL and CDQL corresponds to how the reward is defined in both algorithms, thus a comparison in terms of the value of the reward is not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' TABLE IV: RL vs non-RL latency comparison Latency improvement (%) Digital-Analog BF Hybrid-Analog BF Analog-Analog BF RL Scheme F-TTT D-TTT F-TTT D-TTT F-TTT D-TTT HiDQL 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='0 CDQL 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='2 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='7 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' 3 (c), we present the latency performance of our techniques and the baselines: fixed TTT (F-TTT), dynamic TTT (D-TTT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' From the two baselines, the best performance in terms of latency is achieved by the dynamic TTT parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' On the other hand, we obtained an average 9% improvement when comparing HiDQL and CDQL in terms of latency results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' However, based on more detailed look into results, we further focus in Table IV on the latency results improvements over the previous solutions F-TTT and D-TTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' As shown in Table IV, when compared with the dynamic TTT results, a gain of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6% and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1%, for Digital-Analog BF, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1% and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6%, for Hybrid-Analog BF, and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='7% and 39% for Analog-Analog BF were obtained for HiDQL and CDQL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' TABLE V: CI vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' non-CI latency comparison Latency improvement (%) Digital-Analog BF Hybrid-Analog BF Analog-Analog BF CI Non-CI F-TTT 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='57 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='5 D-TTT 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='7 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' CONTEXT INFORMATION AWARE DC HANDOVER In addition to the SCH algorithm used as part of the proposed DC architecture, we foresee the possible advantages of using context information to improve handover time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In [15], the authors give an overview of the typical initial access (IA) cell search algorithms where contrary to LTE, 5G NR provide mechanisms to determine suitable initial directions of transmission to avoid high isotropic losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Among the IA algorithms, context information (CI)-based algorithms allow users to be aware of the location of the surrounded 5G NR antennas through an LTE link and GPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' In this work, we leverage GPS information as context and reduce the sector in which a gNB should sense the space towards an UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Figure 4 depicts the beforementioned mechanism, where given the user position P1 the gNB is able to calculate a reduced sector to obtain the periodic SRS signals from the surrounded UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' We also assume that each UE has line of sight both with the coordinator and the gNBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' The proposed mechanism is described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' As shown in Table V, context awareness directly improves the DC handover mechanism with a considerable latency improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' CONCLUSIONS In this paper, we presented two Reinforcement Learning (RL)-based dual connectivity (DC) solutions that further 90o 0o 180o 270o P1(x1,y1) gNB (X, Y) δ 2 1 3 4 gNB UE ψ ψ P1’(x1,Y) a) b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' 4: a) ±ψ indicates the sweep angle to obtain SINR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' b) δ corresponds to the relative angle used to derive the sweep sector angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Algorithm 2 CI-aware sector sweep selection algorithm Result: Sector’s angle to perform measurements sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Given P1(x1, y1) and (X, Y ), the UE’s and gNB geolocations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, δ = arcsin ⃗ d2 ⃗ d1 and ΘgNB = 2π N ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' where ⃗d2 = −−−→ P1P ′ 1 and ⃗d1 = −−−−−→ P1gNB # Step 1: Detect UE’s relative quadrant q ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Calculate angle φ respect gNB quadrant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' if y1 > Y and x1 > X then # UE in 1st quadrant φ = δ else if y1 > Y and x1 < X then # UE in 2nd quadrant φ = π 2 + δ else if y1 < Y and x1 < X then # UE in 3rd quadrant φ = π + δ else # UE in 4th quadrant φ = 3π 2 + δ end # Step 2: Calculate sector angle (SA) SA ∈ [φ − 2Θ, φ + 2Θ] Digital-Analog Hybrid-Analog Analog-Analog BF Architecture (gNB-UE) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='010 Latency(s) Latency performance Dynamic TTT (C) Dynamic TTT Fixed TTT (C) Fixed TTT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' 5: Context information (C) latency performance results when utilizing Dynamic TTT and Fixed TTT baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' improve latency metric in inter-RAT handover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Additionally, we propose leveraging an initial access cell search algorithm that utilizes context information to reduce the latency in gNB- UE’s measurement collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' We compared two RL solutions named Hierarchical Deep Q-Learning (HiDQL) and Clipped Double Q-Learning (CDQL) with two previously proposed baselines, fixed Threshold to Time (TTT) and dynamic TTT, obtaining a significant gain in both cases in terms of latency with a 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6% and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1%, for Digital-Analog BF, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='1% and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='6%, for Hybrid-Analog BF and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='7% and 39% for Analog-Analog BF when compared with the baselines’ best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' Finally, we presented a context-aware mechanism under line of sight assumptions and obtained a significant improvement in terms of latency when compared with the case of no context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' As future work we forsee the integration of the presented context-aware mechanism and reinforcement learning to further improve the DC handover mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' ACKNOWLEDGMENT This research is supported by Mitacs and Ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content=' REFERENCES [1] 3GPP TR 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='842, “Study on small cell enhancements for E-UTRA and E-UTRAN, v12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE5T4oBgHgl3EQfCA5G/content/2301.05391v1.pdf'} +page_content='0.' metadata={'source': 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+GENERATING FUNCTIONS, II +DAVID BENSON, RADHA KESSAR, AND MARKUS LINCKELMANN +Abstract. We relate the generating functions of the dimensions of the Hochschild coho- +mology in any fixed degree of the symmetric groups with those of blocks of the symmetric +groups. We show that the first Hochschild cohomology of a positive defect block of a sym- +metric group is non-zero, answering in the affirmative a question of the third author. To +do this, we prove a formula expressing the dimension of degree one Hochschild cohomol- +ogy as a sum of dimensions of centres of blocks of smaller symmetric groups. This in turn +is a consequence of a general formula that makes more precise a theorem of our previous +paper describing the generating functions for the dimensions of Hochschild cohomology of +symmetric groups. +1. Introduction +Let p be a prime number and k a field of characteristic p. As a consequence of results in +[5], using the classification of finite simple groups, if G is a finite group of order divisible +by p, then HH1(kG) is non-zero. It is an open question [6, Question 7.4] whether for G a +finite group and B a block of kG, if the defect groups of B are non-trivial, then HH1(B) is +non-zero. This question has been shown to have a positive answer in some cases in [8] and +[7], for instance. We prove that this question has an affirmative answer if G is a symmetric +group Sn on n letters. +Theorem 1.1. Let B a block of kSn with non trivial defect groups. Then HH1(B) ̸= 0. +In fact, we give a precise formula for the dimension of the first Hochschild cohomology +of a block of a symmetric group as a sum of dimensions of the centres of blocks of smaller +symmetric groups (Theorem 1.2), and this easily implies Theorem 1.1. +In order to state the formula, let us recall that to each block B of kSn is associated a non- +zero integer w ⩽ ⌊n/p⌋ called the weight of B, with the property that the Sylow p-subgroups +of Spw are defect groups of B under the natural inclusion Spw ⩽ Sn. In particular, B has +non-trivial defect groups if and only if w > 0. Moreover, by Theorem 7.2 of Chuang and +Rouquier [3] if B and B′ are blocks of possibly different symmetric groups, with the same +weight, then B and B′ are derived equivalent algebras, and consequently, for any r ⩾ 0, we +have dimk HHr(B) = dimk HHr(B′). +For w ⩾ 0, denote by Bpw the principal block of kSpw. Then Bpw has weight w and by +the above, dimk HHr(B) = dimk HHr(Bpw) for any weight w block B of a symmetric group +algebra. Thus Theorem 1.1 is a consequence of the following result. For w ⩾ 0, let ρ(pw, ∅) +equal the number of partitions of pw with empty p-core. +2020 Mathematics Subject Classification. 20C20. +Key words and phrases. Hochschild cohomology, symmetric groups, partition identities, blocks. +1 + +Theorem 1.2. Let B be a weight w-block of a symmetric group algebra over k. If p = 2, +then +dimk HH1(B) = 2 +w−1 +� +j=0 +dimk Z(Bpj) = 2 +w−1 +� +j=0 +ρ(pj, ∅). +If p ⩾ 3, then +dimk HH1(B) = +w−1 +� +j=0 +dimk Z(Bpj) = +w−1 +� +j=0 +ρ(pj, ∅). +The proof of the above formula goes through the following theorem relating generating +functions of dimensions of Hochschild cohomology of blocks of symmetric groups with those +of the entire group algebra and then invoking the results of our previous paper [1]. Denote +by p(n) the number of partitions of n, and by P(t) the generating function �∞ +n=0 p(n)tn. +Note that dimk HH0(kSn) = dimk Z(kSn) = p(n). +Theorem 1.3. Set Z(t) = +∞ +� +n=0 +dimk Z(Bpn) tn. +For any r ⩾ 1, there exists a rational +function φ(t) (depending on p and r) with φ(0) non-zero, such that +∞ +� +n=0 +dimk HHr(Bpn) tn = tφ(t)Z(t) +and +∞ +� +n=0 +dimk HHr(kSn) tn = tpφ(tp)P(t). +Remark 1.4. In [1], we proved that +∞ +� +n=0 +dimk HHr(Bpn) tn = Rp,r(t)P(t) +where Rp,r(t) is a rational function of t. Ken Ono asked us whether Rp,r(t) is a rational +function of tp, and Theorem 1.3 proves that this is the case, with Rp,r(t) = tpφ(tp). +Remark 1.5. The constant coefficient of φ(t) in Theorem 1.3 is equal to y1 = dimk(HHr(Bp)). +Since Bp is derived equivalent to k(Cp ⋊ Cp−1), we have y1 = dimk HHr(k(Cp ⋊ Cp−1)). An +easy calculation, using the centraliser decomposition, shows that for r ⩾ 1 we have y1 = 2 if +r ≡ 0 or −1 modulo 2(p − 1) (which is in particular the case if p = 2) and y1 = 1 otherwise. +Remark 1.6. We note that the above results do not depend on the choice of the field k. If k′ is +an extension field of k and B a block of kSn for some positive integer n, then B′ = k′ ⊗k B is +a block of k′Sn having the same defect groups as B, and for any finite-dimensional k-algebra +A we have a graded k-algebra isomorphism HH∗(k′ ⊗k A) ∼= k′ ⊗k HH∗(A). +2. Proofs. +We begin with an elementary lemma. +Lemma 2.1. Let R be an integral domain and m a positive integer. If h(t) ∈ R[[t]] is such +that h(tm) ∈ R((t)) is a rational function, then h(t) is also a rational function. +2 + +Proof. Let h(t) = �∞ +n=0 hntn, hn ∈ R and suppose that a(t), b(t) ∈ R[t] are such that +∞ +� +n=0 +hntmn = h(tm) = a(t) +b(t) . +If h(t) = 0, then there is nothing to prove. So, we assume that h(t) ̸= 0. Write +a(t) = +m−1 +� +s=0 +as(tm) ts, +b(t) = +m−1 +� +s=0 +bs(tm) ts, +for as(t), bs(t) ∈ R[t], 0 ⩽ s ⩽ m − 1. Comparing coefficients of powers of t, the equality +a(t) = b(t) +∞ +� +n=0 +hn tmn +implies the equality +bs(tm) = as(tm) +∞ +� +n=0 +hn tmn +for each s, 0 ⩽ s ⩽ m − 1. Choose s such that as(t) ̸= 0. Then h(t) = as(t)/bs(t) is a +rational function of t. +□ +Proof of Theorem 1.3. Let r ⩾ 1. For n ⩾ 0, let c(n) denote the number of p-core partitions +of n and for each s, 0 ⩽ s ⩽ p − 1, set Cs(t) = �∞ +n=0 c(np + s) tn. +For w ⩾ 0, set +zpw = dimk Z(Bpw) and ypw = dimk HHr(Bpw). We use the notation +Z(t) = +∞ +� +n=0 +zpn tn +from the statement of Theorem 1.3 and we set +Y (t) = +∞ +� +n=0 +dimk HHr(Bpn) tn = +∞ +� +n=0 +ypn tn. +Note that by [3, Theorem 7.2], ypw is the dimension of the degree r Hochschild cohomology +of any weight w block of a symmetric group algebra. Further, note that by the Nakayama +conjecture, proved by Brauer [2] and Robinson [4], the number of weight w blocks of kSn +equals c(n−pw). Since the Hochschild cohomology of a finite dimensional algebra is the direct +sum of the Hochschild cohomologies of its blocks, we have that for any s with 0 ⩽ s ⩽ p −1, +and any n ⩾ 0, +dimk Z(kSpn+s) = +n +� +w=0 +zpwc(p(n − w) + s), +and +dimk HHr(kSpn+s) = +n +� +w=0 +ypwc(p(n − w) + s). +In other words, for any s, 0 ⩽ s ⩽ p − 1, +(2.2) +∞ +� +n=0 +dimk Z(kSpn+s) tpn+s = tsZ(tp)Cs(tp) +3 + +and +(2.3) +∞ +� +n=0 +dimk HHr(kSpn+s) tpn+s = tsY (tp)Cs(tp). +Since B0 ∼= k, we have z0 = dimk Z(B0) = 1 and from this it follows that Z(t) is invertible +in Z[[t]]. On the other hand, we have y0 = dimk HHr(B0) = 0 and y1 = dimk HHr(Bp) ̸= 0 +(cf. Remark 1.5). In particular, Y (t) is divisible by t but not by t2 in Z[[t]]. So we may +define a power series in t, with non-zero constant term, +φ(t) = (t−1Y (t))Z(t)−1 ∈ Z[[t]], +and then we have +(2.4) +Y (t) = tφ(t)Z(t). +Now, +dimk HHr(kSn) tn = +p−1 +� +s=0 +∞ +� +n=0 +dimk HHr(kSpn+s) tpn+s = Y (tp) +p−1 +� +s=0 +Cs(tp) ts += tpφ(tp) +p−1 +� +s=0 +Z(tp)Cs(tp) ts +where we have used (2.3) for the second equality and (2.4) for the third equality. On the +other hand, by (2.2), +P(t) = +p−1 +� +s=0 +∞ +� +n=0 +dimk Z(kSpn+s) tpn+s = +p−1 +� +s=0 +Z(tp)Cs(tp) ts. +So, +∞ +� +n=0 +dimk HHr(kSn) tn = tpφ(tp)P(t). +Now by Theorem 1.3 of [1] we have that tpφ(tp) is a rational function of t. It then follows +from Lemma 2.1 that φ(t) is a rational function of t, and by the above, φ(0) is non-zero. +□ +Proof of Theorem 1.2. By Theorem 1.2 of [1], +∞ +� +n=0 +dimk HH1(kSn) tn = + + + + + + + +2t2 +1 − t2 P(t) +p = 2, +tp +1 − tp P(t) +p ⩾ 3. +Note that in [1] the base field is taken to be Fp, but the above result clearly holds as well +for any field of characteristic p (cf. Remark 1.6). By Theorem 1.3 and its notation, applied +with r = 1, +∞ +� +n=0 +dimk HH1(Bpn) tn = + + + + + + + +2t +1 − t Z(t) +p = 2, +t +1 − t Z(t) +p ⩾ 3. +4 + +Here, note that h(t) → h(tm) is an injective map on Z[[t]]. This, along with [3, Theorem +7.2] proves the equations +dimk HH1(Bpn) = + + + + + + + + + + + +2 +w−1 +� +j=0 +dimk Z(Bpj) +p = 2, +w−1 +� +j=0 +dimk Z(Bpj) +p ⩾ 3. +of Theorem 1.2. The remaining equations follow from the Nakayama Conjecture, proved +in [2, 4], which states that characters of a symmetric group belong to the same p-block if +and only if the corresponding partitions have the same p-core, implying in particular that +dimk Z(Bpj) = ρ(pj, ∅). +□ +Theorem 1.1 is an immediate consequence of Theorem 1.2. +Acknowledgements. +The first and second authors are grateful to City, University of +London for its hospitality during the research for this paper. The third author acknowledges +support from EPSRC grant EP/T004592/1. +References +1. D. J. Benson, R. Kessar, and M. Linckelmann, Hochschild cohomology of symmetric groups and generating +functions, J. Group Theory, to appear. +2. R. Brauer, On a conjecture by Nakayama, Trans. Roy. Soc. Canada Sect. III 41 (1947), 11–19. +3. J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and sl2-categorification, Ann. of +Math. 167 (2008), 245–298. +4. G. de B. Robinson, On a conjecture by Nakayama, Trans. Roy. Soc. Canada Sect. III 41 (1947), 20–25. +5. P. Fleischmann, I. Janiszczak, and W. Lempken, Finite groups have local non-Schur centralizers, +Manuscripta Math. 80 (1993), no. 2, 213–224. +6. M. Linckelmann, Finite dimensional algebras arising as blocks of finite group algebras, Representations +of Algebras (G. Leuschke, F. Bleher, R. Schiffler, and D. Zacharia, eds.), Contemp. Math., vol. 705, +American Math. Society, 2018, pp. 155–188. +7. W. Murphy, The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic +Mathieu groups, Journal of Group Theory, to appear; arXiv:2110.02941. +8. C.-C. Todea, Nontriviality of the first Hochschild cohomology of some block algebras of finite groups, +J. Pure & Applied Algebra 227 (2023), no. 2, Paper No. 107192, 10pp. +David Benson, Institute of Mathematics, Fraser Noble Building, University of Aberdeen, +King’s College, Aberdeen AB24 3UE, United Kingdom +Radha Kessar, Department of Mathematics, University of Manchester, Oxford Road, +Manchester M193PL, United Kingdom +Markus Linckelmann, School of Science & Technology, Department of Mathematics, +City, University of London, Northampton Square, London EC1V 0HB, United Kingdom +5 + diff --git a/ZdE1T4oBgHgl3EQfcgSg/content/tmp_files/load_file.txt b/ZdE1T4oBgHgl3EQfcgSg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..80d384010588643b0e86672251bc504c3fcd9f4f --- /dev/null +++ b/ZdE1T4oBgHgl3EQfcgSg/content/tmp_files/load_file.txt @@ -0,0 +1,190 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf,len=189 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='03185v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='GR] 9 Jan 2023 HOCHSCHILD COHOMOLOGY OF SYMMETRIC GROUPS AND GENERATING FUNCTIONS, II DAVID BENSON, RADHA KESSAR, AND MARKUS LINCKELMANN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' We relate the generating functions of the dimensions of the Hochschild coho- mology in any fixed degree of the symmetric groups with those of blocks of the symmetric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' We show that the first Hochschild cohomology of a positive defect block of a sym- metric group is non-zero, answering in the affirmative a question of the third author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' To do this, we prove a formula expressing the dimension of degree one Hochschild cohomol- ogy as a sum of dimensions of centres of blocks of smaller symmetric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' This in turn is a consequence of a general formula that makes more precise a theorem of our previous paper describing the generating functions for the dimensions of Hochschild cohomology of symmetric groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Introduction Let p be a prime number and k a field of characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' As a consequence of results in [5], using the classification of finite simple groups, if G is a finite group of order divisible by p, then HH1(kG) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' It is an open question [6, Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='4] whether for G a finite group and B a block of kG, if the defect groups of B are non-trivial, then HH1(B) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' This question has been shown to have a positive answer in some cases in [8] and [7], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' We prove that this question has an affirmative answer if G is a symmetric group Sn on n letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Let B a block of kSn with non trivial defect groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Then HH1(B) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' In fact, we give a precise formula for the dimension of the first Hochschild cohomology of a block of a symmetric group as a sum of dimensions of the centres of blocks of smaller symmetric groups (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2), and this easily implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' In order to state the formula, let us recall that to each block B of kSn is associated a non- zero integer w ⩽ ⌊n/p⌋ called the weight of B, with the property that the Sylow p-subgroups of Spw are defect groups of B under the natural inclusion Spw ⩽ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' In particular, B has non-trivial defect groups if and only if w > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Moreover, by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2 of Chuang and Rouquier [3] if B and B′ are blocks of possibly different symmetric groups, with the same weight, then B and B′ are derived equivalent algebras, and consequently, for any r ⩾ 0, we have dimk HHr(B) = dimk HHr(B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' For w ⩾ 0, denote by Bpw the principal block of kSpw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Then Bpw has weight w and by the above, dimk HHr(B) = dimk HHr(Bpw) for any weight w block B of a symmetric group algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Thus Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='1 is a consequence of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' For w ⩾ 0, let ρ(pw, ∅) equal the number of partitions of pw with empty p-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 20C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Hochschild cohomology, symmetric groups, partition identities, blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 1 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Let B be a weight w-block of a symmetric group algebra over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' If p = 2, then dimk HH1(B) = 2 w−1 � j=0 dimk Z(Bpj) = 2 w−1 � j=0 ρ(pj, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' If p ⩾ 3, then dimk HH1(B) = w−1 � j=0 dimk Z(Bpj) = w−1 � j=0 ρ(pj, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' The proof of the above formula goes through the following theorem relating generating functions of dimensions of Hochschild cohomology of blocks of symmetric groups with those of the entire group algebra and then invoking the results of our previous paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Denote by p(n) the number of partitions of n, and by P(t) the generating function �∞ n=0 p(n)tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Note that dimk HH0(kSn) = dimk Z(kSn) = p(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Set Z(t) = ∞ � n=0 dimk Z(Bpn) tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' For any r ⩾ 1, there exists a rational function φ(t) (depending on p and r) with φ(0) non-zero, such that ∞ � n=0 dimk HHr(Bpn) tn = tφ(t)Z(t) and ∞ � n=0 dimk HHr(kSn) tn = tpφ(tp)P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' In [1], we proved that ∞ � n=0 dimk HHr(Bpn) tn = Rp,r(t)P(t) where Rp,r(t) is a rational function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Ken Ono asked us whether Rp,r(t) is a rational function of tp, and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3 proves that this is the case, with Rp,r(t) = tpφ(tp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' The constant coefficient of φ(t) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3 is equal to y1 = dimk(HHr(Bp)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Since Bp is derived equivalent to k(Cp ⋊ Cp−1), we have y1 = dimk HHr(k(Cp ⋊ Cp−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' An easy calculation, using the centraliser decomposition, shows that for r ⩾ 1 we have y1 = 2 if r ≡ 0 or −1 modulo 2(p − 1) (which is in particular the case if p = 2) and y1 = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' We note that the above results do not depend on the choice of the field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' If k′ is an extension field of k and B a block of kSn for some positive integer n, then B′ = k′ ⊗k B is a block of k′Sn having the same defect groups as B, and for any finite-dimensional k-algebra A we have a graded k-algebra isomorphism HH∗(k′ ⊗k A) ∼= k′ ⊗k HH∗(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' We begin with an elementary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Let R be an integral domain and m a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' If h(t) ∈ R[[t]] is such that h(tm) ∈ R((t)) is a rational function, then h(t) is also a rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Let h(t) = �∞ n=0 hntn, hn ∈ R and suppose that a(t), b(t) ∈ R[t] are such that ∞ � n=0 hntmn = h(tm) = a(t) b(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' If h(t) = 0, then there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' So, we assume that h(t) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Write a(t) = m−1 � s=0 as(tm) ts, b(t) = m−1 � s=0 bs(tm) ts, for as(t), bs(t) ∈ R[t], 0 ⩽ s ⩽ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Comparing coefficients of powers of t, the equality a(t) = b(t) ∞ � n=0 hn tmn implies the equality bs(tm) = as(tm) ∞ � n=0 hn tmn for each s, 0 ⩽ s ⩽ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Choose s such that as(t) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Then h(t) = as(t)/bs(t) is a rational function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Let r ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' For n ⩾ 0, let c(n) denote the number of p-core partitions of n and for each s, 0 ⩽ s ⩽ p − 1, set Cs(t) = �∞ n=0 c(np + s) tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' For w ⩾ 0, set zpw = dimk Z(Bpw) and ypw = dimk HHr(Bpw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' We use the notation Z(t) = ∞ � n=0 zpn tn from the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3 and we set Y (t) = ∞ � n=0 dimk HHr(Bpn) tn = ∞ � n=0 ypn tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Note that by [3, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2], ypw is the dimension of the degree r Hochschild cohomology of any weight w block of a symmetric group algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Further, note that by the Nakayama conjecture, proved by Brauer [2] and Robinson [4], the number of weight w blocks of kSn equals c(n−pw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Since the Hochschild cohomology of a finite dimensional algebra is the direct sum of the Hochschild cohomologies of its blocks, we have that for any s with 0 ⩽ s ⩽ p −1, and any n ⩾ 0, dimk Z(kSpn+s) = n � w=0 zpwc(p(n − w) + s), and dimk HHr(kSpn+s) = n � w=0 ypwc(p(n − w) + s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' In other words, for any s, 0 ⩽ s ⩽ p − 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2) ∞ � n=0 dimk Z(kSpn+s) tpn+s = tsZ(tp)Cs(tp) 3 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3) ∞ � n=0 dimk HHr(kSpn+s) tpn+s = tsY (tp)Cs(tp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Since B0 ∼= k, we have z0 = dimk Z(B0) = 1 and from this it follows that Z(t) is invertible in Z[[t]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' On the other hand, we have y0 = dimk HHr(B0) = 0 and y1 = dimk HHr(Bp) ̸= 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' In particular, Y (t) is divisible by t but not by t2 in Z[[t]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' So we may define a power series in t, with non-zero constant term, φ(t) = (t−1Y (t))Z(t)−1 ∈ Z[[t]], and then we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='4) Y (t) = tφ(t)Z(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Now, dimk HHr(kSn) tn = p−1 � s=0 ∞ � n=0 dimk HHr(kSpn+s) tpn+s = Y (tp) p−1 � s=0 Cs(tp) ts = tpφ(tp) p−1 � s=0 Z(tp)Cs(tp) ts where we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3) for the second equality and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='4) for the third equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2), P(t) = p−1 � s=0 ∞ � n=0 dimk Z(kSpn+s) tpn+s = p−1 � s=0 Z(tp)Cs(tp) ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' So, ∞ � n=0 dimk HHr(kSn) tn = tpφ(tp)P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Now by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3 of [1] we have that tpφ(tp) is a rational function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' It then follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='1 that φ(t) is a rational function of t, and by the above, φ(0) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2 of [1], ∞ � n=0 dimk HH1(kSn) tn = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 2t2 1 − t2 P(t) p = 2, tp 1 − tp P(t) p ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Note that in [1] the base field is taken to be Fp, but the above result clearly holds as well for any field of characteristic p (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='3 and its notation, applied with r = 1, ∞ � n=0 dimk HH1(Bpn) tn = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 2t 1 − t Z(t) p = 2, t 1 − t Z(t) p ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 4 Here, note that h(t) → h(tm) is an injective map on Z[[t]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' This, along with [3, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2] proves the equations dimk HH1(Bpn) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 2 w−1 � j=0 dimk Z(Bpj) p = 2, w−1 � j=0 dimk Z(Bpj) p ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' The remaining equations follow from the Nakayama Conjecture, proved in [2, 4], which states that characters of a symmetric group belong to the same p-block if and only if the corresponding partitions have the same p-core, implying in particular that dimk Z(Bpj) = ρ(pj, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' □ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='1 is an immediate consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' The first and second authors are grateful to City, University of London for its hospitality during the research for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' The third author acknowledges support from EPSRC grant EP/T004592/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Benson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Kessar, and M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Todea, Nontriviality of the first Hochschild cohomology of some block algebras of finite groups, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' Pure & Applied Algebra 227 (2023), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 2, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' 107192, 10pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} +page_content=' David Benson, Institute of Mathematics, Fraser Noble Building, University of Aberdeen, King’s College, Aberdeen AB24 3UE, United Kingdom Radha Kessar, Department of Mathematics, University of Manchester, Oxford Road, Manchester M193PL, United Kingdom Markus Linckelmann, School of Science & Technology, Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, United Kingdom 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE1T4oBgHgl3EQfcgSg/content/2301.03185v1.pdf'} diff --git a/ZtE0T4oBgHgl3EQf4ALw/content/tmp_files/2301.02734v1.pdf.txt b/ZtE0T4oBgHgl3EQf4ALw/content/tmp_files/2301.02734v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..15220317d8597dd3c67806d71753a2d204993924 --- /dev/null +++ b/ZtE0T4oBgHgl3EQf4ALw/content/tmp_files/2301.02734v1.pdf.txt @@ -0,0 +1,1585 @@ +Political, economic, and governance attitudes of blockchain users +Lucia M. Korpas1, Seth Frey2,3, Joshua Tan1,4* +1The Metagovernance Project, Brookline, MA, USA +2Department of Communication, University of California Davis, Davis, CA, USA +3The Ostrom Workshop, Indiana University, Bloomington, IN, USA +4University of Oxford, Oxford, Oxfordshire, UK +* Correspondence: +Joshua Tan +joshua.z.tan@gmail.com +Abstract +We present a survey to evaluate crypto-political, crypto-economic, and crypto-governance +sentiment in people who are part of a blockchain ecosystem. Based on 3710 survey +responses, we describe their beliefs, attitudes, and modes of participation in crypto and +investigate how self-reported political affiliation and blockchain ecosystem affiliation are +associated with these. We observed polarization in questions on perceptions of the +distribution of economic power, personal attitudes towards crypto, normative beliefs about +the +distribution of power in governance, and external regulation of blockchain +technologies. Differences in political self-identification correlated with opinions on +economic fairness, gender equity, decision-making power and how to obtain favorable +regulation, while blockchain affiliation correlated with opinions on governance and +regulation of crypto and respondents’ semantic conception of crypto and personal goals +for their involvement. We also find that a theory-driven constructed political axis is +supported by the data and investigate the possibility of other groupings of respondents or +beliefs arising from the data. +1 +Introduction +As blockchain technology has evolved over more than a decade, cryptocurrencies and +crypto-economic systems have had a growing impact on the world. Millions of people have involved +themselves in crypto1: as of 2021, around 15 percent of American adults have reported owning +cryptocurrency (Perrin 2021), and many other countries have even higher adoption rates (Buchholz +2021). The past few years have seen the growth of decentralized apps and the crypto startup industry. +Correspondingly, governments are beginning to take regulatory actions. Also, even as blockchain +ecosystems move towards less computationally-intensive consensus mechanisms, the ongoing +environmental impact of blockchain use is huge. +Given the impact of crypto-economic activity on individuals and on shared resources, it is +increasingly important to understand how its users are relating to the technology. While the hard data +of cryptocurrency transactions and account balances is often publicly available by design, users’ +1 Throughout the text, we use the term “crypto” to encompass blockchain technologies such as +cryptocurrencies and the communities and ideologies which drive their development and use. +1 + +motivations for engaging with crypto are more opaque. There is little existing data on the stated +beliefs or attitudes of the variety of people using blockchain technologies. What do blockchain users +believe about the economic, political, and social relevance of crypto? While there has been attention +to the attitudes of the general population towards cryptocurrencies and blockchain technology (Perrin +2021; “Global State of Crypto, 2022” 2022), there is also a need to understand the beliefs of active +participants of blockchain ecosystems. +What do blockchain ecosystem participants believe about how the technology is being – or should be +– developed, used, and regulated? Are there discrete types of crypto contributors, or is there a +spectrum of beliefs? What specific beliefs are most relevant in distinguishing respondents between +types or along axes? This work is a first step in the development of a framework for thinking about +this spectrum or grouping of beliefs in crypto. +We report the results of a large-scale survey of participants in the blockchain economy. The survey +was designed to shed light on respondents’ socioeconomic and sociopolitical beliefs relating to +crypto, economic modes of engagement with crypto, and attitudes towards governance of blockchain +technology. We describe the distributions of these responses and their relationships to self-reported +political ideology and specific crypto ecosystems such as Bitcoin and Ethereum. +We also evaluate the survey instrument itself: are the questions able to assess distinct and relevant +facets of beliefs? Can we identify underlying factors which describe broader groupings of beliefs? +Using factor analysis methods, we find that a political axis and corresponding typology, informed by +the Pew Research Center’s Political Typology Quiz, meaningfully describes variation between +respondents. +2 +Background +While there is no existing political theory of crypto per se, there are substantial ethnographic studies +of crypto communities (and related digital communities) that address the political dimensions of +crypto. For example, ethnographic studies have informed the creation of a proposed political +typology of blockchain projects (Husain, Franklin, and Roep 2020), reflecting earlier ideas on the +“intrinsic” political values of technical artifacts (Winner 1980). In this vein, cryptocurrencies have +been characterized as realizations of crypto-anarchist values such as privacy and autonomy (Chohan +2017; Beltramini 2021), following in the footsteps of earlier cypherpunk writings (Hughes 1993; +May 1994) as well as the original Bitcoin whitepaper (Nakamoto 2008). Other ethnographies have +described issues of on- and off-chain governance (De Filippi and Loveluck 2016) and the political +motivations and cultural context of projects such as Bitcoin (Golumbia 2016) and Ethereum (Brody +and Couture 2021). +A previous industry survey, conducted by CoinDesk in 2018, contained several questions related to +politics and governance (Ryan 2018; Bauerle and Ryan 2018), though the questions focused more +specifically on individual projects and topical questions such as reactions to SEC rulings on the +securitization status of Ethereum. +Distinct from questions about political values, the topic of blockchain governance—including the +relationship between blockchains and traditional governments—is one of the most salient and +polarizing questions in crypto, one that has led to the creation, forking, and dissolution of many +projects. While we cannot recount all the major positions here (some of which are reflected in the +survey itself; see “Methodology”), there is a broad distinction between approaches that emphasize +on-chain governance and those that emphasize off-chain governance. A number of academic analyses +2 + +have studied these different approaches to blockchain governance (Reijers, O’Brolcháin, and Haynes +2016; Liu et al. 2021; van Pelt et al. 2021), along with a vastly greater number of industry manifestos +and opinion pieces (Zamfir 2019; Szabo 1996). +3 +Methodology +3.1 +Survey questions +The survey consists of 19 questions related to respondents’ crypto-related beliefs and activities, with +three types of questions interspersed: those eliciting opinions about the political dimensions of crypto +activity (“crypto-political”), those eliciting economics opinions (“crypto-economic”), and those +eliciting attitudes about the governance of crypto projects (“crypto-governance”). All questions were +multiple choice, with 2-4 possible selections, and the respondent could opt not to answer. See Table 1 +for the full list of questions. +The survey questions and provided choices included both a formal portion drawing from existing +political survey instruments and a more exploratory portion intended to elicit beliefs relevant to a +general crypto-political typology. In particular, a few of the questions selected (Q11-13, Q15, Q19), +were based on questions from Pew’s Political Typology Quiz (Nadeem 2021) and intended to relate +to political sentiment. Other questions (e.g., Q1, Q17) were developed in collaboration with a number +of community members in crypto, drawing on the culture, memes, and references common in crypto. +Altogether, the content was designed to elicit respondents' primary modes of economic engagement +with crypto, their political sentiment, and opinions as to how crypto communities themselves should +be governed. +3.2 +Construction of political “types” and identification of “axes” of belief +Our choice to identify separate “axes” of economic, political, and governance beliefs were based on +discussion with community members and in analogy to existing classifications such as the traditional +“left-right” political axis. For one of these, the political axis, we also leveraged our study design to +group and relate questions more directly by defining a continuous construct intended to assess +respondents' crypto-political leanings. We identified a subset of questions as most relevant to political +orientation, and computed a score for each participant by summing the responses to these questions +(coded with values in the range [-1, 2] as described in Table 1) in analogy to the Pew methodology +(Nadeem 2021). The lowest and highest scores on this political “axis” were designed to highlight +extreme positions of collectivist and anarcho-capitalist approaches to using blockchain technology. +Five discrete types were defined by thresholds in the score according to Table 2: +crypto-anarcho-capitalist, crypto-libertarian, centrist, crypto-communitarian, and crypto-leftist; these +types were developed both with definitions from the Pew typologies and with input from the +community. +3.3 +Recruitment +We relied on a convenience (self-selected) sample of participants in the crypto community. +Participants were recruited by distributing the survey through blockchain-focused forums and +listservs, conferences (LisCon and ETHDenver), social media posts, and articles published on +blockchain-focused news sites. +We motivated voluntary participant engagement with two strategies. We presented the survey as a +quiz that assigned respondents one out of an entertaining typology of “types” on the basis of their +3 + +responses (“crypto-leftist,” “cryptopunk,” etc.) immediately upon completion of the survey. Stylized +as “factions”, the crypto-political types corresponded to the political types we defined based on the +Pew typology, while the crypto-economic and crypto-governance types were constructed by using +thresholds to partition respondents into five ad hoc types (for more detail, see Section 1.1 of the +Supplementary Material). We also incentivized survey completion with the opportunity to receive a +non-fungible token (NFT) corresponding to their assigned “type”, contingent upon their provision of +a valid Ethereum wallet address or ENS name. +3.4 +Analysis +To survey the overall landscape of crypto-political beliefs, we observed the distribution of choices +selected by respondents. We aggregated these responses for each question, including the null +response of no choice selected, and computed the margins of error for a 95% confidence interval, +assuming a random sample of the population. +To investigate how political self-identification and participation in specific blockchain ecosystems +related to beliefs, we grouped participants by their responses to the corresponding questions. We then +determined which questions displayed a statistically significant difference in the distribution of +responses between these groups. +We also wanted to understand which questions were most meaningful in differentiating respondents. +To this end, we first performed a check on the extent to which each question measured a distinct +belief by computing the correlation between responses to different questions as Cramer’s V (a +version of Pearson’s chi-squared statistic scaled to provide a measure of association). Then we used +principal component analysis (PCA) to identify which of the 48 choices provided across 17 questions +explained the most variance between respondents. Specifically, we looked at the component loading +for each feature, i.e., contribution to the first principal component. +For use with PCA, we normalized the feature data (shifted to a mean of 0 with unit variance). Note +that for questions where only two choices were provided, the alternative answer contributed with +equal magnitude (though opposite sign). We omitted questions 2 and 19, relating to specific +blockchain ecosystems. We also omitted answers from respondents that did not answer all questions. +We also wished to evaluate to what extent our crypto-political types, delineated from the political +score we defined, corresponded with patterns in beliefs across respondents. From the PCA results, we +can identify to what extent the assigned types or classes directly correlate with any of the first few +components. +4 +Results +4.1 +Responses and respondents +Between September 27, 2021 and March 4, 2022, the survey received 3710 responses. In 3418 (92%) +of these, all questions were answered. For questions presented to all respondents, the percentage of +respondents who chose not to answer each question was between 0.5% and 1.5% across all questions. +The survey took on average 8 minutes and 40 seconds to complete. +4.2 +Responses to questions: political, economic, and governance attitudes +Overall, respondents were varied in their perceptions of the distribution of economic power in crypto +and their personal attitudes towards crypto. They were also split between the most common +4 + +Figure 1. Responses to (A) question 11, (B) question 12, and (C) question 13 on perceptions of the +distribution of economic power in crypto, with 95% confidence intervals. Though optimistic beliefs +about the current state of crypto-economics were slightly more prevalent, dissatisfaction with the fair +distribution and attainability of crypto-economic wealth was nearly as frequent. +responses to two questions on the distribution of power in governance of crypto. There was +somewhat more agreement on broad beliefs towards external regulation of crypto, though +respondents disagreed on some of the specifics and in matters of degree. The largest majorities were +observed in questions relating to the social implications of crypto. +Perceptions of the distribution of economic power in crypto were closely split between the two +choices provided for each question (Figure 1). By a few percentage points, a slightly higher +proportion of respondents believed that most crypto teams make “a fair and reasonable amount of +profit” rather than “too much profit” (Q11, Fig. 1(A)) and that the economic system in crypto “is +generally fair to most of its participants” rather than “unfairly favors powerful interests” (Q12, Fig. +1(B)). A majority (58%) believed that “most people who want to get ahead in crypto can make it if +they’re willing to work hard” (Q13, Fig. 1(C)). +5 + +(A) +11.Whichstatementcomesclosesttoyourviews? +Cryptoteamsmaketoomuchprofit +Mostcryptoteamsmakeafairand +reasonableamountof profit. +0 +20 +40 +60 +80 +100 +Percentage +(B) +12.Whichstatementcomesclosesttoyourviews? +Theeconomicsystemincryptois +generallyfairtomostof its +participants. +Theeconomicsystemincryptounfairly +favorspowerfulinterests +0 +20 +40 +60 +80 +100 +Percentage +(C) +13.Whichstatementcomesclosesttoyourviews? +Mostpeoplewhowantto getaheadin +crypto can make it if they'rewillingto +workhard. +In crypto,hardworkanddetermination +arenoguaranteeofsuccessformost +people. +0 +20 +40 +60 +80 +100 +PercentageFigure 2. Responses to (A) question 3 and (B) question 4 on personal attitudes towards blockchain, +with 95% confidence intervals. Together, these responses show that both a desire for sociopolitical +change and an interest in personal financial gain were common factors in participants’ interest in +blockchain technologies. +Figure 3. Responses to (A) question 5 and (B) question 16 on blockchain governance, with 95% +confidence intervals. +Personal attitudes towards crypto were also diverse (Figure 2). Respondents were divided on whether +they regarded crypto as “mainly a political philosophy and/or lifestyle” or “mainly an economic +technology”, with a slight majority favoring the latter (Q3, Fig. 2(A)). There was no majority in +respondents’ goals for their own involvement in crypto: the most common goal was “to create social +change and/or disrupt the industry” (39%), followed by “to make as much money as possible” (29%) +(Q4, Fig. 2(B)). +Normative beliefs about the distribution of power in governance of crypto appear to be in some +tension (Figure 3). Most respondents favored a hands-off approach to the governance of crypto, with +45% believing that “most or all cryptogovernance should be on-chain” and 30% believing “crypto +does not need (human) governance” (Q5, Fig. 3(A)). However, a majority of respondents believed +that “a wide variety of on- and off-chain stakeholders” should have decision-making power over a +blockchain (though the next most common response was “the token holders and/or node operators, +i.e., voters, as determined by the protocol”) (Q16, Fig. 3(B)). Note that while the most common +responses to each of these questions are not incompatible, their coexistence indicates a possible +tension in the community between maximizing on-chain governance and empowering off-chain +stakeholders. +6 + +(A) +3.Whichstatementcomes closest toyourviews? +(B) +4.whichstatement comes closestto yourviews? +Mygoal in crypto is to earna living +and/orbuildmycareer. +Cryptoismainlyapoliticalphilosophy +and/orlifestyle +Mygoal in crypto is to createsocial +changeand/ordisrupttheindustry. +My goal in crypto is to make as much +moneyaspossible. +Cryptoismainlyan economictechnology. +Mygoal in crypto is to havefun. +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +0 +Percentage +Percentage(A) +(B) +16.Whoshouldhavedecision-makingpowerovera +5.Whichstatementcomesclosesttoyourviews? +blockchain? +Mostorall cryptogovernanceshouldbe +Thetokenholdersand/ornodeoperators, +i.e.voters, as determined by the +on-chain. +protocol +A widevariety of on-and off-chain +Cryptodoesnotneed(human)governance) +stakeholders inciuding token holders, +letthealgorithmsrunastheywere +nodeoperators,applicationdevelopers, +designed. +foundations,andusers +Mostorall cryptogovernanceshouldbe +The core developers and technical staff +off-chain. +ofablockchain +Howevercryptogoverns itself, it should +Thepublic,electedrepresentatives, +also beregulatedbythegovernment. +and/ornational leaders +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +Percentage +PercentageFigure 4. Responses to (A) question 7 and (B) question 14 on external regulation of blockchain +technologies, with 95% confidence intervals. +Regarding external regulation of blockchain technologies, respondents were somewhat more +consistent (Figure 4). A majority of respondents believed at least some good will come of +government regulation of crypto, though nearly 40% asserted that “government regulation of crypto +will almost always do more harm than good” (Q7, Fig. 4(A)). In line with the above, when asked +what the most important thing the crypto community can do to get more favorable regulation of +cryptocurrencies from national governments, a plurality of respondents sought a cooperative +relationship with government, choosing to “work hand-in-hand with regulators to identify a solution +that works for both government and industry,” versus adopting an evasive approach to “adapt our +technology and practices in order to minimize potential conflicts with the law” or even an +antagonistic one to “mount a public pressure campaign on politicians” or to “keep doing what we’re +doing, legal or not” (Q14, Fig. 4(B)). Also, more than three-quarters of respondents believed that +“having a central bank run a cryptocurrency is a bad idea” (Q8). Overall, though a majority of +respondents were willing to accept or even collaborate on regulation, large minorities strongly +disagreed, and distaste for direct government involvement in implementations of crypto technology +was common. +On the social implications of crypto, most respondents were in agreement, believing that blockchain +and DeFi are “beneficial technologies that, on balance, will help most members of society” (Q10). +Even so, more than a quarter of respondents believed that crypto “has a gender problem” (Q15). +Also, around a quarter of respondents indicated privacy is “the most important feature of blockchain +and crypto” (Q6). +We asked two additional questions on political orientation and blockchain ecosystem affiliation +(Figure 5). Only 14 percent of respondents considered themselves “conservative or right-wing” (532 +respondents) with the remaining participants split equally (with no statistically significant difference) +between “liberal or left-wing” (1550) and “neither” (1599; Q18, Fig. 5(A)). Nearly all participants +(97%) stated an affiliation with at least one blockchain ecosystem or community (Q19, Fig. 5(B)), +supporting our use of this dataset to focus on users of blockchain technology (rather than the general +public). In particular, of the 3591 respondents who indicated affiliation with at least one blockchain, +2175 (61%) selected affiliation with Ethereum and 1120 (31%) with Bitcoin (Fig. 5b). Note that these +are not mutually exclusive groups (789 indicated affiliation with both); furthermore, though a +majority of respondents only specified one affiliation, less than a quarter believe that “there is one +(layer 1) blockchain that is the best” (Q1). In the following two subsections, we discuss the relation +of these distributions with respondents’ beliefs in more depth. +7 + +(A) +(B) +14.To get more favorable regulation of +cryptocurrencies from national governments,the +mostimportantthingthecryptocommunitycando +7.Whichstatementcomesclosesttoyourviews? +is: +Governmentregulation of cryptocando +Hirelawyersand lobbyists;organizethe +some good,e.g.it can help force +communitytomounta publicpressure +H +blockchainstobecomemore +campaign on politicians. +decentralized. +Adapt ourtechnologyandpractices in +ordertominimizepotentialconflicts +withthe law. +Governmentregulation of crypto will +almost always do more harm than good. +Keep on doing what we're doing, legal or +not. +Governmentregulation of cryptois +critical to protect the public interest +H +Workhand-in-handwithregulatorsto +inthesetechnologies. +identify a solution that works for both +一 +government and industry. +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +Percentage +PercentageFigure 5. Responses to (A) question 18 on political orientation and (B) question 19 on blockchain +ecosystem affiliations, with 95% confidence intervals. +The distribution of responses for the questions not covered in this section are included in the +Supplementary Material (Supplementary Figures S1-S4). +4.3 +Differences between respondents by self-reported political orientation +To examine the differences in opinion between the left-of-center, right-of-center, and unaligned +groups, we compared the distribution of answers selected by respondents affiliated with each group +(Q18). We found that perceptions of economic fairness and gender equity elicited the clearest +differences between the three political orientation groups, with economic fairness especially +differentiating left-of-center respondents from the other two groups. Beliefs about governance, +regulation, and personal goals in crypto differentiated right-of-center respondents from the other two +groups. Differences between political orientation groups were ubiquitous: all but one question had at +least one statistically significant difference between the responses groups. +The economic fairness questions (Q11, Q12, and Q13) were among those with the greatest +differentiation between the three groups. Somewhat surprisingly, unlike nonaligned and +right-of-center respondents, a majority of left-of-center respondents believe that “most crypto teams +Figure 6. Responses, grouped by self-reported political affiliation, to question 12 on crypto-economic +fairness, with 95% confidence intervals. Taken together with questions 11 and 13, this distribution +shows that left-of-center respondents overall held a different set of beliefs about wealth distribution +and economic opportunity than other respondents. +8 + +(A) +(B) +19.OPTIONAL:Doyouaffiliatewithanyofthe +18.Doyouconsideryourself: +followingecosystemsorcommunities? +Ethereum +Liberal or left-wing +Bitcoin +Solana +Neither +Polkadot +Cardano +Conservativeorright-wing +H +Other +0 +20 +40 +60 +80 +100 +0 +500 +1000 +1500 +2000 +Percentage +Frequency12.Whichstatementcomesclosesttoyourviews? +left +right +Theeconomicsystemincryptounfairly +neither +favorspowerfulinterests +Theeconomicsystemincryptois +generallyfairto most of its +participants. +0 +20 +40 +60 +80 +100 +Percentagemake a fair and reasonable amount of profit” (Q11) and “the economic system in crypto is generally +fair to most of its participants” (Q12, Fig. 6). Though a majority of both right-of-center and +nonaligned respondents believed instead that “the economic system in crypto unfairly favors +powerful interests”, right-of-center respondents were more likely than nonaligned respondents to +choose this answer (Q12). However, left-of-center respondents were more likely than right-of-center +or nonaligned respondents to believe that “hard work and determination are no guarantees of +success” in crypto (Q13). +Question 12 was one of three questions for which all three groups had a statistically different +distribution of responses. Another was on gender equity: right-of-center respondents were least likely +to believe “crypto has a gender problem,” nonaligned respondents somewhat more likely, and +left-of-center respondents most likely, with about half of left-of-center respondents selecting this +answer (Q15). This spread shows that self-reported political alignment relates to not only economic +but also social issues in the use of blockchain technology. +Differences also arose between the groups in the most common answer to questions on +decision-making power and how to obtain favorable regulation. When asked who should hold +decision-making power over a blockchain, right-of-center respondents were more likely to choose +“the token holders and/or node operators” than “a wide variety of on- and off-chain stakeholders”; +the reverse was true for left-of-center and nonaligned respondents, with left-of-center respondents +more likely than other respondents to choose a variety of stakeholders (Q16, Fig. 7). Concerning how +to obtain favorable regulation, left-of-center and nonaligned respondents were most likely to choose +“work hand-in-hand with regulators” out of the available choices, and more likely to do so than +right-of-center respondents; in contrast, right-of-center respondents were, within confidence +intervals, evenly split between three of the four available choices (Q14). +Other statistically significant differences occurred in the distribution of responses, where one of the +three groups differed from the other two. Right-of-center respondents were most likely to choose +“make as much money as possible” as their goal and less likely to select “create social change and/or +disrupt the industry”; the reverse was true for left-of-center and nonaligned respondents (Q4). +However, left-of-center respondents were less likely than others to believe crypto needs to prioritize +“building art and community” to grow (Q9). Also, a smaller proportion of left-of-center respondents +than other respondents believed that privacy is “the most important feature of blockchain” (Q6). +Figure 7. Responses, grouped by self-reported political affiliation, to question 16 on decision-making +power, with 95% confidence intervals. Together with question 14, this distribution indicates that +right-of-center respondents were more likely than other respondents to hold beliefs aligned with +minimizing external influence on blockchain governance and development. +9 + +16.Whoshouldhavedecision-makingpowerovera +blockchain? +left +Thepublic,electedrepresentatives +right +and/ornationalleaders +neither +Thecoredevelopersandtechnicalstaff +ofablockchain +Thetokenholdersand/ornodeoperators, +i.e.voters.asdeterminedbythe +protocol +Awidevarietyofon-andoff-chain +stakeholdersincludingtokenholders, +nodeoperators,applicationdevelopers, +foundations,andusers +0 +20 +40 +60 +80 +100 +PercentageLeft-of-center respondents were less likely to believe that “crypto does not need (human) +governance,” while nonaligned respondents were less likely to believe “however crypto governs +itself, it should also be regulated by the government” (Q5). Left-of-center respondents were also +more polarized on government regulation: they were less likely to believe it “can do some good,” and +more likely to believe it is either “critical to protect the public interest” or “will always do more harm +than good” (Q7). +4.4 +Differences between respondents by Bitcoin and Ethereum affiliation +At present, dynamics in the crypto community are largely driven by actors in two ecosystems: +Bitcoin and Ethereum. To examine differences in opinion between the 61% of respondents affiliated +with Ethereum and the 31% (non-exclusive) affiliated with Bitcoin, we compared the distribution of +answers selected by respondents affiliated with each of the two blockchains. We found an overall +quite similar distribution of responses regardless of affiliation, with a few statistically significant +differences arising in beliefs about cryptogovernance, the semantics of the term crypto, personal +goals in crypto, and stated political orientation. +Governance and regulation of crypto were a key topic distinguishing Bitcoin affiliates from Ethereum +affiliates (Figure 8). Bitcoin affiliation was associated with a higher likelihood of believing that +“crypto does not need (human) governance” (Q5, Fig. 8(A)) and that “token holders and/or node +operators” should have decision-making power over a blockchain, whereas Ethereum was associated +with “a wide variety of on- and off-chain stakeholders” (Q16, Fig. 8(B)). Somewhat surprisingly, +Bitcoin affiliation was also associated with a higher likelihood of believing that government +regulation of crypto “can do some good” (Q7), although there was no statistically significant +difference in opinions on how to obtain favorable regulation (Q14). Thus, it appears that Bitcoin +affiliation is associated with a higher rate of wanting to maximize on-chain governance but also of +tolerance of external regulation, perhaps in particular that which “can help force blockchains to +become more decentralized,” as is included in the wording of question 7. +Respondents’ semantic conception of crypto and their personal goals for their involvement also had +some relation to blockchain affiliation: Bitcoin affiliation was associated with a higher likelihood of +believing “crypto is mainly an economic technology” (Q3) and identifying with the statement “my +goal in crypto is to make as much money as possible” (Q4). Ethereum affiliation was associated with +a higher likelihood of believing that “the economic system in crypto unfairly favors powerful +interests” (Q12) and that “crypto has a gender problem” (Q15). +Figure 8. Responses, grouped by blockchain affiliation, to (A) question 5 and (B) question 16 on +blockchain governance, with 95% confidence intervals. These distributions indicate that Bitcoin +affiliates were more likely to favor a narrow definition of governance and its participants. +10 + +(A) +(B) +16.Who should have decision-making power over a +5.Which statement comes closest toyourviews? +blockchain? +eth +eth +Most orall cryptogovernanceshouldbe +Thepublic,electedrepresentatives, +btc +btc +off-chain. +and/ornationalleaders +However crypto governs itself, it should +工 +The core developers and technical staff +alsoberegulatedbythegovernment +ofablockchain +Crypto doesnotneed(human)governance; +Thetokenholdersand/ornodeoperators, +letthealgorithmsrunastheywere +i.e.voters, as determined by the +工 +工 +designed. +protocol +A wide varietyof on-and off-chain +Most orall cryptogovernance should be +stakeholders including token holders, +H +on-chain. +node operators,applicationdevelopers, +工 +foundations,andusers +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +Percentage +PercentageFigure 9. Responses, grouped by blockchain affiliation, to question 18 on self-reported political +affiliation, with 95% confidence intervals. While there was a statistically significant difference +between affiliates of the two blockchains in identifying as right-of-center or nonaligned, Cramer’s V +indicates that the strength of association between blockchain affiliation and political orientation was +low. +For question 18 on political orientation, Bitcoin affiliation correlated with a higher likelihood of +selecting “conservative or right-wing” and lower likelihood of selecting “neither” (Figure 9). There +was no statistically significant difference between the proportions of respondents who chose “liberal +or left-wing”. Given that we were interested in analyzing blockchain affiliation separately from stated +political orientation, we additionally checked for the strength of association between Q18 and a +reduced version of Q19 with the options “Bitcoin”, “Ethereum”, and “Neither” (not mutually +exclusive). Cramer’s V was low (less than 0.15) for all combinations of responses, indicating at most +very weak association between the two questions (Supplementary Figure S5). This gives us +confidence that Bitcoin and Ethereum affiliation were not strongly associated with stated political +orientation. +4.5 +Validation of survey instrument +To assess any correlations between responses to different questions, we computed the correlation +matrix for all pairs of questions (Supplementary Fig. S6). Of the 153 unique pairs, most showed little +if any association (V < 0.1); the strength of association was weak for 52 questions (0.1 <= V < 0.3), +and one question pair related to wealth distribution (Q11-Q12) showed a moderate strength of +association (0.33). The prevalence of weak or no association between distinct questions supports our +assertion that each question addresses a distinct facet of a respondent’s beliefs or actions. This allows +us to assess the relative importance of the specific statements provided in the answer choices to +explain differences between respondents. +4.6 +Feature selection and factor analysis +To identify the beliefs which most contributed to explaining variance between respondents and to test +our hypothesis, we computed the PCA vectors for individual choices (features) and examined the +first principal component. Beliefs above a threshold of magnitude 0.18, corresponding to the loading +each response would have if all questions contributed equally to the component, were labeled as +important. The features with the largest contributions to the first principal component were the +following (listed in descending order of importance): +11 + +18.Do you consideryourself +eth +btc +Conservative or right-wing +Liberalorleft-wing +Neither +0 +20 +40 +60 +80 +100 +Percentage- +“The economic system in crypto unfairly favors powerful interests.” (Q12) +- +“Crypto has a gender problem.” (Q15) +- +“Government regulation of crypto will almost always do more harm than good.” (Q7) +- +“[I consider myself] liberal or left-wing.” (Q18) +- +“Crypto teams make too much profit.” (Q11) +- +“In crypto, hard work and determination are no guarantee of success for most people.” (Q13) +- +“However crypto governs itself, it should also be regulated by the government.” (Q5) +- +“Blockchain and DeFi are predatory technologies that, on balance, will harm most members +of society.” (Q10) +All three questions relating to wealth distribution and economic fairness (Q11-13) contributed more +to explaining variance than most other questions. Polarized opinions on government regulation (Q5 +and Q7) and one specific political affiliation (Q18) also featured here. Altogether, 5 of the 8 +questions that we had coded as defining an axis of political belief had a large contribution to this +leading component. +The same analysis can be done for the remaining principal components. The features with the largest +component loading for the next two principal components are “Privacy is the most important feature +of blockchain and crypto” (Q6) for the second principal component and “Crypto is mainly a political +philosophy and/or lifestyle” (Q3) for the third. These choices, and their corresponding questions, are +therefore among the more salient in explaining variance between respondents. +Altogether, however, the variance explained by only the first few components was relatively low +(21% for the first three components) and less than 10% was explained by the first component alone. +Taken together with weak associations between questions as described in Section 4.5, this implies +that the number of latent variables required to describe respondents' beliefs is large. Indeed, factor +analysis using PCA and feature agglomeration yielded a null result, meaning that features did not +cluster into a few interpretable groupings (see Section 1.3 of the Supplementary Material). Even so, +we find that the first principal component axis corroborates a theory-first constructed axis, as +described in the following section. +4.7 +Validation of constructed crypto-political axis +For each respondent, a political score was calculated using the values in Table 1 and a type was +assigned according to the score thresholds described in Table 2. The feature selection and factor +analysis results can be used to evaluate the validity of this constructed crypto-political axis. +The distribution of scores and types assigned to participants who completed the survey is shown in +Figure 10. On the left-of-center side, 20% of respondents were identified as “crypto-communitarian” +or “crypto-leftist”, while 9% of respondents were given the “crypto-centrist” label. The most +commonly assigned type was the “crypto-libertarian” types, with nearly half of respondents receiving +this designation; overall, right-of-center types (“crypto-libertarian” and “crypto-anarcho-capitalist”) +dominated with 71% of respondents. This distribution is unimodal, low skewness, and centered +around the median possible score. However, because the range of possible scores was not centered +around zero, we find that a majority of respondents were labeled as crypto-politically +“right-of-center”. For a summary of how this distribution differed with political self-identification +and blockchain affiliation, Section 1.2 of the Supplementary Material. +12 + +Figure 10. Distribution of assigned crypto-political scores and corresponding sentiment types. +Despite a 14% minority of respondents identifying as ideologically conservative or right-wing, our +measure placed 71% in the right-of-center libertarian and anarchocapitalist categories. +Figure 11. Box plots showing the distribution of values for each political type along each of the first +three principal components produced by PCA, with the mean scores indicated by a white triangle. +There is essentially no overlap between crypto-leftist and crypto-ancap types for component 0. +Partitioning the respondents by the types we identified for them, and plotting them in the first three +PCA components, we find, again, that the first principal component succeeds at capturing the +political dimension of respondent variation, while the next two components are less informative +(Figure 11). The low overlap between the interquartile ranges for adjacent types indicates that the +continuous construct we defined and the defined types which discretize it help to explain differences +between respondents’ beliefs. Thus, the constructed political axis seems to reflect true variation in the +population and may be of use in future work characterizing the ideological structure of the crypto +community. +5 +Discussion +Though optimistic beliefs about the current state of cryptoeconomics were slightly more prevalent, +the survey responses indicate nearly as much dissatisfaction with the fair distribution and attainability +of cryptoeconomic wealth. Both a desire for sociopolitical change and an interest in personal +financial gain were common factors in participants’ interest in blockchain technologies. Respondents +13 + +400 +Type +Value count +300 +Crypto-leftist +Crypto-communitarian +Crypto-centrist +200 +Crypto-libertarian +Crypto-anarchocapitalist +100 +0 +-6 +-4 +-2 +0 +2 +4 +6 +8 +10 +Politics score6 +4 +Type +Crypto-leftist +2 +lue +Crypto-communitarian +Crypto-centrist +0 +Crypto-libertarian +Crypto-anarchocapitalist +-2 +-4 +1 +2 +3 +Componentgenerally were optimistic about the social potential of blockchain technology, with some having +reservations about its gender equity and some focusing on its privacy implications. Overall, though a +majority of respondents were willing to accept or even collaborate on regulation, large minorities +strongly disagreed, and distaste for direct government involvement in implementations of crypto +technology was common. +Despite low rates of respondents’ self-identification with “conservative or right-wing” politics, we +observed a prevalence of right-of-center crypto-political types. A broadly similar distribution was +observed in a CoinDesk report published in 2018 (Ryan 2018). The discrepancy between general +political self-identification and our crypto-specific labeling bears further investigation. It may relate +to an association of the term “conservative” with social conservatism, whereas crypto-libertarianism, +the crypto-political type we found to be most common, emphasizes a form of economic +libertarianism. Furthermore, the connotations of “conservative” and “liberal” vary significantly by +geographic region, so the question may have been interpreted differently across respondents based on +their country of residence. +The correlation of the constructed political axis with the first principal component–a commonly-used, +well-validated axis–suggests a primacy of political variation in explaining patterns of responses. +Furthermore, the existence of differences in the distribution of beliefs between self-identified +political orientations indicates that traditional political ideologies have some bearing on how +participants relate to blockchain technology. For example, the “left-of-center” group articulated +distinct beliefs about economic opportunity, fairness of wealth distribution, privacy, and the growth +of crypto (Q11-13, Q6, and Q9), suggesting that left-of-center respondents are more likely to apply +more community-oriented multi-stakeholder values to the blockchain ecosystem. The “nonaligned” +group, on the other hand, articulated distinct beliefs about gender equity and government regulation +(Q15 and Q5), suggesting this group is more clearly defined by lower trust in existing government +institutions2. This lower approval of government regulation suggests that those who identified as +neither left-of-center nor right-of-center are more likely to position themselves as separate from +existing political and governance systems entirely. +We are interested in understanding the extent to which the characteristics of developers and users of +specific blockchains are distinctive of each blockchain. Critics like David Golumbia have argued that +Bitcoin, both in its design and ideological constitution, is principally a conservative movement +interested strictly in Bitcoin’s record of gaining value (Golumbia 2016). Those observations were not +made in opposition to Ethereum or any other blockchain, although Ethereum had been live for a year +at the time of Golumbia’s writing. Our findings indicate that in fact, there are few differences +between Bitcoin and Ethereum users. However, differences in technical implementation between +Bitcoin and Ethereum may relate to differences in opinion on their governance. Unlike Bitcoin, +which has limited support for transactions other than money transfers, Ethereum as an infrastructure +enables the developing and building of various applications and projects. The broader set of use cases +for Ethereum may lead its users to believe a broader set of stakeholders should be involved in its +governance. Furthermore, Ethereum affiliation was associated with a greater sensitivity to perceived +socioeconomic inequity, which may relate to differences in how the blockchains are used alongside +other technology. In Ethereum ecosystems, users linking their own blockchain activity to other +2 We refer in this work to respondents who chose to identify as neither left-of-center nor +right-of-center as “nonaligned”. We choose this term in contrast to a term such as “apolitical” in a +nod to ideas of political agnosticism developed by ethnographers in observing open-source +communities (Gabriella Coleman 2013). +14 + +personally-identifying information, such as Discord handles or Twitter accounts, is not uncommon; +more research is needed to understand whether lower rates of anonymity relate to greater awareness +of actual or perceived social demographics. +Communities organized around crypto are proving to be a laboratory for new ways that humans can +organize collective action, but are not operating in a historical vacuum: it appears that some patterns +observed in early users of other internet technologies have arisen or continue to appear in the +blockchain context as well. To complement this sociological work, further anthropological research +could shed some light on the extent to which the economic and political beliefs held by participants +in crypto echo the ideologies of two earlier movements: the cryptographic hacker and open-source +software communities. The distribution of responses relating to fair rewards for developer teams and +the utility of hard work and in crypto indicates that meritocratic values are prevalent; meritocracy +may play a similar role in blockchain ecosystems, themselves often open source, as it has in prior +open source and hacker communities (Gabriella Coleman 2013; Dunbar-Hester 2019). Privacy has +been at the forefront of concerns in the development of internet technologies since the cypherpunks +(Hughes 1993) and remains prevalent in blockchain (Brunton 2020). Further research is needed to +understand how these values compare to those of open-source software communities and early +adopters of the internet or how they may have changed over time as cryptocurrencies become more +mainstream. +6 +Limitations +6.1 +Survey methodology +Selection bias may arise given that the random sample assumption is limited by how the survey was +distributed. In particular, since the survey was opt-in, people with stronger and potentially more +extreme opinions may have been more motivated to complete the survey. Also, the survey was made +available only in English, and so is likely not representative of the full geographic distribution of +users of blockchain technologies. Presentation of some preliminary findings prior to finalizing data +collection may also have influenced some respondents. Additionally, we do not have a guarantee of +uniqueness of each respondent; moreover, the two recruitment strategies we used may have +motivated respondents to provide multiple responses. +In choosing the wording of each question and answer choice, we made an effort to mitigate response +bias. Still, we have identified some limitations in interpreting questions based on the wording of the +questions. Q5 may have had an insufficient distinction between the two most commonly-selected +choices. In answering Q14, respondents who selected “Keep on doing what we’re doing” may have +rejected the premise of the question rather than believed this was a way to achieve the stated goal. +Additionally, while we intended Q15 to refer to perceptions of gender inequity in participation or +compensation within crypto, the wording of the choices may have been too vague. +Further demographic information would be valuable context for interpreting some questions. Future +work could include a question on the geographic location of respondents, where local regulations and +political attitudes could inform a more detailed analysis of questions on national government +regulation and political affiliation. Interpretation of question 18 on political self-identification may be +similarly limited by differences in how terms such as “liberal” and “conservative” are understood +across the world. +15 + +6.2 +Analysis +To be able to use PCA for the discrete data, we one-hot encoded specific choices. While PCA is +generally better suited to continuous data than boolean data, we find that in this context the results +were cleanly interpretable. We also chose not to include null responses as an additional coded choice +for feature selection or factor analysis. While this does result in using only a subset of the responses +and potentially removing relevant information about respondents’ beliefs, it prevents the null +responses from receiving artificially high importance due to their relative rarity and bypasses the +difficulty in interpreting the null response. +7 +Conclusion +In this work, we have introduced a new survey of blockchain users’ political, economic, and +governance opinions with respect to crypto. Based on 3710 survey responses, we find that users were +spread across a variety of perceptions of the distribution of economic power, normative beliefs about +the distribution of power in governance, and opinions on the role of external regulation of crypto, +though they were broadly in agreement that crypto has a net-positive impact on the world. Equal +numbers of respondents self-identified as liberal or non-aligned, while only about a third as many +respondents self-identified as conservative; this self-reported political affiliation was associated with +differences in opinions on most questions, but especially on economic fairness, decision-making +power, and how to obtain favorable regulation. In contrast, we observed few differences in opinions +between respondents affiliated with Bitcoin and with Ethereum, on issues of blockchain governance +and regulation and on personal attitudes towards crypto. While the full field of beliefs elides neat +interpretation in terms of underlying factors, we found that the existence of a political dimension was +supported both by a theory-driven construct and by a common, well-validated analytical method +(PCA). +While this dataset is an important step towards understanding the distribution of crypto users’ beliefs +about blockchain technology and its utility, open questions remain as to why users believe what they +do about crypto and how their beliefs match up with reality. For example, considering the question of +who should have decision-making authority over a blockchain: Is the large-minority opinion that +token-holding voters should control a blockchain underlied by a belief that minimizing human input +to governance will make it more efficient and less flawed? Is there a disconnect between the common +normative beliefs of what should be happening in cryptogovernance and which types of stakeholders +actually can and do participate in governance of major blockchains? +Although our research found only a few instances where affiliation with a specific blockchain was +associated with differences in beliefs, further research is needed to better understand whether specific +architectures or ecosystems within crypto differ in the values or goals embedded in them. Future +interdisciplinary work could shed some light on the extent to which participants have common +understandings of core signifiers such as decentralization and autonomy. +Given that our work takes inspiration from the long-running Pew political survey, we see the need for +a regular survey of cryptopolitical sentiment, with an added demographic panel. This could facilitate +the identification and comparison of ideologies and modes of participation within newer chains such +as Solana, L2s such as Polygon, and even large DAOs. +16 + +8 +Conflict of Interest +The authors declare that the research was conducted in the absence of any commercial or financial +relationships that could be construed as a potential conflict of interest. +9 +Author Contributions +LMK conducted data analysis and wrote manuscript. SF advised on methodology and advised on +manuscript. JT designed and deployed the survey instrument and advised on manuscript. +10 +Funding +Lucia Korpas and Joshua Tan were supported by grants from the Filecoin Foundation and from One +Project. +11 +Acknowledgments +The authors would like to acknowledge Michael Zargham for technical discussion, Ann Brody for +qualitative discussion, and Tyler Sullberg and Nathan Schneider for feedback on the manuscript. +12 +Data Availability Statement +The dataset generated and analyzed for this study can be found in the Metagovernance Project’s +Govbase repository on Airtable +(https://airtable.com/shrgnUrj0dqzZDsOd/tblvwbt4KFm8MOSUQ/viw82nVNrdHFrowoo). The +Python code used to conduct the analysis and produce the figures can be found in the GitHub +repository for this work (https://github.com/metagov/cryptopolitics-paper). +17 + +13 +References +Bauerle, Nolan, and Peter Ryan. 2018. “CoinDesk Releases Q2 2018 State of Blockchain Report.” +CoinDesk. 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January 2019. +https://medium.com/cryptolawreview/against-szabos-law-for-a-new-crypto-legal-system-d00 +d0f3d3827. +19 + +14 +Tables +Table 1 +Question +number +Question text +Choice text +Contribution +to political +score +1Which statement +comes closest to +your views? +There is one (layer 1) blockchain that is the best. +There is no one best blockchain. +2Which blockchain +is the best? +Bitcoin +Ethereum +Solana +Cardano +Polkadot +Other +3Which statement +comes closest to +your views? +Crypto is mainly an economic technology. +Crypto is mainly a political philosophy and/or lifestyle. +4Which statement +comes closest to +your views? +My goal in crypto is to have fun. +My goal in crypto is to make as much money as possible. +My goal in crypto is to create social change and/or disrupt the +industry. +My goal in crypto is to earn a living and/or build my career. +5Which statement +comes closest to +your views? +Most or all cryptogovernance should be on-chain. +Most or all cryptogovernance should be off-chain. +Crypto does not need (human) governance; let the algorithms run as +they were designed. +However crypto governs itself, it should also be regulated by the +government. +6Which statement +comes closest to +your views? +Privacy is the most important feature of blockchain and crypto. +2 +Privacy is nice, but it’s not the most important feature of blockchain +and crypto. +0 +7Which statement +comes closest to +your views? +Government regulation of crypto will almost always do more harm +than good. +1 +Government regulation of crypto can do some good, e.g. it can help +force blockchains to become more decentralized. +Government regulation of crypto is critical to protect the public +interest in these technologies. +-1 +8Which statement +comes closest to +your views? +Having a central bank run a cryptocurrency is a good idea. +Having a central bank run a cryptocurrency is a bad idea. +9In order to grow, +the crypto +ecosystem should: +Build art and community. +-1 +Help people around the world earn a living. +-1 +Build useful tech that solve real problems for a set of users. +1 +20 + +Provide financial instruments for maximum wealth creation. +1 +10Which statement +comes closest to +your views? +Blockchain and DeFi are beneficial technologies that, on balance, +will help most members of society. +Blockchain and DeFi are predatory technologies that, on balance, +will harm most members of society. +11Which statement +comes closest to +your views? +Most crypto teams make a fair and reasonable amount of profit. +1 +Crypto teams make too much profit. +-1 +12Which statement +comes closest to +your views? +The economic system in crypto is generally fair to most of its +participants. +1 +The economic system in crypto unfairly favors powerful interests. +-1 +13Which statement +comes closest to +your views? +Most people who want to get ahead in crypto can make it if they're +willing to work hard. +1 +In crypto, hard work and determination are no guarantee of success +for most people. +-1 +14To get more +favorable +regulation of +cryptocurrencies +from national +governments, the +most important +thing the crypto +community can do +is: +Adapt our technology and practices in order to minimize potential +conflicts with the law. +Work hand-in-hand with regulators to identify a solution that works +for both government and industry. +Hire lawyers and lobbyists; organize the community to mount a +public pressure campaign on politicians. +Keep on doing what we’re doing, legal or not. +1 +15Which statement +comes closest to +your views? +Crypto has a gender problem. +0 +Crypto does not have a gender problem. +1 +16Who should have +decision-making +power over a +blockchain? +The public, elected representatives, and/or national leaders +A wide variety of on- and off-chain stakeholders including token +holders, node operators, application developers, foundations, and +users +The token holders and/or node operators, i.e. voters, as determined +by the protocol +The core developers and technical staff of a blockchain +17I'm here for... +the memes +the jobs +the tech +the airdrops +18I identify as: +Liberal or left-wing +-1 +Conservative or right-wing +1 +Neither +0 +21 + +19OPTIONAL: Do +you affiliate with +any of the +following +ecosystems or +communities? +Bitcoin +Ethereum +Solana +Cardano +Polkadot +Other +Table 2 +Politics score +Assigned type +[-7,9] +Overall possible range of +scores +>= 5 +Crypto-anarchocapitalist +0 < x < 5 +Crypto-libertarian +x = 0 +Crypto-centrist +-3 < x < 0 +Crypto-communitarian +<= -3 +Crypto-leftist +22 + +Supplementary Material +1 +Supplementary Information +1.1 +Recruitment strategy: Ad-hoc typology +Based on their responses to the survey, each respondent was assigned a political faction, an economic +faction, and a governance class. These were computed immediately upon completion of the survey +according to numerical weights assigned to each answer choice and formulas defined for each +faction. The faction definitions and names were developed with input from the community, and +served largely as a way to (1) recruit participants and (2) help participants interpret their results. +Depending on the response to Question 3 – whether the respondent considered crypto to be primarily +a political philosophy or an economic technology – each respondent was correspondingly presented +with either their political faction or economic faction as their overall assigned faction. The five +political factions (“crypto-leftist”, “DAOist”, “true neutral”, “crypto-libertarian”, and +“crypto-ancap”) were an alternate naming scheme for our constructed political axis. The five +economic factions (“earner”, “cryptopunk”, “NPC”, “techtrepreneur”, “degen”) were intended to +represent the respondents’ primary mode of economic engagement with crypto. Additionally, we +defined four “classes” (Szabian, Gavinist, Zamfirist, and Walchian) that were intended to capture +respondents’ beliefs about governance and government regulation, inspired by the positions +articulated by Nick Szabo, Gavin Wood, Vlad Zamfir, and Angela Walch. +The mechanism for assigning the factions based on respondents’ answers was a weighted sum of +their responses to each question; every answer choice in every question adds points to one or more +factions, and the faction assigned is the one with the most points, with some thresholding to account +for respondents who did respond to all questions. In cases where not enough responses were given, +the assignment defaulted to “true neutral” or “NPC”. Questions 6, 7, 9, 11, 12, 13, 14, 15, and 18 +were used in computing the political faction, questions 4, 9, 17 in computing the economic faction, +and questions 5, 7, 8, 14, and 16 in computing the governance class. +The political faction names and definitions were conceptualized and iterated through a +community-based effort at crypto conferences and online, especially within the Metagovernance +Project Slack community. The names of the factions reflected both popular slang in crypto (degen, +cryptopunk, crypto-libertarian, crypto-leftist, DAOist, crypto-ancap) as well as a few inventions of +the authors when there was no existing slang or word for that archetype (earner, techtrepreneur, true +neutral, NPC). The visual representations of the factions, which were used as the underlying images +of a series of NFTs, played off various memes (e.g. shiba inu / doge representing degens) and cultural +icons (e.g. Shrek representing crypto-libertarians) common in crypto and on the internet more +broadly. +Within the main text, the political “factions” were renamed to the described “types”. +1.2 +Differences in assigned political types by self-reported political orientation and +blockchain ecosystem affiliation +There was a statistically significant difference in the overall distribution of assigned cryptopolitical +factions across the left-of-center, right-of-center, and nonaligned groups. Right-of-center respondents +were the only group more likely to be assigned the crypto-anarcho-capitalist faction than the +crypto-libertarian faction, and less likely than the other two groups to be assigned true neutral, +23 + +DAOist, or crypto-leftist. Unlike the other two groups, left-of-center respondents were more likely to +be assigned any of crypto-centrist, crypto-communitarian, or crypto-leftist than to be assigned +crypto-anarcho-capitalist. Nonaligned respondents generally split the difference between the +proportions of left-of-center and right-of-center respondents assigned each faction except for +crypto-libertarian, which they were more likely to be assigned than either of the other two groups. +Note that while Q18 was used directly in assigning the political faction, it was one of nine such +questions, each with similar weights in the overall cryptopolitical score. +A higher percentage of Bitcoin-affiliated respondents than Ethereum-affiliated respondents received +the “crypto-anarcho-capitalist” label. +1.3 +Are there clusters of respondents or features? +We were interested in understanding whether the range of responses would be better captured by the +idea of “types” (groupings of respondents, consistent with data-driven clustering methods) or “axes” +(grouping of questions, consistent with factor analysis techniques). We investigated this by extending +our PCA analysis with feature agglomeration, based on a feature set of 48 choices provided across 17 +questions. +We used feature agglomeration to hierarchically generate clusters of features, where each feature +corresponded to the selection of a provided choice. Given that these features are boolean in nature, +we used the Dice distance metric to determine distances between features; this can be understood as +the fraction of the sample for which the two features intersect or overlap, as a representation of the +shared information between them. To determine which clusters should be merged and when, we used +the complete linkage criterion, which relates to the maximum distances between all features of the +two clusters. +From the feature agglomeration results, we ask whether the assigned political types directly correlate +with any of the identified factors (i.e., to what extent does our categorization of beliefs as political, +economic, or governance-related meaningfully describe joint variations in the choices selected?). At +the threshold where three clusters were present, the clusters contained A=31, B=10, and C=7 choices. +The cluster that shared the most features in common with the important principal component features +listed above was the cluster with 10 features: the specific choices listed above for beliefs on +economic fairness, gender, and political affiliation (Q11-Q13, Q15, and Q18) appeared in this feature. +The two specific choices relating to government regulation (Q5 and Q7) were present together in the +cluster with 7 features Although cluster B aligned fairly well with the first (politically-focused) +principal component, our exploration of the other clusters, and the hierarchy that produced them, did +not indicate much qualitative support for this approach. This null assessment is supported by our +mapping of respondents into feature space: both factor analysis methods shown in Supplementary +Figure S7 shows responses organized around a single centroid, not the multiple clusters that would be +expected in the presence of clear respondent types. +Given that the feature agglomeration method allowed us to traverse the entire hierarchy of feature +clusters, we also looked at the agglomerated clusters to see whether three stable feature clusters arise +from the full feature set. Based on the distances required for clusters to merge, it appears instead that +five distinct latent variables may best describe the distribution of respondents’ beliefs, as shown in +Supplementary Figure S8. That said, the smaller two out of the three clusters identified above +persisted unchanged at the five-cluster threshold, suggesting that the underlying factors for each of +those two clusters have more explanatory power. +24 + +2 +Supplementary Figures and Tables +2.1 +Supplementary Figures +Supplementary Figure S1. Distribution of responses to questions 1, 2, and 5. Note that question 2 +was presented only to respondents who selected the response “There is one (layer one) blockchain +that is the best” for question 1, and that respondents were given the option to select Ethereum, +Bitcoin, Solana, Polkadot, Cardano, or a fill-in-the-blank “Other”. +25 + +1. Which statement comes closest to your views? +Thereisone(layer1)blockchainthat +is the best. +Thereis no one best blockchain. +0 +20 +40 +60 +80 +100 +Percentage2. Which blockchain is the best? +Ethereum +Bitcoin +Solana +Cardano +NEAR +Polkadot +AVAX +Polygon +Terra +DeSo +Avalanche +lota +Stellar +Monero +Cosmos +Rose +Binance +Tezos +MINA +Hex, EOS +0 +20 +40 +60 +80 +100 +Percentage5. Which statement comes closest to your views? +Most or all cryptogovernance should be +on-chain. +Crypto does not need (human) governance; +letthealgorithmsrunastheywere +designed. +Most orall cryptogovernance should be +off-chain. +Howevercrypto governsitself,itshould +also be regulated by the government. +0 +20 +40 +60 +80 +100 +PercentageSupplementary Figure S2. Distribution of responses to questions 6, 8, and 9. +26 + +6. Which statement comes closest to your views? +Privacy is the most important feature of +blockchainand crypto, +Privacy is nice,but it's not themost +important featureof blockchain and +crypto. +0 +20 +40 +60 +80 +100 +Percentage8. Which statement comes closest to your views? +Having a central bank run a +cryptocurrency is a good idea. +Having a central bank run a +cryptocurrency is a bad idea. +0 +20 +40 +60 +80 +100 +Percentage9. In order to grow, the crypto ecosystem should: +Providefinancial instrumentsfor +maximumwealthcreation +Build useful tech that solve real +problems for a set of users. +Build art and community +Help people around the world earn a +living +0 +20 +40 +60 +80 +100 +PercentageSupplementary Figure S3. Distribution of responses to questions 10, 15, and 17. +27 + +1o. Which statement comes closest to your views? +Blockchain and DeFi are beneficial +technologiesthat, on balance, will help +most members of society. +Blockchainand DeFiarepredatory +technologies that, on balance, will harm +most members of society. +0 +20 +40 +60 +80 +100 +Percentage15. Which statement comes closest to your views? +Crypto does not have a gender problem. +Cryptohas agenderproblem. +0 +20 +40 +60 +80 +100 +Percentage17. I'm here for.. +the tech +H +the airdrops +the memes +the jobs +H +0 +20 +40 +60 +80 +100 +PercentageSupplementary Figure S4. Distribution of responses to question 19, and distribution in the number +of those multiple-select choices that respondents who answered the question chose. Note that +respondents were given the option to select Ethereum, Bitcoin, Solana, Polkadot, Cardano, or a +fill-in-the-blank “Other”. For plot legibility, only the ecosystems listed by at least 10 respondents are +shown here. +28 + +19. OPTIONAL: Do you affiliate with any of the +following ecosystems or communities? +Ethereum +Bitcoin +Solana +Polkadot +Not specified +Cardano +Avalanche +Q19_all +Cosmos +AVAX +None +Polygon +Terra +NEAR +Fantom +IOTA +Tezos +XRP +0 +500 +1000 +1500 +2000 +FrequencyQ19 count +One +H +Two or more +H +None +0 +20 +40 +60 +80 +100 +PercentageSupplementary Figure S5. Correlation (Cramer’s V) between choices to questions 18 and 19. The +low correlation between choices belonging to different questions indicates that political +self-identification and blockchain affiliation are largely independent. +29 + +answer) +Ethereum +18=Neither +19=Neither +Bitcoin +1.0 +9 +Q18=(No answer) +0.8 +Q18=Conservative or right-wing +Q18=Liberal or left-wing +Cramer's +Q18=Neither +0.4 +Q19=Bitcoin +Q19=Ethereum +0.2 +O19=Neither +一 +0.0Supplementary Figure S6. Correlation (Cramer’s V) between all questions. Gray indicates that the +correlation between questions was not computed; this is used when one question was only presented +to the respondents upon a particular selection for the other question. No clusters are clearly +observable: the questions are not systematically related in macro groupings. +30 + +1.0 +8 +Q1 +Q2 +Q3 +0.8 +Q4 +Q5 +Q6 +Q7 +0.6 +> +Q8 +Cramer's +Q9 +Q10 +Q11 +0.4 +Q12 +Q13 +Q14 +Q15 +Q16 +0.2 +Q17 +Q18 +-0.0(A) +(B) +Supplementary Figure S7. (A) Projection of the respondents onto the first two principal +components. (B) Projection of the respondents onto the feature clusters B (10 features) and C (7 +features). In both cases, there is a continuous (rather than discrete) distribution of respondents along +the two axes. The unimodal distribution of respondents along the two axes does not suggest the +presence of clusters of respondents by their responses. +31 + +PCA with political factions: n components = 3 (O, 1) +6 +4 +Component 2 +2 +0 +-2 +-4 +-4 +-2 +0 +2 +4 +6 +Component 1n components = 3 (1, 2) +1.0 +Count +40 +0.8 +80 +B +120 +Feature cluster +160 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +Feature cluster CSupplementary Figure S8. Dendrogram describing the hierarchical agglomeration of feature +clusters. The colors delineate the first five clusters that branch off from the one-cluster case (i.e., the +one containing all features). The three-cluster case involves grouping some of these together. +32 + +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +11433143422112 +331322 +22 +434 +1213 +2143414 +23 +Number of points in node (or index of point if no parenthesis) \ No newline at end of file diff --git a/ZtE0T4oBgHgl3EQf4ALw/content/tmp_files/load_file.txt b/ZtE0T4oBgHgl3EQf4ALw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d6cf3d4d02d8bcc01c58a6536a79f0a4b45f7f5 --- /dev/null +++ b/ZtE0T4oBgHgl3EQf4ALw/content/tmp_files/load_file.txt @@ -0,0 +1,756 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf,len=755 +page_content='Political, economic, and governance attitudes of blockchain users Lucia M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Korpas1, Seth Frey2,3, Joshua Tan1,4* 1The Metagovernance Project, Brookline, MA, USA 2Department of Communication, University of California Davis, Davis, CA, USA 3The Ostrom Workshop, Indiana University, Bloomington, IN, USA 4University of Oxford, Oxford, Oxfordshire, UK Correspondence: Joshua Tan joshua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='tan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='com Abstract We present a survey to evaluate crypto-political, crypto-economic, and crypto-governance sentiment in people who are part of a blockchain ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Based on 3710 survey responses, we describe their beliefs, attitudes, and modes of participation in crypto and investigate how self-reported political affiliation and blockchain ecosystem affiliation are associated with these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We observed polarization in questions on perceptions of the distribution of economic power, personal attitudes towards crypto, normative beliefs about the distribution of power in governance, and external regulation of blockchain technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Differences in political self-identification correlated with opinions on economic fairness, gender equity, decision-making power and how to obtain favorable regulation, while blockchain affiliation correlated with opinions on governance and regulation of crypto and respondents’ semantic conception of crypto and personal goals for their involvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We also find that a theory-driven constructed political axis is supported by the data and investigate the possibility of other groupings of respondents or beliefs arising from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 Introduction As blockchain technology has evolved over more than a decade, cryptocurrencies and crypto-economic systems have had a growing impact on the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Millions of people have involved themselves in crypto1: as of 2021, around 15 percent of American adults have reported owning cryptocurrency (Perrin 2021), and many other countries have even higher adoption rates (Buchholz 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The past few years have seen the growth of decentralized apps and the crypto startup industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Correspondingly, governments are beginning to take regulatory actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Also, even as blockchain ecosystems move towards less computationally-intensive consensus mechanisms, the ongoing environmental impact of blockchain use is huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Given the impact of crypto-economic activity on individuals and on shared resources, it is increasingly important to understand how its users are relating to the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' While the hard data of cryptocurrency transactions and account balances is often publicly available by design, users’ 1 Throughout the text, we use the term “crypto” to encompass blockchain technologies such as cryptocurrencies and the communities and ideologies which drive their development and use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 motivations for engaging with crypto are more opaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' There is little existing data on the stated beliefs or attitudes of the variety of people using blockchain technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' What do blockchain users believe about the economic, political, and social relevance of crypto?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' While there has been attention to the attitudes of the general population towards cryptocurrencies and blockchain technology (Perrin 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' “Global State of Crypto, 2022” 2022), there is also a need to understand the beliefs of active participants of blockchain ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' What do blockchain ecosystem participants believe about how the technology is being – or should be – developed, used, and regulated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Are there discrete types of crypto contributors, or is there a spectrum of beliefs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' What specific beliefs are most relevant in distinguishing respondents between types or along axes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' This work is a first step in the development of a framework for thinking about this spectrum or grouping of beliefs in crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We report the results of a large-scale survey of participants in the blockchain economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The survey was designed to shed light on respondents’ socioeconomic and sociopolitical beliefs relating to crypto, economic modes of engagement with crypto, and attitudes towards governance of blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We describe the distributions of these responses and their relationships to self-reported political ideology and specific crypto ecosystems such as Bitcoin and Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We also evaluate the survey instrument itself: are the questions able to assess distinct and relevant facets of beliefs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Can we identify underlying factors which describe broader groupings of beliefs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Using factor analysis methods, we find that a political axis and corresponding typology, informed by the Pew Research Center’s Political Typology Quiz, meaningfully describes variation between respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2 Background While there is no existing political theory of crypto per se, there are substantial ethnographic studies of crypto communities (and related digital communities) that address the political dimensions of crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' For example, ethnographic studies have informed the creation of a proposed political typology of blockchain projects (Husain, Franklin, and Roep 2020), reflecting earlier ideas on the “intrinsic” political values of technical artifacts (Winner 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In this vein, cryptocurrencies have been characterized as realizations of crypto-anarchist values such as privacy and autonomy (Chohan 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Beltramini 2021), following in the footsteps of earlier cypherpunk writings (Hughes 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' May 1994) as well as the original Bitcoin whitepaper (Nakamoto 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Other ethnographies have described issues of on- and off-chain governance (De Filippi and Loveluck 2016) and the political motivations and cultural context of projects such as Bitcoin (Golumbia 2016) and Ethereum (Brody and Couture 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' A previous industry survey, conducted by CoinDesk in 2018, contained several questions related to politics and governance (Ryan 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Bauerle and Ryan 2018), though the questions focused more specifically on individual projects and topical questions such as reactions to SEC rulings on the securitization status of Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Distinct from questions about political values, the topic of blockchain governance—including the relationship between blockchains and traditional governments—is one of the most salient and polarizing questions in crypto, one that has led to the creation, forking, and dissolution of many projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' While we cannot recount all the major positions here (some of which are reflected in the survey itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' see “Methodology”), there is a broad distinction between approaches that emphasize on-chain governance and those that emphasize off-chain governance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' A number of academic analyses 2 have studied these different approaches to blockchain governance (Reijers, O’Brolcháin, and Haynes 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' van Pelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2021), along with a vastly greater number of industry manifestos and opinion pieces (Zamfir 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Szabo 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 3 Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='1 Survey questions The survey consists of 19 questions related to respondents’ crypto-related beliefs and activities, with three types of questions interspersed: those eliciting opinions about the political dimensions of crypto activity (“crypto-political”), those eliciting economics opinions (“crypto-economic”), and those eliciting attitudes about the governance of crypto projects (“crypto-governance”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' All questions were multiple choice, with 2-4 possible selections, and the respondent could opt not to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' See Table 1 for the full list of questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The survey questions and provided choices included both a formal portion drawing from existing political survey instruments and a more exploratory portion intended to elicit beliefs relevant to a general crypto-political typology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In particular, a few of the questions selected (Q11-13, Q15, Q19), were based on questions from Pew’s Political Typology Quiz (Nadeem 2021) and intended to relate to political sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Other questions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=', Q1, Q17) were developed in collaboration with a number of community members in crypto, drawing on the culture, memes, and references common in crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=" Altogether, the content was designed to elicit respondents' primary modes of economic engagement with crypto, their political sentiment, and opinions as to how crypto communities themselves should be governed." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 Construction of political “types” and identification of “axes” of belief Our choice to identify separate “axes” of economic, political, and governance beliefs were based on discussion with community members and in analogy to existing classifications such as the traditional “left-right” political axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=" For one of these, the political axis, we also leveraged our study design to group and relate questions more directly by defining a continuous construct intended to assess respondents' crypto-political leanings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We identified a subset of questions as most relevant to political orientation, and computed a score for each participant by summing the responses to these questions (coded with values in the range [-1, 2] as described in Table 1) in analogy to the Pew methodology (Nadeem 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The lowest and highest scores on this political “axis” were designed to highlight extreme positions of collectivist and anarcho-capitalist approaches to using blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Five discrete types were defined by thresholds in the score according to Table 2: crypto-anarcho-capitalist, crypto-libertarian, centrist, crypto-communitarian, and crypto-leftist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' these types were developed both with definitions from the Pew typologies and with input from the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='3 Recruitment We relied on a convenience (self-selected) sample of participants in the crypto community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Participants were recruited by distributing the survey through blockchain-focused forums and listservs, conferences (LisCon and ETHDenver), social media posts, and articles published on blockchain-focused news sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We motivated voluntary participant engagement with two strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We presented the survey as a quiz that assigned respondents one out of an entertaining typology of “types” on the basis of their 3 responses (“crypto-leftist,” “cryptopunk,” etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=') immediately upon completion of the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Stylized as “factions”, the crypto-political types corresponded to the political types we defined based on the Pew typology, while the crypto-economic and crypto-governance types were constructed by using thresholds to partition respondents into five ad hoc types (for more detail, see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='1 of the Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We also incentivized survey completion with the opportunity to receive a non-fungible token (NFT) corresponding to their assigned “type”, contingent upon their provision of a valid Ethereum wallet address or ENS name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='4 Analysis To survey the overall landscape of crypto-political beliefs, we observed the distribution of choices selected by respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We aggregated these responses for each question, including the null response of no choice selected, and computed the margins of error for a 95% confidence interval, assuming a random sample of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' To investigate how political self-identification and participation in specific blockchain ecosystems related to beliefs, we grouped participants by their responses to the corresponding questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We then determined which questions displayed a statistically significant difference in the distribution of responses between these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We also wanted to understand which questions were most meaningful in differentiating respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' To this end, we first performed a check on the extent to which each question measured a distinct belief by computing the correlation between responses to different questions as Cramer’s V (a version of Pearson’s chi-squared statistic scaled to provide a measure of association).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Then we used principal component analysis (PCA) to identify which of the 48 choices provided across 17 questions explained the most variance between respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Specifically, we looked at the component loading for each feature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=', contribution to the first principal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' For use with PCA, we normalized the feature data (shifted to a mean of 0 with unit variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Note that for questions where only two choices were provided, the alternative answer contributed with equal magnitude (though opposite sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We omitted questions 2 and 19, relating to specific blockchain ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We also omitted answers from respondents that did not answer all questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We also wished to evaluate to what extent our crypto-political types, delineated from the political score we defined, corresponded with patterns in beliefs across respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' From the PCA results, we can identify to what extent the assigned types or classes directly correlate with any of the first few components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='1 Responses and respondents Between September 27, 2021 and March 4, 2022, the survey received 3710 responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In 3418 (92%) of these, all questions were answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' For questions presented to all respondents, the percentage of respondents who chose not to answer each question was between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='5% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='5% across all questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The survey took on average 8 minutes and 40 seconds to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 Responses to questions: political, economic, and governance attitudes Overall, respondents were varied in their perceptions of the distribution of economic power in crypto and their personal attitudes towards crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' They were also split between the most common 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses to (A) question 11, (B) question 12, and (C) question 13 on perceptions of the distribution of economic power in crypto, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Though optimistic beliefs about the current state of crypto-economics were slightly more prevalent, dissatisfaction with the fair distribution and attainability of crypto-economic wealth was nearly as frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' responses to two questions on the distribution of power in governance of crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' There was somewhat more agreement on broad beliefs towards external regulation of crypto, though respondents disagreed on some of the specifics and in matters of degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The largest majorities were observed in questions relating to the social implications of crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Perceptions of the distribution of economic power in crypto were closely split between the two choices provided for each question (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' By a few percentage points, a slightly higher proportion of respondents believed that most crypto teams make “a fair and reasonable amount of profit” rather than “too much profit” (Q11, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1(A)) and that the economic system in crypto “is generally fair to most of its participants” rather than “unfairly favors powerful interests” (Q12, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' A majority (58%) believed that “most people who want to get ahead in crypto can make it if they’re willing to work hard” (Q13, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 5 (A) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whichstatementcomesclosesttoyourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Cryptoteamsmaketoomuchprofit Mostcryptoteamsmakeafairand reasonableamountof profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 Percentage (B) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whichstatementcomesclosesttoyourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Theeconomicsystemincryptois generallyfairtomostof its participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Theeconomicsystemincryptounfairly favorspowerfulinterests 0 20 40 60 80 100 Percentage (C) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whichstatementcomesclosesttoyourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=" Mostpeoplewhowantto getaheadin crypto can make it if they'rewillingto workhard." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In crypto,hardworkanddetermination arenoguaranteeofsuccessformost people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 PercentageFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses to (A) question 3 and (B) question 4 on personal attitudes towards blockchain, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Together, these responses show that both a desire for sociopolitical change and an interest in personal financial gain were common factors in participants’ interest in blockchain technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses to (A) question 5 and (B) question 16 on blockchain governance, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Personal attitudes towards crypto were also diverse (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Respondents were divided on whether they regarded crypto as “mainly a political philosophy and/or lifestyle” or “mainly an economic technology”, with a slight majority favoring the latter (Q3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' There was no majority in respondents’ goals for their own involvement in crypto: the most common goal was “to create social change and/or disrupt the industry” (39%), followed by “to make as much money as possible” (29%) (Q4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Normative beliefs about the distribution of power in governance of crypto appear to be in some tension (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Most respondents favored a hands-off approach to the governance of crypto, with 45% believing that “most or all cryptogovernance should be on-chain” and 30% believing “crypto does not need (human) governance” (Q5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 3(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' However, a majority of respondents believed that “a wide variety of on- and off-chain stakeholders” should have decision-making power over a blockchain (though the next most common response was “the token holders and/or node operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=', voters, as determined by the protocol”) (Q16, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 3(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Note that while the most common responses to each of these questions are not incompatible, their coexistence indicates a possible tension in the community between maximizing on-chain governance and empowering off-chain stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 6 (A) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whichstatementcomes closest toyourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (B) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='whichstatement comes closestto yourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Mygoal in crypto is to earna living and/orbuildmycareer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Cryptoismainlyapoliticalphilosophy and/orlifestyle Mygoal in crypto is to createsocial changeand/ordisrupttheindustry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' My goal in crypto is to make as much moneyaspossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Cryptoismainlyan economictechnology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Mygoal in crypto is to havefun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 20 40 60 80 100 0 20 40 60 80 100 0 Percentage Percentage(A) (B) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whoshouldhavedecision-makingpowerovera 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whichstatementcomesclosesttoyourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' blockchain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Mostorall cryptogovernanceshouldbe Thetokenholdersand/ornodeoperators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='voters, as determined by the on-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' protocol A widevariety of on-and off-chain Cryptodoesnotneed(human)governance) stakeholders inciuding token holders, letthealgorithmsrunastheywere nodeoperators,applicationdevelopers, designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' foundations,andusers Mostorall cryptogovernanceshouldbe The core developers and technical staff off-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' ofablockchain Howevercryptogoverns itself, it should Thepublic,electedrepresentatives, also beregulatedbythegovernment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' and/ornational leaders 0 20 40 60 80 100 0 20 40 60 80 100 Percentage PercentageFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses to (A) question 7 and (B) question 14 on external regulation of blockchain technologies, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Regarding external regulation of blockchain technologies, respondents were somewhat more consistent (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' A majority of respondents believed at least some good will come of government regulation of crypto, though nearly 40% asserted that “government regulation of crypto will almost always do more harm than good” (Q7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In line with the above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' when asked what the most important thing the crypto community can do to get more favorable regulation of cryptocurrencies from national governments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' a plurality of respondents sought a cooperative relationship with government,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' choosing to “work hand-in-hand with regulators to identify a solution that works for both government and industry,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' versus adopting an evasive approach to “adapt our technology and practices in order to minimize potential conflicts with the law” or even an antagonistic one to “mount a public pressure campaign on politicians” or to “keep doing what we’re doing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' legal or not” (Q14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Also, more than three-quarters of respondents believed that “having a central bank run a cryptocurrency is a bad idea” (Q8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Overall, though a majority of respondents were willing to accept or even collaborate on regulation, large minorities strongly disagreed, and distaste for direct government involvement in implementations of crypto technology was common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' On the social implications of crypto, most respondents were in agreement, believing that blockchain and DeFi are “beneficial technologies that, on balance, will help most members of society” (Q10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Even so, more than a quarter of respondents believed that crypto “has a gender problem” (Q15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Also, around a quarter of respondents indicated privacy is “the most important feature of blockchain and crypto” (Q6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We asked two additional questions on political orientation and blockchain ecosystem affiliation (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Only 14 percent of respondents considered themselves “conservative or right-wing” (532 respondents) with the remaining participants split equally (with no statistically significant difference) between “liberal or left-wing” (1550) and “neither” (1599;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Q18, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 5(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Nearly all participants (97%) stated an affiliation with at least one blockchain ecosystem or community (Q19, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 5(B)), supporting our use of this dataset to focus on users of blockchain technology (rather than the general public).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In particular, of the 3591 respondents who indicated affiliation with at least one blockchain, 2175 (61%) selected affiliation with Ethereum and 1120 (31%) with Bitcoin (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Note that these are not mutually exclusive groups (789 indicated affiliation with both);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' furthermore, though a majority of respondents only specified one affiliation, less than a quarter believe that “there is one (layer 1) blockchain that is the best” (Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In the following two subsections, we discuss the relation of these distributions with respondents’ beliefs in more depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 7 (A) (B) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='To get more favorable regulation of cryptocurrencies from national governments,the mostimportantthingthecryptocommunitycando 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whichstatementcomesclosesttoyourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' is: Governmentregulation of cryptocando Hirelawyersand lobbyists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='organizethe some good,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='it can help force communitytomounta publicpressure H blockchainstobecomemore campaign on politicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' decentralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Adapt ourtechnologyandpractices in ordertominimizepotentialconflicts withthe law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Governmentregulation of crypto will almost always do more harm than good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=" Keep on doing what we're doing, legal or not." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Governmentregulation of cryptois critical to protect the public interest H Workhand-in-handwithregulatorsto inthesetechnologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' identify a solution that works for both 一 government and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 0 20 40 60 80 100 Percentage PercentageFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses to (A) question 18 on political orientation and (B) question 19 on blockchain ecosystem affiliations, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The distribution of responses for the questions not covered in this section are included in the Supplementary Material (Supplementary Figures S1-S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='3 Differences between respondents by self-reported political orientation To examine the differences in opinion between the left-of-center, right-of-center, and unaligned groups, we compared the distribution of answers selected by respondents affiliated with each group (Q18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We found that perceptions of economic fairness and gender equity elicited the clearest differences between the three political orientation groups, with economic fairness especially differentiating left-of-center respondents from the other two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Beliefs about governance, regulation, and personal goals in crypto differentiated right-of-center respondents from the other two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Differences between political orientation groups were ubiquitous: all but one question had at least one statistically significant difference between the responses groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The economic fairness questions (Q11, Q12, and Q13) were among those with the greatest differentiation between the three groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Somewhat surprisingly, unlike nonaligned and right-of-center respondents, a majority of left-of-center respondents believe that “most crypto teams Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses, grouped by self-reported political affiliation, to question 12 on crypto-economic fairness, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Taken together with questions 11 and 13, this distribution shows that left-of-center respondents overall held a different set of beliefs about wealth distribution and economic opportunity than other respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 8 (A) (B) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='OPTIONAL:Doyouaffiliatewithanyofthe 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Doyouconsideryourself: followingecosystemsorcommunities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Ethereum Liberal or left-wing Bitcoin Solana Neither Polkadot Cardano Conservativeorright-wing H Other 0 20 40 60 80 100 0 500 1000 1500 2000 Percentage Frequency12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whichstatementcomesclosesttoyourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' left right Theeconomicsystemincryptounfairly neither favorspowerfulinterests Theeconomicsystemincryptois generallyfairto most of its participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 Percentagemake a fair and reasonable amount of profit” (Q11) and “the economic system in crypto is generally fair to most of its participants” (Q12, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Though a majority of both right-of-center and nonaligned respondents believed instead that “the economic system in crypto unfairly favors powerful interests”, right-of-center respondents were more likely than nonaligned respondents to choose this answer (Q12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' However, left-of-center respondents were more likely than right-of-center or nonaligned respondents to believe that “hard work and determination are no guarantees of success” in crypto (Q13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Question 12 was one of three questions for which all three groups had a statistically different distribution of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Another was on gender equity: right-of-center respondents were least likely to believe “crypto has a gender problem,” nonaligned respondents somewhat more likely, and left-of-center respondents most likely, with about half of left-of-center respondents selecting this answer (Q15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' This spread shows that self-reported political alignment relates to not only economic but also social issues in the use of blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Differences also arose between the groups in the most common answer to questions on decision-making power and how to obtain favorable regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' When asked who should hold decision-making power over a blockchain, right-of-center respondents were more likely to choose “the token holders and/or node operators” than “a wide variety of on- and off-chain stakeholders”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' the reverse was true for left-of-center and nonaligned respondents, with left-of-center respondents more likely than other respondents to choose a variety of stakeholders (Q16, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Concerning how to obtain favorable regulation, left-of-center and nonaligned respondents were most likely to choose “work hand-in-hand with regulators” out of the available choices, and more likely to do so than right-of-center respondents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' in contrast, right-of-center respondents were, within confidence intervals, evenly split between three of the four available choices (Q14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Other statistically significant differences occurred in the distribution of responses, where one of the three groups differed from the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Right-of-center respondents were most likely to choose “make as much money as possible” as their goal and less likely to select “create social change and/or disrupt the industry”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' the reverse was true for left-of-center and nonaligned respondents (Q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' However, left-of-center respondents were less likely than others to believe crypto needs to prioritize “building art and community” to grow (Q9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Also, a smaller proportion of left-of-center respondents than other respondents believed that privacy is “the most important feature of blockchain” (Q6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses, grouped by self-reported political affiliation, to question 16 on decision-making power, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Together with question 14, this distribution indicates that right-of-center respondents were more likely than other respondents to hold beliefs aligned with minimizing external influence on blockchain governance and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Whoshouldhavedecision-makingpowerovera blockchain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' left Thepublic,electedrepresentatives right and/ornationalleaders neither Thecoredevelopersandtechnicalstaff ofablockchain Thetokenholdersand/ornodeoperators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='voters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='asdeterminedbythe protocol Awidevarietyofon-andoff-chain stakeholdersincludingtokenholders, nodeoperators,applicationdevelopers, foundations,andusers 0 20 40 60 80 100 PercentageLeft-of-center respondents were less likely to believe that “crypto does not need (human) governance,” while nonaligned respondents were less likely to believe “however crypto governs itself, it should also be regulated by the government” (Q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Left-of-center respondents were also more polarized on government regulation: they were less likely to believe it “can do some good,” and more likely to believe it is either “critical to protect the public interest” or “will always do more harm than good” (Q7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='4 Differences between respondents by Bitcoin and Ethereum affiliation At present, dynamics in the crypto community are largely driven by actors in two ecosystems: Bitcoin and Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' To examine differences in opinion between the 61% of respondents affiliated with Ethereum and the 31% (non-exclusive) affiliated with Bitcoin, we compared the distribution of answers selected by respondents affiliated with each of the two blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We found an overall quite similar distribution of responses regardless of affiliation, with a few statistically significant differences arising in beliefs about cryptogovernance, the semantics of the term crypto, personal goals in crypto, and stated political orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Governance and regulation of crypto were a key topic distinguishing Bitcoin affiliates from Ethereum affiliates (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Bitcoin affiliation was associated with a higher likelihood of believing that “crypto does not need (human) governance” (Q5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 8(A)) and that “token holders and/or node operators” should have decision-making power over a blockchain, whereas Ethereum was associated with “a wide variety of on- and off-chain stakeholders” (Q16, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 8(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Somewhat surprisingly, Bitcoin affiliation was also associated with a higher likelihood of believing that government regulation of crypto “can do some good” (Q7), although there was no statistically significant difference in opinions on how to obtain favorable regulation (Q14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Thus, it appears that Bitcoin affiliation is associated with a higher rate of wanting to maximize on-chain governance but also of tolerance of external regulation, perhaps in particular that which “can help force blockchains to become more decentralized,” as is included in the wording of question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Respondents’ semantic conception of crypto and their personal goals for their involvement also had some relation to blockchain affiliation: Bitcoin affiliation was associated with a higher likelihood of believing “crypto is mainly an economic technology” (Q3) and identifying with the statement “my goal in crypto is to make as much money as possible” (Q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Ethereum affiliation was associated with a higher likelihood of believing that “the economic system in crypto unfairly favors powerful interests” (Q12) and that “crypto has a gender problem” (Q15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses, grouped by blockchain affiliation, to (A) question 5 and (B) question 16 on blockchain governance, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' These distributions indicate that Bitcoin affiliates were more likely to favor a narrow definition of governance and its participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 10 (A) (B) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Who should have decision-making power over a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Which statement comes closest toyourviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' blockchain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' eth eth Most orall cryptogovernanceshouldbe Thepublic,electedrepresentatives, btc btc off-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' and/ornationalleaders However crypto governs itself, it should 工 The core developers and technical staff alsoberegulatedbythegovernment ofablockchain Crypto doesnotneed(human)governance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Thetokenholdersand/ornodeoperators, letthealgorithmsrunastheywere i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='voters, as determined by the 工 工 designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' protocol A wide varietyof on-and off-chain Most orall cryptogovernance should be stakeholders including token holders, H on-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' node operators,applicationdevelopers, 工 foundations,andusers 20 40 60 80 100 0 20 40 60 80 100 Percentage PercentageFigure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Responses, grouped by blockchain affiliation, to question 18 on self-reported political affiliation, with 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' While there was a statistically significant difference between affiliates of the two blockchains in identifying as right-of-center or nonaligned, Cramer’s V indicates that the strength of association between blockchain affiliation and political orientation was low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' For question 18 on political orientation, Bitcoin affiliation correlated with a higher likelihood of selecting “conservative or right-wing” and lower likelihood of selecting “neither” (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' There was no statistically significant difference between the proportions of respondents who chose “liberal or left-wing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Given that we were interested in analyzing blockchain affiliation separately from stated political orientation, we additionally checked for the strength of association between Q18 and a reduced version of Q19 with the options “Bitcoin”, “Ethereum”, and “Neither” (not mutually exclusive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Cramer’s V was low (less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='15) for all combinations of responses, indicating at most very weak association between the two questions (Supplementary Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' This gives us confidence that Bitcoin and Ethereum affiliation were not strongly associated with stated political orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='5 Validation of survey instrument To assess any correlations between responses to different questions, we computed the correlation matrix for all pairs of questions (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Of the 153 unique pairs, most showed little if any association (V < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' the strength of association was weak for 52 questions (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='1 <= V < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='3), and one question pair related to wealth distribution (Q11-Q12) showed a moderate strength of association (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The prevalence of weak or no association between distinct questions supports our assertion that each question addresses a distinct facet of a respondent’s beliefs or actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' This allows us to assess the relative importance of the specific statements provided in the answer choices to explain differences between respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='6 Feature selection and factor analysis To identify the beliefs which most contributed to explaining variance between respondents and to test our hypothesis, we computed the PCA vectors for individual choices (features) and examined the first principal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Beliefs above a threshold of magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='18, corresponding to the loading each response would have if all questions contributed equally to the component, were labeled as important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The features with the largest contributions to the first principal component were the following (listed in descending order of importance): 11 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='Do you consideryourself eth btc Conservative or right-wing Liberalorleft-wing Neither 0 20 40 60 80 100 Percentage- “The economic system in crypto unfairly favors powerful interests.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (Q12) “Crypto has a gender problem.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (Q15) “Government regulation of crypto will almost always do more harm than good.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (Q7) “[I consider myself] liberal or left-wing.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (Q18) “Crypto teams make too much profit.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (Q11) “In crypto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' hard work and determination are no guarantee of success for most people.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (Q13) “However crypto governs itself,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' it should also be regulated by the government.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (Q5) “Blockchain and DeFi are predatory technologies that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' on balance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' will harm most members of society.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (Q10) All three questions relating to wealth distribution and economic fairness (Q11-13) contributed more to explaining variance than most other questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Polarized opinions on government regulation (Q5 and Q7) and one specific political affiliation (Q18) also featured here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Altogether, 5 of the 8 questions that we had coded as defining an axis of political belief had a large contribution to this leading component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The same analysis can be done for the remaining principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The features with the largest component loading for the next two principal components are “Privacy is the most important feature of blockchain and crypto” (Q6) for the second principal component and “Crypto is mainly a political philosophy and/or lifestyle” (Q3) for the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' These choices, and their corresponding questions, are therefore among the more salient in explaining variance between respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Altogether, however, the variance explained by only the first few components was relatively low (21% for the first three components) and less than 10% was explained by the first component alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Taken together with weak associations between questions as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content="5, this implies that the number of latent variables required to describe respondents' beliefs is large." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Indeed, factor analysis using PCA and feature agglomeration yielded a null result, meaning that features did not cluster into a few interpretable groupings (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='3 of the Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Even so, we find that the first principal component axis corroborates a theory-first constructed axis, as described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='7 Validation of constructed crypto-political axis For each respondent, a political score was calculated using the values in Table 1 and a type was assigned according to the score thresholds described in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The feature selection and factor analysis results can be used to evaluate the validity of this constructed crypto-political axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The distribution of scores and types assigned to participants who completed the survey is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' On the left-of-center side, 20% of respondents were identified as “crypto-communitarian” or “crypto-leftist”, while 9% of respondents were given the “crypto-centrist” label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The most commonly assigned type was the “crypto-libertarian” types, with nearly half of respondents receiving this designation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' overall, right-of-center types (“crypto-libertarian” and “crypto-anarcho-capitalist”) dominated with 71% of respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' This distribution is unimodal, low skewness, and centered around the median possible score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' However, because the range of possible scores was not centered around zero, we find that a majority of respondents were labeled as crypto-politically “right-of-center”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' For a summary of how this distribution differed with political self-identification and blockchain affiliation, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 of the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 12 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Distribution of assigned crypto-political scores and corresponding sentiment types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Despite a 14% minority of respondents identifying as ideologically conservative or right-wing, our measure placed 71% in the right-of-center libertarian and anarchocapitalist categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Box plots showing the distribution of values for each political type along each of the first three principal components produced by PCA, with the mean scores indicated by a white triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' There is essentially no overlap between crypto-leftist and crypto-ancap types for component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Partitioning the respondents by the types we identified for them, and plotting them in the first three PCA components, we find, again, that the first principal component succeeds at capturing the political dimension of respondent variation, while the next two components are less informative (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The low overlap between the interquartile ranges for adjacent types indicates that the continuous construct we defined and the defined types which discretize it help to explain differences between respondents’ beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Thus, the constructed political axis seems to reflect true variation in the population and may be of use in future work characterizing the ideological structure of the crypto community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 5 Discussion Though optimistic beliefs about the current state of cryptoeconomics were slightly more prevalent, the survey responses indicate nearly as much dissatisfaction with the fair distribution and attainability of cryptoeconomic wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Both a desire for sociopolitical change and an interest in personal financial gain were common factors in participants’ interest in blockchain technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Respondents 13 400 Type Value count 300 Crypto-leftist Crypto-communitarian Crypto-centrist 200 Crypto-libertarian Crypto-anarchocapitalist 100 0 6 4 2 0 2 4 6 8 10 Politics score6 4 Type Crypto-leftist 2 lue Crypto-communitarian Crypto-centrist 0 Crypto-libertarian Crypto-anarchocapitalist 2 4 1 2 3 Componentgenerally were optimistic about the social potential of blockchain technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' with some having reservations about its gender equity and some focusing on its privacy implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Overall, though a majority of respondents were willing to accept or even collaborate on regulation, large minorities strongly disagreed, and distaste for direct government involvement in implementations of crypto technology was common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Despite low rates of respondents’ self-identification with “conservative or right-wing” politics, we observed a prevalence of right-of-center crypto-political types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' A broadly similar distribution was observed in a CoinDesk report published in 2018 (Ryan 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The discrepancy between general political self-identification and our crypto-specific labeling bears further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' It may relate to an association of the term “conservative” with social conservatism, whereas crypto-libertarianism, the crypto-political type we found to be most common, emphasizes a form of economic libertarianism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Furthermore, the connotations of “conservative” and “liberal” vary significantly by geographic region, so the question may have been interpreted differently across respondents based on their country of residence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The correlation of the constructed political axis with the first principal component–a commonly-used, well-validated axis–suggests a primacy of political variation in explaining patterns of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Furthermore, the existence of differences in the distribution of beliefs between self-identified political orientations indicates that traditional political ideologies have some bearing on how participants relate to blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' For example, the “left-of-center” group articulated distinct beliefs about economic opportunity, fairness of wealth distribution, privacy, and the growth of crypto (Q11-13, Q6, and Q9), suggesting that left-of-center respondents are more likely to apply more community-oriented multi-stakeholder values to the blockchain ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The “nonaligned” group, on the other hand, articulated distinct beliefs about gender equity and government regulation (Q15 and Q5), suggesting this group is more clearly defined by lower trust in existing government institutions2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' This lower approval of government regulation suggests that those who identified as neither left-of-center nor right-of-center are more likely to position themselves as separate from existing political and governance systems entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We are interested in understanding the extent to which the characteristics of developers and users of specific blockchains are distinctive of each blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Critics like David Golumbia have argued that Bitcoin, both in its design and ideological constitution, is principally a conservative movement interested strictly in Bitcoin’s record of gaining value (Golumbia 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Those observations were not made in opposition to Ethereum or any other blockchain, although Ethereum had been live for a year at the time of Golumbia’s writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Our findings indicate that in fact, there are few differences between Bitcoin and Ethereum users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' However, differences in technical implementation between Bitcoin and Ethereum may relate to differences in opinion on their governance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Unlike Bitcoin, which has limited support for transactions other than money transfers, Ethereum as an infrastructure enables the developing and building of various applications and projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The broader set of use cases for Ethereum may lead its users to believe a broader set of stakeholders should be involved in its governance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Furthermore, Ethereum affiliation was associated with a greater sensitivity to perceived socioeconomic inequity, which may relate to differences in how the blockchains are used alongside other technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In Ethereum ecosystems, users linking their own blockchain activity to other 2 We refer in this work to respondents who chose to identify as neither left-of-center nor right-of-center as “nonaligned”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We choose this term in contrast to a term such as “apolitical” in a nod to ideas of political agnosticism developed by ethnographers in observing open-source communities (Gabriella Coleman 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 14 personally-identifying information, such as Discord handles or Twitter accounts, is not uncommon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' more research is needed to understand whether lower rates of anonymity relate to greater awareness of actual or perceived social demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Communities organized around crypto are proving to be a laboratory for new ways that humans can organize collective action, but are not operating in a historical vacuum: it appears that some patterns observed in early users of other internet technologies have arisen or continue to appear in the blockchain context as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' To complement this sociological work, further anthropological research could shed some light on the extent to which the economic and political beliefs held by participants in crypto echo the ideologies of two earlier movements: the cryptographic hacker and open-source software communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The distribution of responses relating to fair rewards for developer teams and the utility of hard work and in crypto indicates that meritocratic values are prevalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' meritocracy may play a similar role in blockchain ecosystems, themselves often open source, as it has in prior open source and hacker communities (Gabriella Coleman 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Dunbar-Hester 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Privacy has been at the forefront of concerns in the development of internet technologies since the cypherpunks (Hughes 1993) and remains prevalent in blockchain (Brunton 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Further research is needed to understand how these values compare to those of open-source software communities and early adopters of the internet or how they may have changed over time as cryptocurrencies become more mainstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 6 Limitations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='1 Survey methodology Selection bias may arise given that the random sample assumption is limited by how the survey was distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In particular, since the survey was opt-in, people with stronger and potentially more extreme opinions may have been more motivated to complete the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Also, the survey was made available only in English, and so is likely not representative of the full geographic distribution of users of blockchain technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Presentation of some preliminary findings prior to finalizing data collection may also have influenced some respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Additionally, we do not have a guarantee of uniqueness of each respondent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' moreover, the two recruitment strategies we used may have motivated respondents to provide multiple responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In choosing the wording of each question and answer choice, we made an effort to mitigate response bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Still, we have identified some limitations in interpreting questions based on the wording of the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Q5 may have had an insufficient distinction between the two most commonly-selected choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In answering Q14, respondents who selected “Keep on doing what we’re doing” may have rejected the premise of the question rather than believed this was a way to achieve the stated goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Additionally, while we intended Q15 to refer to perceptions of gender inequity in participation or compensation within crypto, the wording of the choices may have been too vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Further demographic information would be valuable context for interpreting some questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Future work could include a question on the geographic location of respondents, where local regulations and political attitudes could inform a more detailed analysis of questions on national government regulation and political affiliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Interpretation of question 18 on political self-identification may be similarly limited by differences in how terms such as “liberal” and “conservative” are understood across the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 Analysis To be able to use PCA for the discrete data, we one-hot encoded specific choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' While PCA is generally better suited to continuous data than boolean data, we find that in this context the results were cleanly interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We also chose not to include null responses as an additional coded choice for feature selection or factor analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' While this does result in using only a subset of the responses and potentially removing relevant information about respondents’ beliefs, it prevents the null responses from receiving artificially high importance due to their relative rarity and bypasses the difficulty in interpreting the null response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 7 Conclusion In this work, we have introduced a new survey of blockchain users’ political, economic, and governance opinions with respect to crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Based on 3710 survey responses, we find that users were spread across a variety of perceptions of the distribution of economic power, normative beliefs about the distribution of power in governance, and opinions on the role of external regulation of crypto, though they were broadly in agreement that crypto has a net-positive impact on the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Equal numbers of respondents self-identified as liberal or non-aligned, while only about a third as many respondents self-identified as conservative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' this self-reported political affiliation was associated with differences in opinions on most questions, but especially on economic fairness, decision-making power, and how to obtain favorable regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In contrast, we observed few differences in opinions between respondents affiliated with Bitcoin and with Ethereum, on issues of blockchain governance and regulation and on personal attitudes towards crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' While the full field of beliefs elides neat interpretation in terms of underlying factors, we found that the existence of a political dimension was supported both by a theory-driven construct and by a common, well-validated analytical method (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' While this dataset is an important step towards understanding the distribution of crypto users’ beliefs about blockchain technology and its utility, open questions remain as to why users believe what they do about crypto and how their beliefs match up with reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' For example, considering the question of who should have decision-making authority over a blockchain: Is the large-minority opinion that token-holding voters should control a blockchain underlied by a belief that minimizing human input to governance will make it more efficient and less flawed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Is there a disconnect between the common normative beliefs of what should be happening in cryptogovernance and which types of stakeholders actually can and do participate in governance of major blockchains?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Although our research found only a few instances where affiliation with a specific blockchain was associated with differences in beliefs, further research is needed to better understand whether specific architectures or ecosystems within crypto differ in the values or goals embedded in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Future interdisciplinary work could shed some light on the extent to which participants have common understandings of core signifiers such as decentralization and autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Given that our work takes inspiration from the long-running Pew political survey, we see the need for a regular survey of cryptopolitical sentiment, with an added demographic panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' This could facilitate the identification and comparison of ideologies and modes of participation within newer chains such as Solana, L2s such as Polygon, and even large DAOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 16 8 Conflict of Interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 9 Author Contributions LMK conducted data analysis and wrote manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' SF advised on methodology and advised on manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' JT designed and deployed the survey instrument and advised on manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 10 Funding Lucia Korpas and Joshua Tan were supported by grants from the Filecoin Foundation and from One Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 11 Acknowledgments The authors would like to acknowledge Michael Zargham for technical discussion, Ann Brody for qualitative discussion, and Tyler Sullberg and Nathan Schneider for feedback on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 12 Data Availability Statement The dataset generated and analyzed for this study can be found in the Metagovernance Project’s Govbase repository on Airtable (https://airtable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='com/shrgnUrj0dqzZDsOd/tblvwbt4KFm8MOSUQ/viw82nVNrdHFrowoo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The Python code used to conduct the analysis and produce the figures can be found in the GitHub repository for this work (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='com/metagov/cryptopolitics-paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 17 13 References Bauerle, Nolan, and Peter Ryan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2018.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='pewresearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='org/fact-tank/2021/11/11/16-of-americans-say-they-have-ever-inves ted-in-traded-or-used-cryptocurrency/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 18 Reijers, Wessel, Fiachra O’Brolcháin, and Paul Haynes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' “Governance in Blockchain Technologies & Social Contract Theories.” Ledger 1 (December): 134–51.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' “Left, Right and Center: Crypto Isn’t Just for Libertarians Anymore.” CoinDesk, July 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='coindesk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='com/markets/2018/07/27/left-right-and-center-crypto-isnt-just-for-libert arians-anymore/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Szabo, Nick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' “Smart Contracts: Building Blocks for Digital Markets.” EXTROPY: The Journal of Transhumanist Thought 18 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Winner, Langdon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' “Do Artifacts Have Politics?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Daedalus, Winter 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Zamfir, Vlad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' “Against Szabo’s Law, For A New Crypto Legal System.” Crypto Law Review (blog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' January 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' https://medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='com/cryptolawreview/against-szabos-law-for-a-new-crypto-legal-system-d00 d0f3d3827.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 19 14 Tables Table 1 Question number Question text Choice text Contribution to political score 1Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' There is one (layer 1) blockchain that is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' There is no one best blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2Which blockchain is the best?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Bitcoin Ethereum Solana Cardano Polkadot Other 3Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Crypto is mainly an economic technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Crypto is mainly a political philosophy and/or lifestyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 4Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' My goal in crypto is to have fun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' My goal in crypto is to make as much money as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' My goal in crypto is to create social change and/or disrupt the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' My goal in crypto is to earn a living and/or build my career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 5Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Most or all cryptogovernance should be on-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Most or all cryptogovernance should be off-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Crypto does not need (human) governance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' let the algorithms run as they were designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' However crypto governs itself, it should also be regulated by the government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 6Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Privacy is the most important feature of blockchain and crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 2 Privacy is nice, but it’s not the most important feature of blockchain and crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 7Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Government regulation of crypto will almost always do more harm than good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 Government regulation of crypto can do some good, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' it can help force blockchains to become more decentralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Government regulation of crypto is critical to protect the public interest in these technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 8Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Having a central bank run a cryptocurrency is a good idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Having a central bank run a cryptocurrency is a bad idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 9In order to grow, the crypto ecosystem should: Build art and community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 Help people around the world earn a living.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 Build useful tech that solve real problems for a set of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 20 Provide financial instruments for maximum wealth creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 10Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Blockchain and DeFi are beneficial technologies that, on balance, will help most members of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Blockchain and DeFi are predatory technologies that, on balance, will harm most members of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 11Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Most crypto teams make a fair and reasonable amount of profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 Crypto teams make too much profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 12Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The economic system in crypto is generally fair to most of its participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 The economic system in crypto unfairly favors powerful interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 13Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=" Most people who want to get ahead in crypto can make it if they're willing to work hard." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 In crypto, hard work and determination are no guarantee of success for most people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 14To get more favorable regulation of cryptocurrencies from national governments, the most important thing the crypto community can do is: Adapt our technology and practices in order to minimize potential conflicts with the law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Work hand-in-hand with regulators to identify a solution that works for both government and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Hire lawyers and lobbyists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' organize the community to mount a public pressure campaign on politicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Keep on doing what we’re doing, legal or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 15Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Crypto has a gender problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 Crypto does not have a gender problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1 16Who should have decision-making power over a blockchain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The public, elected representatives, and/or national leaders A wide variety of on- and off-chain stakeholders including token holders, node operators, application developers, foundations, and users The token holders and/or node operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=" voters, as determined by the protocol The core developers and technical staff of a blockchain 17I'm here for." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' the memes the jobs the tech the airdrops 18I identify as: Liberal or left-wing 1 Conservative or right-wing 1 Neither 0 21 19OPTIONAL: Do you affiliate with any of the following ecosystems or communities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Bitcoin Ethereum Solana Cardano Polkadot Other Table 2 Politics score Assigned type [-7,9] Overall possible range of scores >= 5 Crypto-anarchocapitalist 0 < x < 5 Crypto-libertarian x = 0 Crypto-centrist 3 < x < 0 Crypto-communitarian <= -3 Crypto-leftist 22 Supplementary Material 1 Supplementary Information 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='1 Recruitment strategy: Ad-hoc typology Based on their responses to the survey, each respondent was assigned a political faction, an economic faction, and a governance class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' These were computed immediately upon completion of the survey according to numerical weights assigned to each answer choice and formulas defined for each faction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The faction definitions and names were developed with input from the community, and served largely as a way to (1) recruit participants and (2) help participants interpret their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Depending on the response to Question 3 – whether the respondent considered crypto to be primarily a political philosophy or an economic technology – each respondent was correspondingly presented with either their political faction or economic faction as their overall assigned faction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The five political factions (“crypto-leftist”, “DAOist”, “true neutral”, “crypto-libertarian”, and “crypto-ancap”) were an alternate naming scheme for our constructed political axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The five economic factions (“earner”, “cryptopunk”, “NPC”, “techtrepreneur”, “degen”) were intended to represent the respondents’ primary mode of economic engagement with crypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Additionally, we defined four “classes” (Szabian, Gavinist, Zamfirist, and Walchian) that were intended to capture respondents’ beliefs about governance and government regulation, inspired by the positions articulated by Nick Szabo, Gavin Wood, Vlad Zamfir, and Angela Walch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The mechanism for assigning the factions based on respondents’ answers was a weighted sum of their responses to each question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' every answer choice in every question adds points to one or more factions, and the faction assigned is the one with the most points, with some thresholding to account for respondents who did respond to all questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In cases where not enough responses were given, the assignment defaulted to “true neutral” or “NPC”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Questions 6, 7, 9, 11, 12, 13, 14, 15, and 18 were used in computing the political faction, questions 4, 9, 17 in computing the economic faction, and questions 5, 7, 8, 14, and 16 in computing the governance class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The political faction names and definitions were conceptualized and iterated through a community-based effort at crypto conferences and online, especially within the Metagovernance Project Slack community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The names of the factions reflected both popular slang in crypto (degen, cryptopunk, crypto-libertarian, crypto-leftist, DAOist, crypto-ancap) as well as a few inventions of the authors when there was no existing slang or word for that archetype (earner, techtrepreneur, true neutral, NPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The visual representations of the factions, which were used as the underlying images of a series of NFTs, played off various memes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' shiba inu / doge representing degens) and cultural icons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Shrek representing crypto-libertarians) common in crypto and on the internet more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Within the main text, the political “factions” were renamed to the described “types”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 Differences in assigned political types by self-reported political orientation and blockchain ecosystem affiliation There was a statistically significant difference in the overall distribution of assigned cryptopolitical factions across the left-of-center, right-of-center, and nonaligned groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Right-of-center respondents were the only group more likely to be assigned the crypto-anarcho-capitalist faction than the crypto-libertarian faction, and less likely than the other two groups to be assigned true neutral, 23 DAOist, or crypto-leftist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Unlike the other two groups, left-of-center respondents were more likely to be assigned any of crypto-centrist, crypto-communitarian, or crypto-leftist than to be assigned crypto-anarcho-capitalist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Nonaligned respondents generally split the difference between the proportions of left-of-center and right-of-center respondents assigned each faction except for crypto-libertarian, which they were more likely to be assigned than either of the other two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Note that while Q18 was used directly in assigning the political faction, it was one of nine such questions, each with similar weights in the overall cryptopolitical score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' A higher percentage of Bitcoin-affiliated respondents than Ethereum-affiliated respondents received the “crypto-anarcho-capitalist” label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='3 Are there clusters of respondents or features?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We were interested in understanding whether the range of responses would be better captured by the idea of “types” (groupings of respondents, consistent with data-driven clustering methods) or “axes” (grouping of questions, consistent with factor analysis techniques).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We investigated this by extending our PCA analysis with feature agglomeration, based on a feature set of 48 choices provided across 17 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' We used feature agglomeration to hierarchically generate clusters of features, where each feature corresponded to the selection of a provided choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Given that these features are boolean in nature, we used the Dice distance metric to determine distances between features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' this can be understood as the fraction of the sample for which the two features intersect or overlap, as a representation of the shared information between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' To determine which clusters should be merged and when, we used the complete linkage criterion, which relates to the maximum distances between all features of the two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' From the feature agglomeration results, we ask whether the assigned political types directly correlate with any of the identified factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=', to what extent does our categorization of beliefs as political, economic, or governance-related meaningfully describe joint variations in the choices selected?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' At the threshold where three clusters were present, the clusters contained A=31, B=10, and C=7 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The cluster that shared the most features in common with the important principal component features listed above was the cluster with 10 features: the specific choices listed above for beliefs on economic fairness, gender, and political affiliation (Q11-Q13, Q15, and Q18) appeared in this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The two specific choices relating to government regulation (Q5 and Q7) were present together in the cluster with 7 features Although cluster B aligned fairly well with the first (politically-focused) principal component, our exploration of the other clusters, and the hierarchy that produced them, did not indicate much qualitative support for this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' This null assessment is supported by our mapping of respondents into feature space: both factor analysis methods shown in Supplementary Figure S7 shows responses organized around a single centroid, not the multiple clusters that would be expected in the presence of clear respondent types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Given that the feature agglomeration method allowed us to traverse the entire hierarchy of feature clusters, we also looked at the agglomerated clusters to see whether three stable feature clusters arise from the full feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Based on the distances required for clusters to merge, it appears instead that five distinct latent variables may best describe the distribution of respondents’ beliefs, as shown in Supplementary Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' That said, the smaller two out of the three clusters identified above persisted unchanged at the five-cluster threshold, suggesting that the underlying factors for each of those two clusters have more explanatory power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 24 2 Supplementary Figures and Tables 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='1 Supplementary Figures Supplementary Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Distribution of responses to questions 1, 2, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Note that question 2 was presented only to respondents who selected the response “There is one (layer one) blockchain that is the best” for question 1, and that respondents were given the option to select Ethereum, Bitcoin, Solana, Polkadot, Cardano, or a fill-in-the-blank “Other”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Thereisone(layer1)blockchainthat is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Thereis no one best blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 Percentage2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Which blockchain is the best?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Ethereum Bitcoin Solana Cardano NEAR Polkadot AVAX Polygon Terra DeSo Avalanche lota Stellar Monero Cosmos Rose Binance Tezos MINA Hex, EOS 0 20 40 60 80 100 Percentage5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Most or all cryptogovernance should be on-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Crypto does not need (human) governance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' letthealgorithmsrunastheywere designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Most orall cryptogovernance should be off-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Howevercrypto governsitself,itshould also be regulated by the government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 PercentageSupplementary Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Distribution of responses to questions 6, 8, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=" Privacy is the most important feature of blockchainand crypto, Privacy is nice,but it's not themost important featureof blockchain and crypto." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 Percentage8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Having a central bank run a cryptocurrency is a good idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Having a central bank run a cryptocurrency is a bad idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 Percentage9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In order to grow, the crypto ecosystem should: Providefinancial instrumentsfor maximumwealthcreation Build useful tech that solve real problems for a set of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Build art and community Help people around the world earn a living 0 20 40 60 80 100 PercentageSupplementary Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Distribution of responses to questions 10, 15, and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 27 1o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Blockchain and DeFi are beneficial technologiesthat, on balance, will help most members of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Blockchainand DeFiarepredatory technologies that, on balance, will harm most members of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 Percentage15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Which statement comes closest to your views?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Crypto does not have a gender problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Cryptohas agenderproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 0 20 40 60 80 100 Percentage17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=" I'm here for." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='. the tech H the airdrops the memes the jobs H 0 20 40 60 80 100 PercentageSupplementary Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Distribution of responses to question 19, and distribution in the number of those multiple-select choices that respondents who answered the question chose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Note that respondents were given the option to select Ethereum, Bitcoin, Solana, Polkadot, Cardano, or a fill-in-the-blank “Other”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' For plot legibility, only the ecosystems listed by at least 10 respondents are shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 28 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' OPTIONAL: Do you affiliate with any of the following ecosystems or communities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Ethereum Bitcoin Solana Polkadot Not specified Cardano Avalanche Q19_all Cosmos AVAX None Polygon Terra NEAR Fantom IOTA Tezos XRP 0 500 1000 1500 2000 FrequencyQ19 count One H Two or more H None 0 20 40 60 80 100 PercentageSupplementary Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Correlation (Cramer’s V) between choices to questions 18 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The low correlation between choices belonging to different questions indicates that political self-identification and blockchain affiliation are largely independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 29 answer) Ethereum 18=Neither 19=Neither Bitcoin 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='0 9 Q18=(No answer) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content="8 Q18=Conservative or right-wing Q18=Liberal or left-wing Cramer's Q18=Neither 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='4 Q19=Bitcoin Q19=Ethereum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 O19=Neither 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='0Supplementary Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Correlation (Cramer’s V) between all questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Gray indicates that the correlation between questions was not computed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' this is used when one question was only presented to the respondents upon a particular selection for the other question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' No clusters are clearly observable: the questions are not systematically related in macro groupings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='0 8 Q1 Q2 Q3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='8 Q4 Q5 Q6 Q7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content="6 > Q8 Cramer's Q9 Q10 Q11 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='4 Q12 Q13 Q14 Q15 Q16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 Q17 Q18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='0(A) (B) Supplementary Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (A) Projection of the respondents onto the first two principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' (B) Projection of the respondents onto the feature clusters B (10 features) and C (7 features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' In both cases, there is a continuous (rather than discrete) distribution of respondents along the two axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The unimodal distribution of respondents along the two axes does not suggest the presence of clusters of respondents by their responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 31 PCA with political factions: n components = 3 (O, 1) 6 4 Component 2 2 0 2 4 4 2 0 2 4 6 Component 1n components = 3 (1, 2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='0 Count 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='8 80 B 120 Feature cluster 160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='8 Feature cluster CSupplementary Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' Dendrogram describing the hierarchical agglomeration of feature clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The colors delineate the first five clusters that branch off from the one-cluster case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=', the one containing all features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' The three-cluster case involves grouping some of these together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content=' 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQf4ALw/content/2301.02734v1.pdf'} +page_content='8 0.' metadata={'source': 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--- /dev/null +++ b/ZtE0T4oBgHgl3EQfnAG2/content/tmp_files/2301.02507v1.pdf.txt @@ -0,0 +1,1111 @@ +arXiv:2301.02507v1 [cs.DM] 6 Jan 2023 +Perturbation results for +distance-edge-monitoring numbers∗ +Chenxu Yang †, +Ralf Klasing ‡, +Changxiang He §, +Yaping Mao ¶ +Abstract +Foucaud et al. recently introduced and initiated the study of a new graph-theoretic +concept in the area of network monitoring. Let G = (V, E) be a graph. A set of vertices +M ⊆ V (G) is a distance-edge-monitoring set of G if any edges in G can be monitored by +a vertex in M. The distance-edge-monitoring number dem(G) is the minimum cardinality +of a distance-edge-monitoring set of G. In this paper, we first show that dem(G \ e) − +dem(G) ≤ 2 for any graph G and edge e ∈ E(G). Moreover, the bound is sharp. Next, +we construct two graphs G and H to show that dem(G) − dem(G − u) and dem(H − +v) − dem(H) can be arbitrarily large, where u ∈ V (G) and v ∈ V (H). We also study the +relationship between dem(H) and dem(G) for H ⊂ G. In the end, we give an algorithm to +judge whether the distance-edge-monitoring set still remain in the resulting graph when +any edge of a graph G is deleted. +Keywords: Distance; Perturbation results; Distance-edge-monitoring set. +AMS subject classification 2020: 05C12; 11J83; 35A30; 51K05. +1 +Introduction +In 2022, Foucaud et al. [10] introduced a new graph-theoretic concept called distance-edge- +monitoring set, which means network monitoring using distance probes. Networks are naturally +modeled by finite undirected simple connected graphs, whose vertices represent computers and +whose edges represent connections between them. When a connection (an edge) fails in the +network, we can detect this failure, and thus achieve the purpose of monitoring the network. +∗Supported by the National Science Foundation of China (No. 12061059), the Qinghai Key Laboratory of +Internet of Things Project (2017-ZJ-Y21) +†School of Computer, Qinghai Normal University, Xining, Qinghai 810008, China. cxuyang@aliyun.com +‡Corresponding author: Universit´e de Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR 5800, Talence, France. +ralf.klasing@labri.fr +§College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China. +changxianghe@163.com +¶Academy of Plateau Science and Sustainability, Xining, Qinghai 810008, China. maoyaping@ymail.com +1 + +Probes are made up of vertices we choose in the network. At any given moment, a probe of the +network can measure its graph distance to any other vertex of the network. Whenever an edge +of the network fails, one of the measured distances changes, so the probes are able to detect the +failure of any edge. Probes that measure distances in graphs are present in real-life networks. +They are e.g. useful in the fundamental task of routing [7, 11] and are also frequently used for +problems concerning network verification [1, 3, 4]. +In a network, we can put as few detectors as possible to monitor all the edges, a natural +question is whether the detectors placed in the original graph are still sufficient and need to +be supplemented or reduced when some nodes or edges in the original graph are subjected to +external interference and damage, we refer to [8, 9, 13, 14, 16]. This kind of problem is usually +called perturbation problem. +Graphs considered are finite, undirected and simple. +Let G = (V, E) be a graph with +vertex set V and edge set E, respectively. +The neighborhood set of a vertex v ∈ V (G) is +NG(v) = {u ∈ V (G) | uv ∈ E(G)}. The degree of a vertex v in G is denoted by d(v) = |NG(v)|. +Let δ(G) and ∆(G) be the minimum and maximum degree of a graph G, respectively. For +any subset X of V (G), let G[X] denote the subgraph induced by X, and E[X] the edge set of +G[X]; similarly, for any subset F of E(G), let G[F] denote the subgraph induced by F. We +use G − X to denote the subgraph of G obtained by removing all the vertices of X together +with the edges incident with them from G; similarly, we use G \ F to denote the subgraph of +G obtained by removing all the edges of F from G. If X = {v} and F = {e}, we simply write +G − v and G \ e for G − {v} and G \ {e}, respectively. +In a graph G, the base graph Gb of a graph G is the graph obtained from G by iteratively +removing vertices of degree 1. Clearly, dem(G) = dem(Gb). A x-y path with length dG(x, y) in +G is a x-y geodesic. +1.1 +Distance-edge-monitoring sets and numbers +Foucaud et al. [10] introduced a new graph-theoretic concept called distance-edge-monitoring +sets, which is relevant to network monitoring. +Definition 1. For a set M of vertices and an edge e of a graph G, let P(M, e) be the set of +pairs (x, y) with a vertex x of M and a vertex y of V (G) such that dG(x, y) ̸= dG\e(x, y). In +other words, e belongs to all shortest paths between x and y in G. +Definition 2. For a vertex x, let EM(x) be the set of edges e such that there exists a vertex v in +G with (x, v) ∈ P({x}, e), that is EM(x) = {e | e ∈ E(G) and ∃v ∈ V (G) such that dG(x, v) ̸= +dG\e(x, v)} or EM(x) = {e | e ∈ E(G)and P({x}, e) ̸= ∅}. If e ∈ EM(x), we say that e is +monitored by x. +Finding a particular vertex set M and placing a detector on that set to monitor all edge +sets in G, has practical applications in sensor and network systems. +2 + +Definition 3. A set M of vertices of a graph G is distance-edge-monitoring set if every edge +e of G is monitored by some vertex of M, that is, the set P(M, e) is nonempty. Equivalently, +� +x∈M +EM(x) = E(G). +One may wonder to know the existence of such an edge detection set M. The answer is +affirmative. If we take M = V (G), then +E(G) ⊂ +� +x∈V (G) +N(x) ⊂ +� +x∈V (G) +EM(x). +Therefore, we consider the smallest cardinal of M and give the following parameter. +Definition 4. The distance-edge-monitoring number dem(G) of a graph G is defined as the +smallest size of a distance-edge-monitoring set of G, that is +dem(G) = min +� +|M| +��� +� +x∈M +EM(x) = E(G) +� +. +The vertices of M represent distance probes in a network modeled by G. Distance-edge- +monitoring sets are very effective in network fault tolerance testing. For example, a distance- +edge-monitoring set can detect a failing edge, and it can correctly locate the failing edge by +distance from x to y, because the distance from x to y will increases when the edge e fails. +Foucaud et al. [10] showed that 1 ≤ dem(G) ≤ n − 1 for any G with order n, and graphs +with dem(G) = 1, n − 1 was characterized in [10]. +Theorem 1.1. [10] Let G be a connected graph with at least one edge. Then dem(G) = 1 if +and only if G is a tree. +Theorem 1.2. [10] dem(G) = n − 1 if and only if G is the complete graph of order n. +Theorem 1.3. [10] Let G be a connected graph with a vertex x of G and for any y ∈ N(x), +then, we have xy ∈ EM(x). +Theorem 1.4. [10] For a vertex x of a graph G, the set of edges EM(x) induces a forest. +A vertex set M is called a vertex cover of G if M ∩{u, v} ̸= ∅ for uv ∈ E(G), The minimum +cardinality of a vertex cover M in G is the vertex covering number of G, denoted by β(G). +Theorem 1.5. [10] In any graph G of order n, any vertex cover of G is a distance-edge- +monitoring set, and thus dem(G) ≤ β(G). +1.2 +Research backgrounds and our results +Perturbation problems in graph theory are as follows. +3 + +Problem 1. Let G be a graph, and let e ∈ E(G) and v ∈ V (G). Let f(G) be a graph parameter. +(i) The relation between f(G) and f(G − v); +(ii) The relation between f(G) and f(G \ e). +Chartrand et al. [6] studied the perturbation problems on the metric dimension. Monson et +al. [13] studied the effects of vertex deletion and edge deletion on the clique partition number +in 1996. In 2015, Eroh et al. [9] considered the effect of vertex or edge deletion on the metric +dimension of graphs. Wei et al. [14] gave some results on the edge metric dimension of graphs. +Delen et al. [8] study the effect of vertex and edge deletion on the independence number of +graphs. +We first consider the existence of graphs with given values of distance-edge-monitoring +numbers. +Problem 2. Let r, t, n be three integers with 1 ≤ r, s ≤ n−1, and let G be a connected of order +n. +(1) Is there a connected graph G of order n such that dem(G) = r? +(2) Is there a connected subgraph H ⊂ G such that dem(H) = s and dem(G) = r? +In Section 2, we give the answers to Problem 2. +Proposition 1.1. For two integers r, n with 1 ≤ r ≤ n − 1, there exists a connected graph G +of order n such that dem(G) = r. +Corollary 1.6. Given given three integers s, t, n with 1 ≤ s ≤ t ≤ n−1, there exists a connected +graph H ⊂ G of such that dem(H) = s and dem(G) = t. +In Section 3, we focus on Problem 1 (ii) and study the difference between dem(G \ e) and +dem(G). +Theorem 1.7. Let G be a graph. For any edge e ∈ E(G), we have +dem(G \ e) − dem(G) ≤ 2. +Moreover, this bound is sharp. +Let G be a graph. Denote by G + E the graph with V (G + E) = V (G) and E(G + E) = +E(G) ∪ E. We construct graphs having the following properties in Section 3. +Theorem 1.8. For any positive integer k, i with 2 ≤ i ≤ k, there exists a graph G and an +edge set Ei+1 (|Ei+1| = i + 1) such that dem(G + Ei+1) − dem(G) = i. +Furthermore, let +Gi = G + Ei(1 ≤ i ≤ k), we have dem(Gi) = i. +A feedback edge set of a graph G is a set of edges such that removing them from G leaves +a forest. The smallest size of a feedback edge set of G is denoted by fes(G) (it is sometimes +called the cyclomatic number of G). +4 + +Theorem 1.9. [10] If fes(G) ≤ 2, then dem(G) ≤ fes(G) + 1. Moreover, if fes(G) ≤ 1, then +equality holds. +Theorem 1.9 implies the following corollary, and the proof will be given in Section 3. +Corollary 1.10. Let Tn be a tree of order n. For edges e1, e2 ∈ E(Tn), we have +(1) dem(Tn + e1) = dem(Tn) + 1. +(2) dem(Tn + e1 + e2) = 2 or 3. +Furthermore, there exists a tree T i +n with two edges ei +1, ei +2 such that dem(Tn +e1 +e2) = i+1, +where i = 1, 2. +The following result in Section 4 shows that there exist a graph G and a subgraph H such +that the difference dem(G) − dem(H) can be arbitrarily large. +Theorem 1.11. For any positive integer k, there exist two graphs G1, G2 and their non- +spanning subgraphs H1, H2 such that +dem(G1) − dem(H1) = k and dem(H2) − dem(G2) = k. +Furthermore, dem(G) − dem(H) can be arbitrarily large, even for H = G − v. +Theorem 1.12. For any positive integer k, there exist two graphs G, H and two vertices u ∈ +V (G), v ∈ V (H) such that +(1) dem(G) − dem(G − u) ≥ k +(2) dem(H − v) − dem(H) ≥ k. +For a graph G with fixed order, the difference between dem(G) and dem(G − v) can be +bounded. +Proposition 1.2. For a graph G with order n (n ≥ 2) and v ∈ V (G), if G − v contain at least +one edge, we have dem(G) − dem(G − v) ≤ n − 2. Moreover, the equality holds if and only if +G is K3. +Theorem 1.13. Let G be a connected graph with order n ≥ 4 and dem(G) = 2. Let E ⊆ E(G). +If dem(G) = dem(G − E), then |E| ≤ 2n − 6. Furthermore, the bound is sharp. +Theorem 1.14. Let T be a spanning tree of Kn. Then 1 ≤ dem(Kn|T) ≤ ⌊n/2⌋. Furthermore, +the bound is sharp. +In Section 5, we focus on the following problem and give an algorithm to judge whether the +distance-edge-monitoring set is still valid in the resulting graph when any edge (or vertex) of a +graph G is deleted. +Problem 3. For any graph G, if some edges or vertices in G break, we want to know if the +original detection set can detect all edges. +For some well-known graphs, the effects of vertex deletion are dem(G) > dem(G − v), +dem(G) = dem(G − v), or dem(G) < dem(G − v), which will be presented in Section 5. +5 + +2 +Results for Problem 2 +A kite K(r, n) is a graph obtained from the complete graph Kr+1 and a path Pn−r by attaching +a vertex of Kr+1 and one end-vertex of Pn−r; see an example of K(7, 12) in Figure 1. +u0 +u1 +u2 +u3 +u4 +u5 +u6 +u7 +w1 +w2 +w3 +w4 +Figure 1: The graph K(7, 12) +We first give the proof of Proposition 1.1. +Proof of Proposition 1.1: Let G = K(r, n) with V (G) = {vi | 1 ≤ i ≤ n} and E(G) = +{vivj | 1 ≤ i < j ≤ r + 1} ∪{vr+svr+s+1 | 1 ≤ s ≤ n − r − 1}. From Theorem 1.2, we have +dem(G) = dem(Gb) = dem(Kr+1) = dem(G) = r. In fact, for the above G, the path Pn−r−1 +can be replaced by Tn−r−1, where Tn−r−1 is any tree of order n − r − 1. +Proposition 1.1 shows that Corollary 1.6 is true, that is, for three integers s, t, n with +1 ≤ s ≤ t ≤ n − 1, there exists a connected graph H ⊂ G of such that dem(H) = s and +dem(G) = t. (For example, let G = K(r, n) and H = K(s, n).) This gives an answer about +Problem 2. +3 +The effect of edge deletions +The following observation is immediate. +Observation 3.1. Let G1, G2, ..., Gm be the connected components of G. Then +dem(G) = dem (G1) + · · · + dem (Gm) . +Furthermore, we suppose that the distance-edge-monitoring number of K1 is 0. +The following proposition will be used in the next sections. +Proposition 3.1. For any uv ∈ E(G) and w ∈ NG(u) ∪ NG(v), if uv /∈ EM(w), then uv is +only monitored by {u, v} in V (G). +6 + +Proof. Since w ∈ NG(u) ∪ NG(v) and uv /∈ EM(w), we have dG(w, u) = dG\uv(w, u) and +dG(w, v) = dG\uv(w, v). +For any x ∈ V (G) − NG[u] ∪ NG[v], the path from x to u must +through w1, where w1 ∈ NG(u) ∪ NG(v). Then dG(x, u) = dG(x, w1) + dG(w1, u) = dG(x, w1) + +dG\uv(w1, u) = dG\uv(x, u). Similarly, dG(x, v) = dG\uv(x, v). For any x ∈ V (G) − {u, v}, we +have uv /∈ EM(x). From Theorem 1.3, uv ∈ EM(u) and uv ∈ EM(v), and hence uv is only +monitored by {u, v} in V (G). +We first give the proof of Theorem 1.7. +Proof of Theorem 1.7: Note that dem(G) = dem (G1) + · · · + dem (Gm), where G1, . . . , Gm +are all the connected components of G. So we assume that G is connected. Let M be an +distance edge monitoring set of G and e = uv. If M is also an distance-edge-monitoring set of +G \ e, then dem(G \ e) ≤ dem(G). Otherwise, M′ = M ∪ {u, v}. It suffices to show that M′ is +an distance-edge-monitoring set of G \ e. +If G \ e has two components, say G1 and G2, then dem(G \ e) = dem (G1) + dem (G2). +Without loss of generality, assume that u ∈ V (G1) and v ∈ V (G2). +For any edge e1 = +x1y1 ∈ E (G1), if there exists a vertex w ∈ V (G1) ∩ M, and we done. Otherwise, there exists +a vertex w ∈ V (G2) ∩ M such that dG\e1 (x1, w) ̸= dG (x1, w) or dG\e1 (y1, w) ̸= dG (y1, w). +Without loss of generality, we suppose that dG\e1 (y1, w) ̸= dG (y1, w) and dG (x1y1, w) = +dG (x1, w). Since dG (y1, w) = dG (y1, x1) + dG (x1, u) + dG(u, w), dG\{e,e1} (x1, u) = dG\e1 (x1, u) +and dG\{e,e1} (y1, x1) > dG\e (y1, x1), it follows that +dG\{e,e1} (u, y1) = dG\{e,e1} (u, x1) + dG\{e,e1} (x1, y1) > dG\e (u, x1) + dG\e (x1, y1) = dG\e (u, y1) +and hence dG\{e,e1} (y1, u) ̸= dG\e1 (y1, u). So e1 is monitored by M1 ∪ {u} in graph G \ e. +This implies that dem (G1) ≤ |(M ∩ V (G1)) ∪ {u}|. Similarly, we can obtain that dem (G2) ≤ +|(M ∩ V (G2)) ∪ {v}|, and hence dem(G \ e) ≤ |M′| ≤ dem(G) + 2. +Suppose that G \ e is connected. If M is also a distance-edge-monitoring set of G \ e, then +we are done. Otherwise, there exists e1 = xy ∈ E(G \ e) such that the edge e1 is not monitored +by M in G \ e. Since M is a distance- edge-monitoring set of G, it follows that there exists a +vertex z ∈ M such that dG\e1(x, z) ̸= dG(x, z) or dG\e1(y, z) ̸= dG(y, z). In addition, e1 is not +monitored by M in G\e, and so the distance from z to x or y is not changed after removing the +edge e1 in G \ e, which means that dG\{e,e1} (y, z) = dG\e (y, z) and dG\{e,e1} (x, z) = dG\e (x, z). +If dG (e1, z) = dG(x, z), then the edge e lies on every z − y geodesic in G, otherwise there exists +z∗ ∈ M such that dG\{e,e1} (x, z∗) ̸= dG (x, z∗) and e does not appear in z∗ − y geodesic in G, +which contracts to the fact that M is not the distance-edge-monitoring set of graph G \ e. +Claim 1. If a geodesic in G from z to y traverses the edge e in the order u, v, then each geodesic +in G from z to y traverses e in the order u, v. +Proof. Assume, to the contrary, that there exists z1 − y geodesic and z2 − y geodesic. We can +assume that the two geodesics z1 − y and z2 − y are P1 = z1 . . . uv . . . y and P2 = z2 . . . vu . . . y. +7 + +The z1 −y geodesic implies that d(u, v) + d(v, y) = d(u, y), and the z2 −y geodesic implies that +d(v, u) + d(u, y) = d(v, y), and hence d(u, v) = 0, a contradiction. +From Claim 1, without loss of generality, we may assume that every geodesic in G from z to y +traverses the edge e in the order u, v. Thus, we have dG(z, y) = dG(z, v)+dG(v, y). We now show +that xy can be monitored by v in G \ e. Note that dG\e1(z, y) ̸= dG(z, y), dG\e(v, y) = dG(v, y) +and dG\e(x, y) = dG(x, y). Then dG\{e,e1} (v, y) = dG\{e,e1} (v, x) + dG\{e,e1} (x, y) = dG\e1 (v, x) + +dG\e1 (x, y) > dG (v, x) + dG (x, y) = dG\e (v, x) + dG\e (x, y) ≥ dG\e(v, y). Since dG\e(v, y) > +dG\{e,e1}(v, y), it follows that e1 can be monitored by v. Since both u and v have the chance +to be chosen as the monitoring vertices, we use them to monitor the edge xy, and therefore +M′ = M ∪{u, v} is a distance edge-monitoring-set of G\e, and thus dem(G\e) ≤ dem(G)+2, +as desired. +To show the sharpness of Theorem 1.7, we consider the following example. +Proposition 3.2. There exist connected graphs G1, G2 of order n such that dem(G1 \ e) − +dem(G1) = 2 and dem(G2) − dem(G2 \ e) = 2. +Proof. Firstly, we consider the graph G1 (n ≥ 8) with vertex set V (G1) = {vi|1 ≤ i ≤ n − 8} ∪ +{ui|1 ≤ i ≤ 8} and edge set E(G1) = {uivi | 1 ≤ i ≤ 8} ∪ {uiui+1 | 1 ≤ i ≤ 7} ∪ {uivi+1 | 1 ≤ i ≤ +7} ∪ {u1u8} ∪ {u1u5} ∪ {v1v8} ∪ {v1v9} ∪ {vivi+1 | 9 ≤ i ≤ n − 9}. Obviously, Gb(G1) = G∗ +8; see +Figure 2. Take M = {u2, u4, v3, v6, u7, v8}. Note that {u1u5, u5v5, u2v2, u2u1, u2u3} ⊆ EM(u2), +v1 +v2 +v3 +v4 +v5 +v6 +v7 +v8 +u1 u2 +u3 +u4 +u5 +u6 +u7 +u8 +v1 +v2 +v3 +v4 +v5 +v6 +v7 +v8 +u1 u2 +u3 +u4 +u5 +u6 +u7 +u8 +dem(G) = 6 +dem(G − e) = 8 +Figure 2: The graph G∗ +8 +{v1u1, u4v3, u4v5, u4v4} ⊆ EM(u4), {v3u3, v2v3, v4v3, v5v4, v2v1} ⊆ EM(v3), {v8v1, u8v8, v8u7} +⊆ EM(v8), {u7u8, u8u1, u6u7, u6u5, u7v7} ⊆ EM(u7), and {v5v6, v6v7, u6v6} ∈ EM(v6). Since +M is a distance-edge-monitoring set of G∗ +8, it follows that for uivi (2 ≤ i ≤ 8) and any +w ∈ N(ui) ∪ N(vi), we have dG\uivi(w, ui) = dG(w, ui) and dG\uivi(w, vi) = dG(w, vi), and so +uivi /∈ EM(w). From Proposition 3.1, the edge uivi (2 ≤ i ≤ 8) is monitored by {ui, vi}, and +hence dem(G∗ +8) = 6. Since G∗ +8\u1u5 ∼= C8□K2, it follows that dem(G∗ +8\u1u5) = dem(C8□K2) = +8, and hence dem(G1 \ u1u5) − dem(Gn) = dem(G∗ +8 \ u1u5) − dem(G8) = 8 − 6 = 2, as desired. +8 + +v1 +v1 +v2 +v2 +v3 +v3 +v4 +v4 +v5 +v5 +v6 +v6 +dem(G) = 4 +dem(G − e) = 2 +Figure 3: The graph G⋄ +6. +Next, we consider the graph G2 (n ≥ 6) with vertex set V (G2) = {vi | 1 ≤ i ≤ n} and +edge set E(G2) = {v1v2, v3v4, v5v6, v1v3, v1v5, v2v4, v2v6, v3v5, v4v6} ∪ {vivi+1 | 6 ≤ i ≤ n − 1}. +Obviously, Gb(G2) = G⋄ +6; see Figure 3. +Take M = {v2, v3, v4, v5}. +Note that {v1v2, v6v2, v4v2} ⊆ EM(v2), {v1v3, v5v3, v4v3} ⊆ +EM(v3), {v6v4} ⊆ EM(v4), {v5v1, v6v5} ⊆ EM(v5). Then M is a distance-edge-monitoring +set of G⋄ +6, and hence dem(G⋄ +6) ≤ 4. For v2i−1v2i (1 ≤ i ≤ 3) and w ∈ N(ui) ∪ N(vi), we +have dG\v2i−1v2i(w, v2i−1) = dG(w, v2i−1) and dG\v2i−1v2i(w, v2i) = dG(w, v2i), and so v2i−1v2i /∈ +EM(w). From Proposition 3.1, the edge v2i−1v2i (1 ≤ i ≤ 3) is monitored by {v2i−1, v2i}, and +hence dem(G⋄ +6) ≥ 3, and so M ∩ {v2i−1, v2i} ̸= ∅ (1 ≤ i ≤ 3). All the M ∈ V (G⋄ +6) is showing +Table 1. Since dem(G⋄ +6) ≥ 4, it follows that dem(G⋄ +6) = 4. +M +E(G) − ∪x∈MEM(x) +v1, v3, v6 +v2v4 +v1, v4, v5 +v2v6 +v1, v4, v6 +v3v5 +v2, v3, v5 +v4v6 +v2, v3, v6 +v1v5 +v2, v4, v5 +v1v3 +v1, v3, v5 +v2v6, v2v4, v4v6 +v2, v4, v6 +v1v3, v1v5, v3v5 +Table 1. The edges not monitored by M(|M| = 3). +For the graph G⋄ +6 \v3v4, if we take M = {v2, v5}, then {v1v2, v6v2, v4v2, v1v3} ⊆ EM(v2) and +{v5v1, v6v5, v3v5, v6v4} ⊆ EM(v5), and hence dem(G⋄ +6 \ v3v4) ≤ 2. Since G⋄ +6 \ v3v4 is not a tree, +it follows from Theorem 1.1 that dem(G⋄ +6 \ v3v4) = 2, and hence dem(G2) − dem(G2 \ v3v4) += dem(G2) − dem(G⋄ +6 \ v3v4) = 4 − 2 = 2, as desired. +Next, we give the proof of Theorem 1.8. +9 + +Proof of Theorem 1.8: Let k, i are integers with 1 ≤ i ≤ k. The graph Gi is obtained by +iteratively adding an edge uivi to a graph Gi−1. Without loss of generality, let + + + +V (G0) = {c} ∪ {uj | 1 ≤ j ≤ k} ∪ {vj | 1 ≤ j ≤ k}; +E(G0) = {cuj, cvj | 1 ≤ j ≤ k}. +and for 1 ≤ i ≤ k, the graph of Gi is defined by +V (Gi) = V (Gi−1) and E(Gi) = E(Gi−1) ∪ {uivi}. +Let G = Gi+1(3 ≤ i ≤ k). Then Gb = Fr(i + 1). Let M = {c} ∪ {ut | 1 ≤ t ≤ i}. From +Theorem 1.3, any edge e ∈ E(G) is monitored by M, and hence dem(G) ≤ i + 1. +For any w ∈ N(uj) ∪ N(vj) = {c}, we have ujvj /∈ ME(c) (1 ≤ j ≤ i + 1 ≤ k). From +Proposition 3.1, the edge ujvj is only monitored by uj or vj. +Then dem(G) ≥ i + 1, and +hence there exists a graph Gi such that |E(Gi) − E(Gi−1)| = 1, dem(Gi) ≤ dem(Gi+1) and +dem(Gi) = i. Since G0 = K1,2k is a tree, it follows from Theorem 1.1 that dem(G0) = 1. Let +Ei+1 = {utvt | 1 ≤ t ≤ i + 1 ≤ k}. Then dem(G + Ei+1) = dem(Gi+1) = i + 1, and hence +dem(G + Ei+1) − dem(G) = i, as desired. +In the end of this section, we give the proof of Corollary 1.10. +Proof of Corollary 1.10: For any tree Tn, Tn + e is a unicyclic graph. From Theorems 1.4 +and 1.1, we have dem(Tn + e1) = dem(Tn) + 1 = 2. Similarly, Tn + e1 + e2 is a tricyclic graph, +and hence dem(Tn + e1 + e2) ≤ 3. Let Tn = Pn with vertex set {v1, · · · , vn} and edge set +{vivi+1 | 1 ≤ i ≤ n − 1}. If e1 = v1v6 and e2 = v2v5, then dem(Tn + e1 + e2) = 3. Let Tn = Sn +with vertex set {v1, · · · , vn} and edge set {v1vi | 2 ≤ i ≤ n}. For any edges e1 and e2, we have +dem(Tn + e1 + e2) = 2. +4 +The effect of vertex deletion +A kipas � +Kn with n ≥ 3 is the graph on n + 1 vertices obtained from the join of K1 and Pn. +Proposition 4.1. For n ≥ 6, we have dem( � +Kn) = ⌊n/2⌋. +Proof. Let V ( � +Kn) = {v0, v1, · · · , vn} and E( � +Kn) = {v0vi | 1 ≤ i ≤ n} ∪ {vivi+1 | 1 ≤ i ≤ n − 1}. +First, we prove that | dem( � +Kn)| ≥ ⌊n/2⌋. For any vertices vi, vj ∈ V ( � +Kn), we have +d(vi, vj) = + + + +1, +if i = 0 or j = 0 or |i − j| = 1; +2, +if i· j ̸= 0 and |i − j| ≥ 2. +For any edge vivi+1 (2 ≤ i ≤ n − 2), we have NG(vi) ∪ NG(vi+1) = {vi−1, v0, vi+2}. Since +dG(vi, v0) = dG−vivi+1(vi+1, v0) = 1, dG−vivi+1(vi, vi−1) = dG(vi, vi−1) = 2 and dG−vivi+1(vi+1, vi+2) +10 + += dG(vi+1, vi+2) = 2, it follows that vivi+1 /∈ EM(vi+1)∪EM(vi−1)∪EM(v0). From Proposition +3.1, the edge e = vivi+1 is monitored by {vi, vi+1}. Similarly, for i = 1, n−1, vivi+1 is monitored +by {vi, vi+1}. Note that the vertex covering number of G is β(G). Since β(Pn) = ⌊n/2⌋, it +follows that dem( � +Kn) ≥ ⌊n/2⌋. +Next, we prove that | dem( � +Kn)| ≤ ⌊n/2⌋. Let M = {vi | 0 ≡ i (mod 2), 1 ≤ i ≤ n}. For +any edge e ∈ E(Pn) ∪ {v0vi | 0 ≡ i (mod 2), 1 ≤ i ≤ n}, it follows from Theorem 1.3 that e +is monitored by M. In addition, for any edge e = v0vi ∈ {v0vi | 1 ≡ i (mod 2), 1 ≤ i ≤ n}. +Since n ≥ 6, it follows that there exists j such that dG(vi, vj) = 2 and dG−v0vi(vi, vj) = 3, where +j = i + 3(1 ≤ i ≤ n − 3) and j = 2(n − 3 ≤ i ≤ n). So, e ∈ EM(vj). +Since any edge e ∈ E( � +Kn) can be monitored by M, it follows that dem( � +Kn) ≤ ⌊n/2⌋, and +hence dem( � +Kn) = ⌊n/2⌋. +Proof of Theorem 1.11 Note that � +K2k+2 = {v0} ∨ P2k+2, it follows from Theorem 1.1, we +have dem(P2k+2) = 1. From Lemma 4.1, we have dem(� +K2k+2) = k+1, and hence dem(� +K2k+2)− +dem(� +K2k+2 − v0) = dem(� +K2k+2) − dem(P2k+2) = k. Let G1 = � +K2k+2 and H1 = P2k+2. Then +dem(G1) − dem(H1) = dem(� +K2k+2) − dem(P2k+2) = k, where H1 is not a spanning subgraph +of G1. +Let G2k+3 be a graph with vertex set V (G2k+3) = {ui | 1 ≤ i ≤ k + 1} ∪ {vi | 0 ≤ i ≤ k + 1} +and edge set E(G2k+3) = {v0ui | 1 ≤ i ≤ k + 1} ∪ {uivi | 1 ≤ i ≤ k + 1}. Obviously, we have +G2k+3 − v0 ∼= (k + 1)K2. From Observation 3.1 and Theorem 1.2, we have dem(G2k+3 − v0) = +dem((k + 1)K2) = (k + 1) dem(K2) = k + 1. Since G2k+3 is a tree, it follows from Theorem +1.1 that dem(G2k+3) = 1, and hence dem(G2k+1 − v0) − dem(G2k+1) = k. Let G2 = G2k+1 and +H2 = (k + 1)K2. Then dem(H2) − dem(G2) = dem((k + 1)K2) − dem(G2k+1) = k, where H2 is +not a spanning subgraph of G2, as desired. +Note that G2k+3 − v0 ∼= (k + 1)K2 is disconnected. For connected graphs, we can also show +that there is a connected subgraph H such that dem(H) −dem(G) can be arbitrarily large; see +Theorem 4.3. +The conical graph C(ℓ, k) is a graph obtained by taking adjacency from a center vertex c +to the first layer of Cartesian product of Pℓ and Ck, where ℓ ≥ 1 and k ≥ 3. The conical graph +C(3, 8) can be seen in Figure 4. +For ℓ = 1, the graph C(1, k) is the wheel graph Wk, which is formed by connecting a single +vertex c to all the vertices of cycle Ck. It is clear that for the conical graph C(ℓ, k), we have +|V (C(ℓ, k))| = kℓ + 1 and |E(C(ℓ, k))| = 2kℓ. +Theorem 4.1. For k ≥ 8 and ℓ ≥ 2, we have +dem(C(ℓ, k)) = + + + +�Ak +i=1 ⌈k/(4i − 2)⌉, +if ℓ ≤ Ak; +�Ak +i=1 ⌈k/(4i − 2)⌉ + 2(ℓ − Ak), +if ℓ ≥ Ak + 1, +where Ak = ⌊k/4 + (1 + (−1)k+1)/8⌋. +11 + +c +u1 +1 +u1 +2 +u1 +3 +u1 +4 +u1 +5 +u1 +6 +u1 +7 +u1 +8 +u2 +1 +u2 +2 +u2 +3 +u2 +4 +u2 +5 +u2 +6 +u2 +7 +u2 +8 +u3 +1 +u3 +2 +u3 +3 +u3 +4 +u3 +5 +u3 +6 +u3 +7 +u3 +8 +Figure 4: The conical graph C(3, 8) +Proof. Let V (C(ℓ, k)) = {c} ∪ {uj +i | 1 ≤ i ≤ k, 1 ≤ j ≤ ℓ} and E(C(ℓ, k)) = (∪ℓ +i=1E(Ci)) ∪ +(∪k +i=1E(Pi)), where E(Ci) = {ui +kui +1} ∪ {ui +jui +j+1 | 1 ≤ j ≤ k − 1} (1 ≤ i ≤ ℓ), E(Pi) = {cu1 +i } ∪ +{uj +iuj+1 +i +| 1 ≤ j ≤ ℓ − 1} (1 ≤ i ≤ k). +Fact 1. For any vertex v ∈ V (Ci), we have +|EM(v) ∩ E(Ci)| = + + + + + + + + + +4i − 2, +if 1 ≤ i ≤ ⌊k/4 + (1 + (−1)k+1)/8⌋; +k − 2, +if k is even and i ≥ ⌈k/4 + (1 + (−1)k+1)/8⌉; +k − 1, +if k is odd and i ≥ ⌈k/4 + (1 + (−1)k+1)/8⌉. +Furthermore, we have |EM(c) ∩ E(Ci)| = 0. +Proof. For any vertex ui +s, ui +t ∈ V (Ci)(1 ≤ s, t ≤ k and i ≤ ⌊k/4 + (1 + (−1)k+1)/8⌋), since +ui +s . . . , u1 +scu1 +t . . . ui−1 +t +ui +t is the path from ui +s to ui +t, and hence dCi(ui +s, ui +t) ≤ 2i. +So we have +{ui +tui +t+1 | s − 2i + 1 ≤ t ≤ s + 2i − 2} ∈ EM(ui +s), where the subscripts are taken mod- +ulo n, that is, ui +k+1 = ui +1. +For any ui +jui +j+1 ∈ E(Ci)− {ui +tui +t+1 | s − 2i + 1 ≤ t ≤ s + 2i − +2} − {ui +s−2iui +s−2i−1, ui +s+2i+1ui +s+2i}. Since d(ui +s, ui +j) = d(ui +s, ui +j+1) = 2i, and dG\ui +jui +j+1(ui +s, ui +j) = +dG\ui +jui +j+1(ui +s, ui +j+1) = 2i, it follows that ui +jui +j+1 /∈ EM(ui). In addition, let e1 = ui +s−2iui +s−2i−1 +and e2 = ui +s+2i+1ui +s+2i. +Then dG\e1(ui +s, ui +s−2i) = dG(ui +s, ui +s−2i), and hence dG\e2(ui +s, ui +s+2i) = +dG(ui +s, ui +s+2i), and so e1, e2 /∈ EM(ui). +Similarly, for any edge e = uj +muj +m+1 ∈ E(Cj)(j ̸= i and 1 ≤ m ≤ k), since dG\e(ui +s, uj +m) += dG(ui +s, uj +m) and dG\e(ui +s, uj +m+1) = dG(ui +s, uj +m+1), it follows that e /∈ EM(ui). For any vertex +ui +s ∈ V (Ci), we have |EM(ui +s)∩E(Ci)| = 4i−2. Note that if n is even number, then 4i−2 ≤ k−2. +and hence, i ≤ ⌈k/4⌉. If n is odd, then 4i − 2 ≤ k − 1, and hence i ≤ ⌈(k + 1)/4⌉. So we have +i ≤ ⌊k/4 + (1 + (−1)k+1)/8⌋. +For i ≥ ⌈k/4+(1+(−1)k)/8⌉ and 1 ≤ s, t ≤ k, if ui +s, ui +t ∈ V (Ci), then dG(ui +s, ui +t) = dCi(ui +s, ui +t). +For any cycle Ci with length k, if k is even, then ui +s+k/2ui +s+k/2−1, ui +s+k/2ui +s+k/2+1 /∈ EM(ui +s), +and so |EM(v) ∩ E(Ci)| = k − 2. If k is odd, then ui +s+(k−1)/2ui +s+(k+1)/2 /∈ EM(ui +s), and so +|EM(v) ∩ E(Ci)| = k − 1, as desired. +12 + +For any edge e3 = ui +sui +(s+1) ∈ E(Ci), we have dG(c, ui +s) = dG(c, ui +(s+1)) = i, and hence +dG\e3(c, ui +s) = dG\e3(c, ui +(s+1)) = i. Since e3 /∈ E(c), it follows that |EM(c) ∩ E(Ci)| = 0, and +hence Fact 1 holds. +Firstly, note that Ak = ⌊k/4 + (1 + (−1)k+1)/8⌋, we consider ℓ ≥ Ak + 1. From Fact 1, we +have dem(G) ≥ �Ak +i=1 ⌈k/(4i − 2)⌉ +2(ℓ − Ak). +Take M = ∪i=k +i=1Mi, where +Mi = + + + +{ui +j|1 ≤ j ≤ k, 1 ≡ j +(mod (4i − 2))}, +if i ≤ Ak. +{ui +1, ui +⌈k/2⌉}, +ifAk + 1 ≤ i ≤ ℓ. +From the selection method of M, for any edge e ∈ E(Ci)(1 ≤ i ≤ ℓ), e is monitored by M. It +suffices to prove that each edge in E(Pi) is monitored by M. For vertex u1 +i , u1 +j ∈ V (G)(1 ≤ +i, j ≤ k), if j /∈ {i + 1, i + 2, i − 1, i − 2}, where the subscripts are taken modulo n, that is, +u1 +−1 = u1 +k and u1 +1 = u1 +k+1. then dG(u1 +i , u1 +j) ̸= dG−cu1 +j(u1 +i , u1 +j), and hence cu1 +j ∈ EM(u1 +i ). Similarly, +for 1 ≤ t ≤ ℓ − 1, since dG(u1 +i , ut +j) ̸= dG−ut +jut−1 +j +(u1 +i , ut +j), it follows that ut +jut−1 +j +∈ EM(u1 +i ), +and hence EM(u1 +i ) = {u1 +i u1 +i+1, u1 +iu1 +i−1} ∪ {cu1} ∪ {up−1 +j +up +j|2 ≤ p ≤ ℓ − 1, 1 ≤ j ≤ n, j /∈ +{i−2, i−1, i+1, i+2}}. Without loss of generality, take u1 +1 ∈ M, then, for any edge e ∈ E(Pi) +(j /∈ 2, 3, k, k − 1), it is monitored by u1 +1. For any edge e ∈ E(Pi) (j ∈ {2, 3, k, k − 1}), if k is +even, then k − 3 − 2 ≥ 3, since k ≥ 9. Obviously, u1 +3, u1 +k−1, u1 +k−3 ∈ M. If k is odd, then it is +monitored by vertex set {u1 +3, u1 +k}, which is a subset of M. For any e ∈ E(Pi) (1 ≤ i ≤ k), it is +monitored by Mi. +Hence, we have dem(G) ≤ �⌊(k+2)/4⌋+1 +i=1 +⌈k/(4i − 2)⌉ + 2(ℓ − Ak). For ℓ ≤ Ak, it is similar +to the case that ℓ ≥ Ak + 1, as desired. +Li et al. obtained the following result about distance-edge-monitoring numbers of Cn□Pm. +Theorem 4.2. [12] Let ℓ and k be two integers with ℓ ≥ 3 and k ≥ 2. Then +dem (Ck□Pℓ) = + + + +k +if k ≥ 2ℓ + 1, +2ℓ +if k < 2ℓ + 1 +Theorem 4.3. For any positive integer k ≥ 8, there exists a connected graph subgraph of G +such that such that +dem(G − v) − dem(G) = ⌊k/2⌋ − ⌈k/6⌉, +where v ∈ V (G). +Proof. Let G = C(ℓ, k), where ℓ = 2 and k ≥ 2ℓ+1. Note that G−v0 = Ck□K2. From Theorem +4.2, we have dem(G − v0) = dem(Ck□K2) = k. From Theorem 4.1, we have dem(C(2, k)) = +�2 +i=1 ⌈k/(4i − 2)⌉ = ⌈k/2⌉ − ⌈k/6⌉, and hence dem(G − v) − dem(G) = k − ⌈k/2⌉ − ⌈k/6⌉ = +⌊k/2⌋ − ⌈k/6⌉, as desired. +13 + +Let G = E(C(ℓ, k)) and H = Ck□Pℓ. From Theorems 4.1 and 4.2, if ℓ ≫ k, then dem(G) +dem(H) ≈ 1. +From Theorems 4.3 and 4.2, if k ≫ ℓ = 2, then dem(G) +dem(H) ≈ 0.3. +Corollary 4.4. There exist two graphs H and G such that +dem(G) +dem(H) ≈ 0.3, +where H is an induced subgraph of G. +Proof of Theorem 1.12: Let G = � +K2k+2. +From Proposition 1.11, there exists a vertex +u ∈ V (G) such that dem(� +K2k+2) − dem(� +K2k+2 − u) = k. Note that G − u = P2k+2 is connected +graph. In addition, let H = C(2, t), k = ⌊t/2⌋ − ⌈t/6⌉ and v ∈ V (H). From Theorem 4.3, we +have dem(C(2, t) − v) − dem(C(2, t)) = k and C(2, t) − v = Ct□K2 is a connected graph. +In fact, G − v is subgraph of G. From Theorem 1.12, for any positive integer k ≥ 3, there +exists a graph G such that dem(G − v) − dem(G) ≥ k ≥ 0. +Let G = C(2, t)(t ≥ 8) and H1 = C(2, t) − v. From Theorem 1.12, we have dem(H1) ≥ +dem(G). Note that G is not tree, it follows from Theorem 1.1 that dem(G) ≥ 2. Let H2 is +a subtree of G. From Theorem 1.1, we have dem(H2) = 1 ≤ 2 ≤ dem(G), and so we have +Corollary 4.5. +Corollary 4.5. Let G be a connected graph. Then there exist two non-spanning subgraphs +H1, H2 ⊂ G such that dem(H1) ≥ dem(G) and dem(H2) ≤ dem(G). +Proof of Proposition 1.2: For any graph G with order n, and G−v contain at least one edge, +we have dem(G) ≤ n−1 and 1 ≤ dem(G−v). Obviously, we obtained dem(G)−dem(G−v) ≤ +n − 2. Furthermore, let G = K3, then dem(G) − dem(G − v) = n − 2. So the upper bound +is sharp. Conversely, since dem(G) − dem(G − v) = n − 2, it follows that dem(G) = n − 1 +and dem(G − v) = 1. From Theorem 1.1, G − v must be a tree. If |V (G − v)| ≥ 3, and note +that dem(G) = n − 1, it follows from Theorem 1.1 that G = Kn (n ≥ 4), which is impossi- +ble. Then |V (G − v)| ≤ 2, and hence G = Kn (n ≤ 3). If G = K2, then G − v = K1, which +contradicts to the fact that G−v contains at least one edge. So, we have G = K3, as desired. +Next, we consider the subgraph H of G. If H ⊂ G such that dem(H) ≤ dem(G), then, what +is the relationship between H and G? A natural question is, what is the maximum number of +edges we can delete from G without changing number of distance-edge monitoring? We give a +partial answer as follows. +The base graph Gb of a graph G is the graph obtained from G by iteratively removing +vertices of degree 1. We know that Gb is subgraph of G, but dem(G) = dem(Gb). So we can +obtain below result, which give a lower bound for E such that dem(G) = dem(G − E), so we +have Observation 4.1. +14 + +Observation 4.1. For any graph G, let E1 = E(G) − E(Gb). If dem(G) = dem(G − E) and +G−E is connect graph with order at least 2, then |E| ≥ |E1|. Furthermore, the bound is sharp. +Proof of Theorem 1.13: Since dem(G) = 2, without loss of generality, let M = {u, v} be a +distance-edge-monitoring set of G. +If uv ∈ E(G), then it follows from Theorem 1.4 that |E(G)| ≤ 2(n − 1) − 1, and hence +dem(G) = dem(G − E) = 2. From Theorem 1.1, G − E must contain an cycle, and hence +|E| ≤ 2(n − 1) − 1 − 3 = 2n − 6. +If uv ̸∈ E(G), then M = {u, v} is a distance-edge-monitoring set of G. +If |EM(u) ∩ +EM(u)| ≥ 1, which is similar to the case u, v is adjacent to in G. +So, we suppose that +EM(u) ∩ EM(u) = ∅. From Theorem 1.4, |E(G)| ≤ 2(n − 1). From Theorem 1.1, G must +contain a cycle and |E| ≤ 2(n − 1) − 3 = 2n − 5. Furthermore, we have the following claim. +Claim 2. |E| ≤ 2n − 6. +Proof. Assume, to the contrary, that |E| = 2n−5. Then G−E must be an cycle of length 3, say +G −E = K3. Without loss of generality, let V (G −E) = {v1, v2, v3}. In addition, the subgraph +induced by the edge set EM(u) and EM(v) are spanning trees in G. If u, v ∈ {v1, v2, v3}, then +uv ∈ E(G), a contradiction. For any vertex w ∈ V (G) − {v1, v2, v3}, if dG(w) = 1, then w +is in one of spanning tree of EM(u) or EM(v), a contradiction. Hence, we have dG(w) ≥ 2. +Especially for two vertices u and v, there exists a vertex u1 ∈ N(u) such that uu1 ∈ EM(v). +From Theorem 1.3, we have uu1 ∈ EM(u), and hence |EM(u)∩EM(u)| ̸= 0, which contradicts +to the fact that EM(u) ∩ EM(u) = ∅. +From Claim 2, we have |E| ≤ 2n − 6. Furthermore, let G = (n − 2)K1 ∨ K2 with ver- +tex set V (G) = {vi | 1 ≤ i ≤ n} and edge set E(G) = {v1v2} ∪ {v1vi, v2vi|3 ≤ i ≤ n}. Let +E = {v1vi, v2vi | 4 ≤ i ≤ n} ⊂ E(G). From Observation 3.1, we have dem(G − E) = dem(K3) ++(n − 1) dem(K1) = 2, and so there exists an edge set E such that dem(G − E) = 2 and +|E| = 2n − 6, as desired. +For H ⊂ G, the distance-edge-monitoring set of subgraph H in G is a set M of vertices of +a graph H such that E(H) ⊆ � +x∈M +EM(x). +Definition 5. The restrict-distance-edge-monitoring number dem(G|H) of a graph G|H is de- +fined as the smallest size of a distance-edge-monitoring set of H in G, that is, +dem(G|H) = min +� +|M| +���E(H) ⊆ +� +x∈M +EM(x), M ⊆ V (H) +� +. +In Figure 5, let G = K4 with V (G) = {v0, v1, v2, v3} and E(G) = {vivj | 0 ≤ i < j ≤ 3}. Let +E(T1) = {v0v1, v0v2, v0v3} and E(T2) = {v0v3, v3v1, v1v2}. Obviously, we have dem(K4|Ti) = i +15 + +v0 +v1 +v2 +v3 +v2 +v1 +v0 +v3 +T1 +T2 +Figure 5: T1 and T2 +(i = 1, 2). The distance-edge-monitoring set of subgraph Ti (i = 1, 2) in K4 is shown in Figure 5, +where the blue vertices form the set M. +In the end of this section, we give the proof of Theorem 1.14 as follows. +Proof of Theorem 1.14: Let G = Kn with vertex set {vi | 1 ≤ i ≤ n} and edge set {vivj | 1 ≤ +i < j ≤ n}. Let T be a spanning tree in Kn. For any edge uv ∈ E(T) and vertex w ∈ +(NG(u) ∪ NG(v)) ∩ V (T), we have dG(w, u) = dG(w, v) = 1 and dG\uv(w, u) = dG\uv(w, v) = 1, +and hence uv /∈ EM(w). From Proposition 3.1, any edge uv ∈ E(T) is only monitored by u or +v, and hence dem(Kn|T) ≥ β(T). From Theorem 1.5, we have dem(Kn|T) ≤ β(T), and hence +dem(Kn|T) = β(T). Since T is tree with order n, it follows that there is no odd cycles, and +hence T is a bipartite graph. Without loss of generality, let V (T) = U ∪ V (|U| ≤ |V |), which +is a bipartite partition of V (T). From the pigeonhole principle, we have |U| ≤ ⌊n +2⌋. For any +uv ∈ E(T), we have {u, v} ∩ U ̸= ∅, and hence β(T) ≤ ⌊n +2⌋. In addition, T contains at least +one edge, and hence 1 ≤ β(T), and so 1 ≤ β(T) ≤ ⌊n +2⌋. +Suppose that T = Sn with edge set E(Sn) = {v1vi | 2 ≤ i ≤ n}. Without loss of generality, +let dT(v1) = n − 1. Then {v1} is the vertex cover set of Sn, and hence β(T) = 1, and so +the lower bound is sharp. Suppose that T = Pn with edge set E(Pn) = {vivi+1 | 1 ≤ i ≤ n}. +Obviously, we have {vi | i ≖ 0 (mod 2), and 1 ≤ i ≤ n, vi ∈ V (Pn)} is vertex cover set of Pn, +and hence β(T) = ⌊n +2⌋, and hence the upper bound is sharp. +Similar to Theorem 1.14, we have below Corollary 4.6. +Corollary 4.6. Let H be a subgraph with order p of G. Then we have +1 ≤ dem(G|H) ≤ p − 1. +Furthermore, the left (right) equality hold if and only if H = Sp−1 (H = Kp−1, respectively). +Theorem 4.7. If H is an induced subgraph of graph G, then we have +dem(G) − dem(G|H) ≤ |V (G)| − |V (H)|. +16 + +Moreover, If both G and H are complete graph, then the bound is sharp. +Proof. For any graph G and H, where H is induced subgraph of G. Let M1 ⊂ V (H) is the +smallest size of a distance-edge-monitoring set of H in G, without loss of generality, suppose +that the restrict distance-edge-monitoring number is r, that is, dem(G|H) = r. +Let M = +(V (G) − V (H)) ∪ M1. We will prove that dem(G) ≤ |M|. For any edge e = uv ∈ E(G), +if u or v in V (G) − V (H), then it follows from Theorem 1.3 that e is monitored by V (G) − +V (H). For any edge e ∈ E(H), since M1 is the a restrict distance-edge-monitoring set, it +follows that e is monitored by M1, and hence M is a distance-edge-monitoring set in G. Since +|M| = |M1| + (|V (G)| − |V (H)|) = |V (G)| − |V (H)| + dem(G|H), it follows that dem(G) ≤ +|M| = dem(G|H) + (|V (G)| − |V (H)|), and so dem(G) − dem(G|H) ≤ (|V (G)| − |V (H)|). +Furthermore, we take G = Kn and H = Km (3 ≤ m ≤ n). From Theorem 1.2, we have +dem(G) = n − 1. For any uv ∈ E(H) and w ∈ N(u) ∪ N(v), we have dG(w, u) = dG(w, v) = 1 +and dG\uv(w, u) = dG\uv(w, v) = 1, and hence uv /∈ EM(w). From Proposition 3.1, any edge +uv ∈ E(H) is only monitored by u or v, and hence dem(Kn|Km) ≥ β(Km) = m − 1. From +Theorem 1.5, we have dem(Kn|Km) ≤ β(Km) = m − 1, and hence dem(Kn|Km) = m − 1. +Obviously, dem(G) − dem(G|H) = (n − 1) − (m − 1) = |V (G)| − |V (H)|, as desired. +5 +Complexity issues +Foucaud et al. proved in [10] that the problem DISTANCE-EDGE-MONITORING SET is +NP-complete. For any graph G, a natural question is whether the original detection set can +detect all edges if some edges or vertices in G are broken. We design the algorithm and then +give the results on the algorithm analysis. +DISTANCE-EDGE-MONITORING SET +Instance: A graph G = (V, E), an edge e ∈ E(G) and a distance-edge-monitoring set M. +Question: Whether M is still a distance-edge-monitoring set for the graph G − e? +Given a graph G, a distance-edge-monitoring set M, and an edge e ∈ E(G), our goal is +to determine whether the original distance-edge-monitoring set M is still valid in the resulting +graph G \ e. The algorithm is shown in Algorithm 1. +The algorithm of how to compute the edge set EM(x) from G was given by Yang et +al. in [15], and is based on the breadth-first spanning tree algorithm. The time complexity is +polynomial. +5.1 +Deleting one edge or vertex from some known graphs +In this section, we study the change for distance-edge-monitoring numbers of some well-known +graphs. +17 + +Algorithm 1 The algorithm for determining M is or not monitor set for G \ e +Input: A graph G, M ⊂ V (G) and e ∈ E(G); +Output: E(G \ e) ⊂ ∪x∈MEM(x) is TRUE or FALSE; +1: M1 ← E(G \ e) +2: for each vertex v ∈ M do +3: +M1 ← M1 − EM(v) +4: if M1 = ∅ then return E(G \ e) ⊂ ∪x∈MEM(x) is TRUE; +5: else return E(G \ e) ⊂ ∪x∈MEM(x) is FALSE; +We denote NVi(G) = {v | dG(v) = i, v ∈ V (G)} and NEa,b(G) = {uv | uv ∈ E(G), dG(u) = +a, dG(v) = b}. +Note that in Pn, we have E(Pn) = NE1,2(Pn) ∪ NE2,2(Pn). If e ∈ NE1,2(Pn), then we have +dem (Pn \ e) = 1, and hence dem (Pn)−dem (Pn \ e) = 0. If e ∈ NE2,2(Pn), then dem (Pn \ e) = +2, and hence dem (Pn) − dem (Pn \ e) = −1. Therefore these arguments imply the following +Corollary 5.1, which imply that it is possible to find classes of graphs G and edge e such that +dem(G \ e) ≤ dem(G) + 1. +Corollary 5.1. Let n be an integer and Pn be a path with order n. For any e ∈ E(Pn), we +have +dem(Pn \ e) = + + + +dem(Pn), +if e ∈ NE1,2(Pn); +dem(Pn) + 1, +if e ∈ NE2,2(Pn). +The following corollary is immediate, which imply that there exists graphs G and edge e +such that dem(G) > dem(G \ e) and dem(G) = dem(G \ e). +Corollary 5.2. Let n ≥ 3 be an integer and for any edge e ∈ E(G), we have +(i) dem(Cn \ e) = dem(Cn) − 1 = 1. +(ii) dem(Kn \ e) = dem(Kn) − 1 = n − 2. +(iii) dem(Kn,n \ e) = dem(Kn,n) = n. +Proof. (i) From Theorem 1.1, we have dem(Cn \ e) = dem(Pn) = 1. Since dem(Cn) = 2, it +follows that dem(Cn \ e) = dem(Cn) − 1 = 1. +(ii) From Theorem 1.2, we have dem(Kn) = n − 1. +For e = uv ∈ E(Kn \ e), if we +take M = V (Kn) − {u, v}, then one end-vertex of e belongs to M for any edge e. +From +Theorem 1.3, we have dem(G \ uv) ≤ n − 2. +For any edge xy ∈ E(G \ uv) and vertex +w ∈ (NG(x) ∪ NG(y)) ∩ V (G − {u}), we have dG(w, x) = dG(w, y) = 1 and dG\xy(w, x) = +dG\xy(w, y) = 1, and hence xy /∈ EM(w). From Proposition 3.1, any edge xy ∈ E(G − {u}) is +only monitored by x or y, and hence dem(G \ uv) ≥ n − 2, and so dem(G \ uv) = n − 2. This +implies that dem(Kn) − dem(Kn \ e) = 1. +(iii) Let G = Kn,n with vertices set V (Kn,n) = {ui | 1 ≤ i ≤ n} ∪{vi | 1 ≤ i ≤ n} and edge +set E(Kn,n) = {uivj | 1 ≤ i, j ≤ n}. Without loss of generality, let e = u1v1, For any vertex +18 + +set M ⊂ V (G) with |M| = n − 1, let U1 = M ∩ {ui | 1 ≤ i ≤ n}, V1 = M ∩ {vi | 1 ≤ i ≤ n}, +and |U1| + |V1| = n − 1. +Note that n ≥ 3. +If |U1| = n − 1 and |V1| = 0, then for any +w ∈ M, there exist ut /∈ M and v2 ∈ V2 such that dG(ut, w) = dG\utv2(ut, w) = 2 and +dG\e(v2, w) = dG\utv2(v2, w) = 1. So utv2 is not monitored by M. If |U1| ≤ n−2 and |V1| ≤ n−2, +then there exist p, q ∈ {1, . . . , n} such that upvq ∈ E(Kn,n), upvq ̸= u1v1 and up, vq /∈ M. If +w ∈ U1, then dG(up, w) = dG\upvq(up, w) = 2 and dG(vq, w) = dG\upvq(vq, w) = 1. If w ∈ V1, then +dG(up, w) = dG\upvq(up, w) = 1 and dG(vq, w) = dG\upvq(vq, w) = 2. So upvq is not monitored +by M. Therefore, M is not an distance-edge-monitoring set, and hence dem(G) ≥ n. From +Theorem 1.5, we have dem(G) ≤ β(G) = n, and hence dem(G) = n. +It is possible to find that there exists a graph G and v ∈ V (G) such that dem(G) > +dem(G − v), dem(G) = dem(G − v), dem(G) < dem(G − v). +The following corollary shows that there exists a graph G such that dem(G) > dem(G − v). +Corollary 5.3. Let n ≥ 3 be an integer. For any v ∈ v(G), we have +(i) dem(Cn − v) = dem(Cn) − 1 = 1; +(ii) dem(Kn − v) = dem(Kn) − 1 = n − 2; +(iii) dem(Kn,n − v) = dem(Kn,n) − 1 = n − 1. +Proposition 5.1. Let n ≥ 5 be an integer. For any v ∈ V (Pn), we have +dem(Pn − v) = + + + +dem(Pn), +if v ∈ NV1(Pn) ∪ {v2, vn−1}; +dem(Pn) + 1, +if v ∈ NV2(Pn) \ {v2, vn−1}, +where Pn = v1v2 · · · vn−1vn. +Proof. Note that V (Pn) = NV1(Pn) ∪ NV2(Pn). If v ∈ NV1(Pn), then dem (Pn − v) = 1. If +v ∈ {v2, vn−1}, then dem (Pn − v) = dem (Pn−1) + dem (K1) = 1, where v(K1) = {v1} or +v(K1) = {vn}, and hence dem (Pn) − dem (Pn − v) = 0. If v ∈ NV2(Pn) \ {v2, vn−1}, then +dem (Pn − v) = 2, and hence dem (Pn) − dem (Pn − v) = −1. +Proposition 5.1 shows that there exists a graph G and v ∈ V (G) such that dem(G) = +dem(G − v) or dem(G) < dem(G − v). +6 +Conclusion +In this paper, we have studied the effect of deleting edges and vertices in a graph G on the +distance-edge-monitoring number. We obtained that for any graph G and edge e ∈ E(G), we +have dem(G − e) − dem(G) ≤ 2. Furthermore, the bound is sharp. In addition, we can find a +graph H and v ∈ V (H) such that dem(H − v) − dem(H) can be arbitrarily large. This fact +gives an answer to the monotonicity of the distance-edge-monitoring number. This means that +there exist two graphs H and G such that H ⊂ G and dem(H) ≥ dem(G). +It is interesting to consider the following problems for future work. +19 + +(1) It is interesting to characterize the graphs dem(H) = dem(G) if H ⊂ G. +(2) For a graph G and E ⊂ E(G), what is the maximum value of |E| such that dem(G) = +dem(G − E)? +(3) For any ǫ > 0, whether the ratio dem(G) +dem(H) ≤ ǫ holds, where H is an induced subgraph +of G. +For further future work, it would be interesting to study distance-edge monitoring sets in +further standard graph classes, including pyramids, Sierpi´nki-type graphs, circulant graphs, +graph products, or line graphs. In addition, characterizing the graphs with dem(G) = n − 2 +would be of interest, as well as clarifying further the relation of the parameter dem(G) to +other standard graph parameters, such as arboricity, vertex cover number and feedback edge +set number. +References +[1] E. Bampas, D. Bil`o, G. Drovandi, L. Gual`a, R. Klasing, G. Proietti, Network verification +via routing table queries, J. Comput. System Sci. 81(1) (2015), 234–248. +[2] J. Baste, F. Beggas, H. Kheddouci, I. Sau, On the parameterized complexity of the edge +monitoring problem, Inform. Process. Lett. 121 (2017), 39–44. +[3] Z. Beerliova, F. Eberhard, T. Erlebach, A. Hall, M. Hoffmann, M. Mihal´ak, L.S. Ram, +Network discovery and verification, IEEE J. Sel. Areas Commun. 24(12) (2006), 2168– +2181. +[4] D. Bil`o, T. Erlebach, M. Mihal´ak, P. Widmayer, Discovery of network properties with +all-shortest-paths queries, Theoret. Comput. Sci. 411(14-15) (2010), 1626–1637. +[5] G. Chartrand, L. Lesniak, P. Zhang, Graphs & digraphs, London: Chapman & Hall, 2015. +[6] G. Chartrand, P. Zhang. The theory and applications of resolvability in graphs–A Survey, +Congr. Numer. 160 (2003), 47–68. +[7] L. Dall’Asta, J. Alvarez-Hamelin, A. Barrat, A. V´azquez, A. Vespignani, Exploring net- +works with traceroute-like probes: Theory and simulations, Theoret. Comput. Sci. 355(1) +(2006), 6–24. +[8] S. Delen, F.E. Zihni, F.O. Erdogan, H.O. Ayna, I.N. Cangul, The effect of vertex and +edge deletion on the independence number of graphs, F.E.J. of Appl. Math. 11(2) (2022), +11–19. +20 + +[9] L. Eroh, P. Feit, C. Kang, E. Yi, The effect of vertex or edge deletion on the metric +dimension of graphs, J. Combin. Optim. 6(4) (2015), 433–444. +[10] F. Foucaud, S. Kao, R. Klasing, M. Miller, J. Ryan, Monitoring the edges of a graph using +distances, Discrete Appl. Math. 319 (2022), 424–438. +[11] R. Govindan, H. Tangmunarunkit, Heuristics for Internet map discovery, in: Proceedings +of the 19th IEEE International Conference on Computer Communications, INFOCOM’00, +2000, pp. 1371–1380. +[12] W. Li, R. Klasing, Y. Mao, B. Ning, Monitoring the edges of product networks using +distances, arXiv:2211.10743, 2022. +[13] S.D. Monson, The effects of vertex deletion and edge deletion on the clique partition +number, Ars Comb. (1996), 42. +[14] M. Wei, J. Yue, L. Chen, The effect of vertex and edge deletion on the edge metric +dimension of graphs. J. Comb. Optim. 44 (2022), 331–342. +[15] C. Yang, R. Klasing, Y. Mao, X. Deng, On the distance-edge-monitoring numbers, +arXiv.2211.04920, 2022. +[16] H.X. Ye, C. Yang, J.M. Xu, Diameter vulnerability of graphs by edge deletion, Discrete +Math. 309 (2009), 1001–1006. +21 + diff --git a/ZtE0T4oBgHgl3EQfnAG2/content/tmp_files/load_file.txt b/ZtE0T4oBgHgl3EQfnAG2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..528287f984b00b690688e1047d8fe65432ee1659 --- /dev/null +++ b/ZtE0T4oBgHgl3EQfnAG2/content/tmp_files/load_file.txt @@ -0,0 +1,816 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf,len=815 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='02507v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='DM] 6 Jan 2023 Perturbation results for distance-edge-monitoring numbers∗ Chenxu Yang †, Ralf Klasing ‡, Changxiang He §, Yaping Mao ¶ Abstract Foucaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G = (V, E) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' A set of vertices M ⊆ V (G) is a distance-edge-monitoring set of G if any edges in G can be monitored by a vertex in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The distance-edge-monitoring number dem(G) is the minimum cardinality of a distance-edge-monitoring set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In this paper, we first show that dem(G \\ e) − dem(G) ≤ 2 for any graph G and edge e ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Moreover, the bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Next, we construct two graphs G and H to show that dem(G) − dem(G − u) and dem(H − v) − dem(H) can be arbitrarily large, where u ∈ V (G) and v ∈ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We also study the relationship between dem(H) and dem(G) for H ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In the end, we give an algorithm to judge whether the distance-edge-monitoring set still remain in the resulting graph when any edge of a graph G is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Keywords: Distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Perturbation results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Distance-edge-monitoring set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' AMS subject classification 2020: 05C12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 11J83;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 35A30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 51K05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 1 Introduction In 2022, Foucaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] introduced a new graph-theoretic concept called distance-edge- monitoring set, which means network monitoring using distance probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Networks are naturally modeled by finite undirected simple connected graphs, whose vertices represent computers and whose edges represent connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' When a connection (an edge) fails in the network, we can detect this failure, and thus achieve the purpose of monitoring the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' ∗Supported by the National Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 12061059), the Qinghai Key Laboratory of Internet of Things Project (2017-ZJ-Y21) †School of Computer, Qinghai Normal University, Xining, Qinghai 810008, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' cxuyang@aliyun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='com ‡Corresponding author: Universit´e de Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR 5800, Talence, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' ralf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='klasing@labri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='fr §College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' changxianghe@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='com ¶Academy of Plateau Science and Sustainability, Xining, Qinghai 810008, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' maoyaping@ymail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='com 1 Probes are made up of vertices we choose in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' At any given moment, a probe of the network can measure its graph distance to any other vertex of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Whenever an edge of the network fails, one of the measured distances changes, so the probes are able to detect the failure of any edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Probes that measure distances in graphs are present in real-life networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' They are e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' useful in the fundamental task of routing [7, 11] and are also frequently used for problems concerning network verification [1, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In a network, we can put as few detectors as possible to monitor all the edges, a natural question is whether the detectors placed in the original graph are still sufficient and need to be supplemented or reduced when some nodes or edges in the original graph are subjected to external interference and damage, we refer to [8, 9, 13, 14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' This kind of problem is usually called perturbation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Graphs considered are finite, undirected and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G = (V, E) be a graph with vertex set V and edge set E, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The neighborhood set of a vertex v ∈ V (G) is NG(v) = {u ∈ V (G) | uv ∈ E(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The degree of a vertex v in G is denoted by d(v) = |NG(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let δ(G) and ∆(G) be the minimum and maximum degree of a graph G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any subset X of V (G), let G[X] denote the subgraph induced by X, and E[X] the edge set of G[X];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' similarly, for any subset F of E(G), let G[F] denote the subgraph induced by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We use G − X to denote the subgraph of G obtained by removing all the vertices of X together with the edges incident with them from G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' similarly, we use G \\ F to denote the subgraph of G obtained by removing all the edges of F from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If X = {v} and F = {e}, we simply write G − v and G \\ e for G − {v} and G \\ {e}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In a graph G, the base graph Gb of a graph G is the graph obtained from G by iteratively removing vertices of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Clearly, dem(G) = dem(Gb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' A x-y path with length dG(x, y) in G is a x-y geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 Distance-edge-monitoring sets and numbers Foucaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] introduced a new graph-theoretic concept called distance-edge-monitoring sets, which is relevant to network monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For a set M of vertices and an edge e of a graph G, let P(M, e) be the set of pairs (x, y) with a vertex x of M and a vertex y of V (G) such that dG(x, y) ̸= dG\\e(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In other words, e belongs to all shortest paths between x and y in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For a vertex x, let EM(x) be the set of edges e such that there exists a vertex v in G with (x, v) ∈ P({x}, e), that is EM(x) = {e | e ∈ E(G) and ∃v ∈ V (G) such that dG(x, v) ̸= dG\\e(x, v)} or EM(x) = {e | e ∈ E(G)and P({x}, e) ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If e ∈ EM(x), we say that e is monitored by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Finding a particular vertex set M and placing a detector on that set to monitor all edge sets in G, has practical applications in sensor and network systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 2 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' A set M of vertices of a graph G is distance-edge-monitoring set if every edge e of G is monitored by some vertex of M, that is, the set P(M, e) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Equivalently, � x∈M EM(x) = E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' One may wonder to know the existence of such an edge detection set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The answer is affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If we take M = V (G), then E(G) ⊂ � x∈V (G) N(x) ⊂ � x∈V (G) EM(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Therefore, we consider the smallest cardinal of M and give the following parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The distance-edge-monitoring number dem(G) of a graph G is defined as the smallest size of a distance-edge-monitoring set of G, that is dem(G) = min � |M| ��� � x∈M EM(x) = E(G) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The vertices of M represent distance probes in a network modeled by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Distance-edge- monitoring sets are very effective in network fault tolerance testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For example, a distance- edge-monitoring set can detect a failing edge, and it can correctly locate the failing edge by distance from x to y, because the distance from x to y will increases when the edge e fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Foucaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] showed that 1 ≤ dem(G) ≤ n − 1 for any G with order n, and graphs with dem(G) = 1, n − 1 was characterized in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] Let G be a connected graph with at least one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dem(G) = 1 if and only if G is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] dem(G) = n − 1 if and only if G is the complete graph of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] Let G be a connected graph with a vertex x of G and for any y ∈ N(x), then, we have xy ∈ EM(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] For a vertex x of a graph G, the set of edges EM(x) induces a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' A vertex set M is called a vertex cover of G if M ∩{u, v} ̸= ∅ for uv ∈ E(G), The minimum cardinality of a vertex cover M in G is the vertex covering number of G, denoted by β(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] In any graph G of order n, any vertex cover of G is a distance-edge- monitoring set, and thus dem(G) ≤ β(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2 Research backgrounds and our results Perturbation problems in graph theory are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 3 Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G be a graph, and let e ∈ E(G) and v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let f(G) be a graph parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (i) The relation between f(G) and f(G − v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (ii) The relation between f(G) and f(G \\ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Chartrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [6] studied the perturbation problems on the metric dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Monson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [13] studied the effects of vertex deletion and edge deletion on the clique partition number in 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In 2015, Eroh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [9] considered the effect of vertex or edge deletion on the metric dimension of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [14] gave some results on the edge metric dimension of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Delen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [8] study the effect of vertex and edge deletion on the independence number of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We first consider the existence of graphs with given values of distance-edge-monitoring numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let r, t, n be three integers with 1 ≤ r, s ≤ n−1, and let G be a connected of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (1) Is there a connected graph G of order n such that dem(G) = r?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (2) Is there a connected subgraph H ⊂ G such that dem(H) = s and dem(G) = r?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In Section 2, we give the answers to Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For two integers r, n with 1 ≤ r ≤ n − 1, there exists a connected graph G of order n such that dem(G) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Given given three integers s, t, n with 1 ≤ s ≤ t ≤ n−1, there exists a connected graph H ⊂ G of such that dem(H) = s and dem(G) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In Section 3, we focus on Problem 1 (ii) and study the difference between dem(G \\ e) and dem(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge e ∈ E(G), we have dem(G \\ e) − dem(G) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Moreover, this bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Denote by G + E the graph with V (G + E) = V (G) and E(G + E) = E(G) ∪ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We construct graphs having the following properties in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any positive integer k, i with 2 ≤ i ≤ k, there exists a graph G and an edge set Ei+1 (|Ei+1| = i + 1) such that dem(G + Ei+1) − dem(G) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, let Gi = G + Ei(1 ≤ i ≤ k), we have dem(Gi) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' A feedback edge set of a graph G is a set of edges such that removing them from G leaves a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The smallest size of a feedback edge set of G is denoted by fes(G) (it is sometimes called the cyclomatic number of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 4 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [10] If fes(G) ≤ 2, then dem(G) ≤ fes(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Moreover, if fes(G) ≤ 1, then equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='9 implies the following corollary, and the proof will be given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let Tn be a tree of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For edges e1, e2 ∈ E(Tn), we have (1) dem(Tn + e1) = dem(Tn) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (2) dem(Tn + e1 + e2) = 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, there exists a tree T i n with two edges ei 1, ei 2 such that dem(Tn +e1 +e2) = i+1, where i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The following result in Section 4 shows that there exist a graph G and a subgraph H such that the difference dem(G) − dem(H) can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any positive integer k, there exist two graphs G1, G2 and their non- spanning subgraphs H1, H2 such that dem(G1) − dem(H1) = k and dem(H2) − dem(G2) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, dem(G) − dem(H) can be arbitrarily large, even for H = G − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any positive integer k, there exist two graphs G, H and two vertices u ∈ V (G), v ∈ V (H) such that (1) dem(G) − dem(G − u) ≥ k (2) dem(H − v) − dem(H) ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For a graph G with fixed order, the difference between dem(G) and dem(G − v) can be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For a graph G with order n (n ≥ 2) and v ∈ V (G), if G − v contain at least one edge, we have dem(G) − dem(G − v) ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Moreover, the equality holds if and only if G is K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G be a connected graph with order n ≥ 4 and dem(G) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let E ⊆ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If dem(G) = dem(G − E), then |E| ≤ 2n − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, the bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let T be a spanning tree of Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then 1 ≤ dem(Kn|T) ≤ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, the bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In Section 5, we focus on the following problem and give an algorithm to judge whether the distance-edge-monitoring set is still valid in the resulting graph when any edge (or vertex) of a graph G is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any graph G, if some edges or vertices in G break, we want to know if the original detection set can detect all edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For some well-known graphs, the effects of vertex deletion are dem(G) > dem(G − v), dem(G) = dem(G − v), or dem(G) < dem(G − v), which will be presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 5 2 Results for Problem 2 A kite K(r, n) is a graph obtained from the complete graph Kr+1 and a path Pn−r by attaching a vertex of Kr+1 and one end-vertex of Pn−r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' see an example of K(7, 12) in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' u0 u1 u2 u3 u4 u5 u6 u7 w1 w2 w3 w4 Figure 1: The graph K(7, 12) We first give the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1: Let G = K(r, n) with V (G) = {vi | 1 ≤ i ≤ n} and E(G) = {vivj | 1 ≤ i < j ≤ r + 1} ∪{vr+svr+s+1 | 1 ≤ s ≤ n − r − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2, we have dem(G) = dem(Gb) = dem(Kr+1) = dem(G) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In fact, for the above G, the path Pn−r−1 can be replaced by Tn−r−1, where Tn−r−1 is any tree of order n − r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 shows that Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='6 is true, that is, for three integers s, t, n with 1 ≤ s ≤ t ≤ n − 1, there exists a connected graph H ⊂ G of such that dem(H) = s and dem(G) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (For example, let G = K(r, n) and H = K(s, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=') This gives an answer about Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 3 The effect of edge deletions The following observation is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=', Gm be the connected components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dem(G) = dem (G1) + · · · + dem (Gm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, we suppose that the distance-edge-monitoring number of K1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The following proposition will be used in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any uv ∈ E(G) and w ∈ NG(u) ∪ NG(v), if uv /∈ EM(w), then uv is only monitored by {u, v} in V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since w ∈ NG(u) ∪ NG(v) and uv /∈ EM(w), we have dG(w, u) = dG\\uv(w, u) and dG(w, v) = dG\\uv(w, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any x ∈ V (G) − NG[u] ∪ NG[v], the path from x to u must through w1, where w1 ∈ NG(u) ∪ NG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dG(x, u) = dG(x, w1) + dG(w1, u) = dG(x, w1) + dG\\uv(w1, u) = dG\\uv(x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Similarly, dG(x, v) = dG\\uv(x, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any x ∈ V (G) − {u, v}, we have uv /∈ EM(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3, uv ∈ EM(u) and uv ∈ EM(v), and hence uv is only monitored by {u, v} in V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We first give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='7: Note that dem(G) = dem (G1) + · · · + dem (Gm), where G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' , Gm are all the connected components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So we assume that G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let M be an distance edge monitoring set of G and e = uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If M is also an distance-edge-monitoring set of G \\ e, then dem(G \\ e) ≤ dem(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Otherwise, M′ = M ∪ {u, v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' It suffices to show that M′ is an distance-edge-monitoring set of G \\ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If G \\ e has two components, say G1 and G2, then dem(G \\ e) = dem (G1) + dem (G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Without loss of generality, assume that u ∈ V (G1) and v ∈ V (G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge e1 = x1y1 ∈ E (G1), if there exists a vertex w ∈ V (G1) ∩ M, and we done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Otherwise, there exists a vertex w ∈ V (G2) ∩ M such that dG\\e1 (x1, w) ̸= dG (x1, w) or dG\\e1 (y1, w) ̸= dG (y1, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Without loss of generality, we suppose that dG\\e1 (y1, w) ̸= dG (y1, w) and dG (x1y1, w) = dG (x1, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since dG (y1, w) = dG (y1, x1) + dG (x1, u) + dG(u, w), dG\\{e,e1} (x1, u) = dG\\e1 (x1, u) and dG\\{e,e1} (y1, x1) > dG\\e (y1, x1), it follows that dG\\{e,e1} (u, y1) = dG\\{e,e1} (u, x1) + dG\\{e,e1} (x1, y1) > dG\\e (u, x1) + dG\\e (x1, y1) = dG\\e (u, y1) and hence dG\\{e,e1} (y1, u) ̸= dG\\e1 (y1, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So e1 is monitored by M1 ∪ {u} in graph G \\ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' This implies that dem (G1) ≤ |(M ∩ V (G1)) ∪ {u}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Similarly, we can obtain that dem (G2) ≤ |(M ∩ V (G2)) ∪ {v}|, and hence dem(G \\ e) ≤ |M′| ≤ dem(G) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Suppose that G \\ e is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If M is also a distance-edge-monitoring set of G \\ e, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Otherwise, there exists e1 = xy ∈ E(G \\ e) such that the edge e1 is not monitored by M in G \\ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since M is a distance- edge-monitoring set of G, it follows that there exists a vertex z ∈ M such that dG\\e1(x, z) ̸= dG(x, z) or dG\\e1(y, z) ̸= dG(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In addition, e1 is not monitored by M in G\\e, and so the distance from z to x or y is not changed after removing the edge e1 in G \\ e, which means that dG\\{e,e1} (y, z) = dG\\e (y, z) and dG\\{e,e1} (x, z) = dG\\e (x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If dG (e1, z) = dG(x, z), then the edge e lies on every z − y geodesic in G, otherwise there exists z∗ ∈ M such that dG\\{e,e1} (x, z∗) ̸= dG (x, z∗) and e does not appear in z∗ − y geodesic in G, which contracts to the fact that M is not the distance-edge-monitoring set of graph G \\ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If a geodesic in G from z to y traverses the edge e in the order u, v, then each geodesic in G from z to y traverses e in the order u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Assume, to the contrary, that there exists z1 − y geodesic and z2 − y geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We can assume that the two geodesics z1 − y and z2 − y are P1 = z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' uv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' y and P2 = z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' vu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 7 The z1 −y geodesic implies that d(u, v) + d(v, y) = d(u, y), and the z2 −y geodesic implies that d(v, u) + d(u, y) = d(v, y), and hence d(u, v) = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Claim 1, without loss of generality, we may assume that every geodesic in G from z to y traverses the edge e in the order u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Thus, we have dG(z, y) = dG(z, v)+dG(v, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We now show that xy can be monitored by v in G \\ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that dG\\e1(z, y) ̸= dG(z, y), dG\\e(v, y) = dG(v, y) and dG\\e(x, y) = dG(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dG\\{e,e1} (v, y) = dG\\{e,e1} (v, x) + dG\\{e,e1} (x, y) = dG\\e1 (v, x) + dG\\e1 (x, y) > dG (v, x) + dG (x, y) = dG\\e (v, x) + dG\\e (x, y) ≥ dG\\e(v, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since dG\\e(v, y) > dG\\{e,e1}(v, y), it follows that e1 can be monitored by v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since both u and v have the chance to be chosen as the monitoring vertices, we use them to monitor the edge xy, and therefore M′ = M ∪{u, v} is a distance edge-monitoring-set of G\\e, and thus dem(G\\e) ≤ dem(G)+2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' To show the sharpness of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='7, we consider the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' There exist connected graphs G1, G2 of order n such that dem(G1 \\ e) − dem(G1) = 2 and dem(G2) − dem(G2 \\ e) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Firstly, we consider the graph G1 (n ≥ 8) with vertex set V (G1) = {vi|1 ≤ i ≤ n − 8} ∪ {ui|1 ≤ i ≤ 8} and edge set E(G1) = {uivi | 1 ≤ i ≤ 8} ∪ {uiui+1 | 1 ≤ i ≤ 7} ∪ {uivi+1 | 1 ≤ i ≤ 7} ∪ {u1u8} ∪ {u1u5} ∪ {v1v8} ∪ {v1v9} ∪ {vivi+1 | 9 ≤ i ≤ n − 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Obviously, Gb(G1) = G∗ 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Take M = {u2, u4, v3, v6, u7, v8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that {u1u5, u5v5, u2v2, u2u1, u2u3} ⊆ EM(u2), v1 v2 v3 v4 v5 v6 v7 v8 u1 u2 u3 u4 u5 u6 u7 u8 v1 v2 v3 v4 v5 v6 v7 v8 u1 u2 u3 u4 u5 u6 u7 u8 dem(G) = 6 dem(G − e) = 8 Figure 2: The graph G∗ 8 {v1u1, u4v3, u4v5, u4v4} ⊆ EM(u4), {v3u3, v2v3, v4v3, v5v4, v2v1} ⊆ EM(v3), {v8v1, u8v8, v8u7} ⊆ EM(v8), {u7u8, u8u1, u6u7, u6u5, u7v7} ⊆ EM(u7), and {v5v6, v6v7, u6v6} ∈ EM(v6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since M is a distance-edge-monitoring set of G∗ 8, it follows that for uivi (2 ≤ i ≤ 8) and any w ∈ N(ui) ∪ N(vi), we have dG\\uivi(w, ui) = dG(w, ui) and dG\\uivi(w, vi) = dG(w, vi), and so uivi /∈ EM(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, the edge uivi (2 ≤ i ≤ 8) is monitored by {ui, vi}, and hence dem(G∗ 8) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since G∗ 8\\u1u5 ∼= C8□K2, it follows that dem(G∗ 8\\u1u5) = dem(C8□K2) = 8, and hence dem(G1 \\ u1u5) − dem(Gn) = dem(G∗ 8 \\ u1u5) − dem(G8) = 8 − 6 = 2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 8 v1 v1 v2 v2 v3 v3 v4 v4 v5 v5 v6 v6 dem(G) = 4 dem(G − e) = 2 Figure 3: The graph G⋄ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Next, we consider the graph G2 (n ≥ 6) with vertex set V (G2) = {vi | 1 ≤ i ≤ n} and edge set E(G2) = {v1v2, v3v4, v5v6, v1v3, v1v5, v2v4, v2v6, v3v5, v4v6} ∪ {vivi+1 | 6 ≤ i ≤ n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Obviously, Gb(G2) = G⋄ 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Take M = {v2, v3, v4, v5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that {v1v2, v6v2, v4v2} ⊆ EM(v2), {v1v3, v5v3, v4v3} ⊆ EM(v3), {v6v4} ⊆ EM(v4), {v5v1, v6v5} ⊆ EM(v5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then M is a distance-edge-monitoring set of G⋄ 6, and hence dem(G⋄ 6) ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For v2i−1v2i (1 ≤ i ≤ 3) and w ∈ N(ui) ∪ N(vi), we have dG\\v2i−1v2i(w, v2i−1) = dG(w, v2i−1) and dG\\v2i−1v2i(w, v2i) = dG(w, v2i), and so v2i−1v2i /∈ EM(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, the edge v2i−1v2i (1 ≤ i ≤ 3) is monitored by {v2i−1, v2i}, and hence dem(G⋄ 6) ≥ 3, and so M ∩ {v2i−1, v2i} ̸= ∅ (1 ≤ i ≤ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' All the M ∈ V (G⋄ 6) is showing Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since dem(G⋄ 6) ≥ 4, it follows that dem(G⋄ 6) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' M E(G) − ∪x∈MEM(x) v1, v3, v6 v2v4 v1, v4, v5 v2v6 v1, v4, v6 v3v5 v2, v3, v5 v4v6 v2, v3, v6 v1v5 v2, v4, v5 v1v3 v1, v3, v5 v2v6, v2v4, v4v6 v2, v4, v6 v1v3, v1v5, v3v5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The edges not monitored by M(|M| = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For the graph G⋄ 6 \\v3v4, if we take M = {v2, v5}, then {v1v2, v6v2, v4v2, v1v3} ⊆ EM(v2) and {v5v1, v6v5, v3v5, v6v4} ⊆ EM(v5), and hence dem(G⋄ 6 \\ v3v4) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since G⋄ 6 \\ v3v4 is not a tree, it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 that dem(G⋄ 6 \\ v3v4) = 2, and hence dem(G2) − dem(G2 \\ v3v4) = dem(G2) − dem(G⋄ 6 \\ v3v4) = 4 − 2 = 2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Next, we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 9 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='8: Let k, i are integers with 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The graph Gi is obtained by iteratively adding an edge uivi to a graph Gi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Without loss of generality, let \uf8f1 \uf8f2 \uf8f3 V (G0) = {c} ∪ {uj | 1 ≤ j ≤ k} ∪ {vj | 1 ≤ j ≤ k};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' E(G0) = {cuj, cvj | 1 ≤ j ≤ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' and for 1 ≤ i ≤ k, the graph of Gi is defined by V (Gi) = V (Gi−1) and E(Gi) = E(Gi−1) ∪ {uivi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G = Gi+1(3 ≤ i ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then Gb = Fr(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let M = {c} ∪ {ut | 1 ≤ t ≤ i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3, any edge e ∈ E(G) is monitored by M, and hence dem(G) ≤ i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any w ∈ N(uj) ∪ N(vj) = {c}, we have ujvj /∈ ME(c) (1 ≤ j ≤ i + 1 ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, the edge ujvj is only monitored by uj or vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dem(G) ≥ i + 1, and hence there exists a graph Gi such that |E(Gi) − E(Gi−1)| = 1, dem(Gi) ≤ dem(Gi+1) and dem(Gi) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since G0 = K1,2k is a tree, it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 that dem(G0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let Ei+1 = {utvt | 1 ≤ t ≤ i + 1 ≤ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dem(G + Ei+1) = dem(Gi+1) = i + 1, and hence dem(G + Ei+1) − dem(G) = i, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In the end of this section, we give the proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='10: For any tree Tn, Tn + e is a unicyclic graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, we have dem(Tn + e1) = dem(Tn) + 1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Similarly, Tn + e1 + e2 is a tricyclic graph, and hence dem(Tn + e1 + e2) ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let Tn = Pn with vertex set {v1, · · · , vn} and edge set {vivi+1 | 1 ≤ i ≤ n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If e1 = v1v6 and e2 = v2v5, then dem(Tn + e1 + e2) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let Tn = Sn with vertex set {v1, · · · , vn} and edge set {v1vi | 2 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edges e1 and e2, we have dem(Tn + e1 + e2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 4 The effect of vertex deletion A kipas � Kn with n ≥ 3 is the graph on n + 1 vertices obtained from the join of K1 and Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For n ≥ 6, we have dem( � Kn) = ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let V ( � Kn) = {v0, v1, · · · , vn} and E( � Kn) = {v0vi | 1 ≤ i ≤ n} ∪ {vivi+1 | 1 ≤ i ≤ n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' First, we prove that | dem( � Kn)| ≥ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any vertices vi, vj ∈ V ( � Kn), we have d(vi, vj) = \uf8f1 \uf8f2 \uf8f3 1, if i = 0 or j = 0 or |i − j| = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 2, if i· j ̸= 0 and |i − j| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge vivi+1 (2 ≤ i ≤ n − 2), we have NG(vi) ∪ NG(vi+1) = {vi−1, v0, vi+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since dG(vi, v0) = dG−vivi+1(vi+1, v0) = 1, dG−vivi+1(vi, vi−1) = dG(vi, vi−1) = 2 and dG−vivi+1(vi+1, vi+2) 10 = dG(vi+1, vi+2) = 2, it follows that vivi+1 /∈ EM(vi+1)∪EM(vi−1)∪EM(v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, the edge e = vivi+1 is monitored by {vi, vi+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Similarly, for i = 1, n−1, vivi+1 is monitored by {vi, vi+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that the vertex covering number of G is β(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since β(Pn) = ⌊n/2⌋, it follows that dem( � Kn) ≥ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Next, we prove that | dem( � Kn)| ≤ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let M = {vi | 0 ≡ i (mod 2), 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge e ∈ E(Pn) ∪ {v0vi | 0 ≡ i (mod 2), 1 ≤ i ≤ n}, it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3 that e is monitored by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In addition, for any edge e = v0vi ∈ {v0vi | 1 ≡ i (mod 2), 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since n ≥ 6, it follows that there exists j such that dG(vi, vj) = 2 and dG−v0vi(vi, vj) = 3, where j = i + 3(1 ≤ i ≤ n − 3) and j = 2(n − 3 ≤ i ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So, e ∈ EM(vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since any edge e ∈ E( � Kn) can be monitored by M, it follows that dem( � Kn) ≤ ⌊n/2⌋, and hence dem( � Kn) = ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='11 Note that � K2k+2 = {v0} ∨ P2k+2, it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, we have dem(P2k+2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, we have dem(� K2k+2) = k+1, and hence dem(� K2k+2)− dem(� K2k+2 − v0) = dem(� K2k+2) − dem(P2k+2) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G1 = � K2k+2 and H1 = P2k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dem(G1) − dem(H1) = dem(� K2k+2) − dem(P2k+2) = k, where H1 is not a spanning subgraph of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G2k+3 be a graph with vertex set V (G2k+3) = {ui | 1 ≤ i ≤ k + 1} ∪ {vi | 0 ≤ i ≤ k + 1} and edge set E(G2k+3) = {v0ui | 1 ≤ i ≤ k + 1} ∪ {uivi | 1 ≤ i ≤ k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Obviously, we have G2k+3 − v0 ∼= (k + 1)K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2, we have dem(G2k+3 − v0) = dem((k + 1)K2) = (k + 1) dem(K2) = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since G2k+3 is a tree, it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 that dem(G2k+3) = 1, and hence dem(G2k+1 − v0) − dem(G2k+1) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G2 = G2k+1 and H2 = (k + 1)K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dem(H2) − dem(G2) = dem((k + 1)K2) − dem(G2k+1) = k, where H2 is not a spanning subgraph of G2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that G2k+3 − v0 ∼= (k + 1)K2 is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For connected graphs, we can also show that there is a connected subgraph H such that dem(H) −dem(G) can be arbitrarily large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The conical graph C(ℓ, k) is a graph obtained by taking adjacency from a center vertex c to the first layer of Cartesian product of Pℓ and Ck, where ℓ ≥ 1 and k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The conical graph C(3, 8) can be seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For ℓ = 1, the graph C(1, k) is the wheel graph Wk, which is formed by connecting a single vertex c to all the vertices of cycle Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' It is clear that for the conical graph C(ℓ, k), we have |V (C(ℓ, k))| = kℓ + 1 and |E(C(ℓ, k))| = 2kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For k ≥ 8 and ℓ ≥ 2, we have dem(C(ℓ, k)) = \uf8f1 \uf8f2 \uf8f3 �Ak i=1 ⌈k/(4i − 2)⌉, if ℓ ≤ Ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' �Ak i=1 ⌈k/(4i − 2)⌉ + 2(ℓ − Ak), if ℓ ≥ Ak + 1, where Ak = ⌊k/4 + (1 + (−1)k+1)/8⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 11 c u1 1 u1 2 u1 3 u1 4 u1 5 u1 6 u1 7 u1 8 u2 1 u2 2 u2 3 u2 4 u2 5 u2 6 u2 7 u2 8 u3 1 u3 2 u3 3 u3 4 u3 5 u3 6 u3 7 u3 8 Figure 4: The conical graph C(3, 8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let V (C(ℓ, k)) = {c} ∪ {uj i | 1 ≤ i ≤ k, 1 ≤ j ≤ ℓ} and E(C(ℓ, k)) = (∪ℓ i=1E(Ci)) ∪ (∪k i=1E(Pi)), where E(Ci) = {ui kui 1} ∪ {ui jui j+1 | 1 ≤ j ≤ k − 1} (1 ≤ i ≤ ℓ), E(Pi) = {cu1 i } ∪ {uj iuj+1 i | 1 ≤ j ≤ ℓ − 1} (1 ≤ i ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any vertex v ∈ V (Ci), we have |EM(v) ∩ E(Ci)| = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 4i − 2, if 1 ≤ i ≤ ⌊k/4 + (1 + (−1)k+1)/8⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' k − 2, if k is even and i ≥ ⌈k/4 + (1 + (−1)k+1)/8⌉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' k − 1, if k is odd and i ≥ ⌈k/4 + (1 + (−1)k+1)/8⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, we have |EM(c) ∩ E(Ci)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any vertex ui s, ui t ∈ V (Ci)(1 ≤ s, t ≤ k and i ≤ ⌊k/4 + (1 + (−1)k+1)/8⌋), since ui s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' , u1 scu1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' ui−1 t ui t is the path from ui s to ui t, and hence dCi(ui s, ui t) ≤ 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So we have {ui tui t+1 | s − 2i + 1 ≤ t ≤ s + 2i − 2} ∈ EM(ui s), where the subscripts are taken mod- ulo n, that is, ui k+1 = ui 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any ui jui j+1 ∈ E(Ci)− {ui tui t+1 | s − 2i + 1 ≤ t ≤ s + 2i − 2} − {ui s−2iui s−2i−1, ui s+2i+1ui s+2i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since d(ui s, ui j) = d(ui s, ui j+1) = 2i, and dG\\ui jui j+1(ui s, ui j) = dG\\ui jui j+1(ui s, ui j+1) = 2i, it follows that ui jui j+1 /∈ EM(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In addition, let e1 = ui s−2iui s−2i−1 and e2 = ui s+2i+1ui s+2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dG\\e1(ui s, ui s−2i) = dG(ui s, ui s−2i), and hence dG\\e2(ui s, ui s+2i) = dG(ui s, ui s+2i), and so e1, e2 /∈ EM(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Similarly, for any edge e = uj muj m+1 ∈ E(Cj)(j ̸= i and 1 ≤ m ≤ k), since dG\\e(ui s, uj m) = dG(ui s, uj m) and dG\\e(ui s, uj m+1) = dG(ui s, uj m+1), it follows that e /∈ EM(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any vertex ui s ∈ V (Ci), we have |EM(ui s)∩E(Ci)| = 4i−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that if n is even number, then 4i−2 ≤ k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' and hence, i ≤ ⌈k/4⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If n is odd, then 4i − 2 ≤ k − 1, and hence i ≤ ⌈(k + 1)/4⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So we have i ≤ ⌊k/4 + (1 + (−1)k+1)/8⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For i ≥ ⌈k/4+(1+(−1)k)/8⌉ and 1 ≤ s, t ≤ k, if ui s, ui t ∈ V (Ci), then dG(ui s, ui t) = dCi(ui s, ui t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any cycle Ci with length k, if k is even, then ui s+k/2ui s+k/2−1, ui s+k/2ui s+k/2+1 /∈ EM(ui s), and so |EM(v) ∩ E(Ci)| = k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If k is odd, then ui s+(k−1)/2ui s+(k+1)/2 /∈ EM(ui s), and so |EM(v) ∩ E(Ci)| = k − 1, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 12 For any edge e3 = ui sui (s+1) ∈ E(Ci), we have dG(c, ui s) = dG(c, ui (s+1)) = i, and hence dG\\e3(c, ui s) = dG\\e3(c, ui (s+1)) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since e3 /∈ E(c), it follows that |EM(c) ∩ E(Ci)| = 0, and hence Fact 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Firstly, note that Ak = ⌊k/4 + (1 + (−1)k+1)/8⌋, we consider ℓ ≥ Ak + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Fact 1, we have dem(G) ≥ �Ak i=1 ⌈k/(4i − 2)⌉ +2(ℓ − Ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Take M = ∪i=k i=1Mi, where Mi = \uf8f1 \uf8f2 \uf8f3 {ui j|1 ≤ j ≤ k, 1 ≡ j (mod (4i − 2))}, if i ≤ Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' {ui 1, ui ⌈k/2⌉}, ifAk + 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From the selection method of M, for any edge e ∈ E(Ci)(1 ≤ i ≤ ℓ), e is monitored by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' It suffices to prove that each edge in E(Pi) is monitored by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For vertex u1 i , u1 j ∈ V (G)(1 ≤ i, j ≤ k), if j /∈ {i + 1, i + 2, i − 1, i − 2}, where the subscripts are taken modulo n, that is, u1 −1 = u1 k and u1 1 = u1 k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' then dG(u1 i , u1 j) ̸= dG−cu1 j(u1 i , u1 j), and hence cu1 j ∈ EM(u1 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Similarly, for 1 ≤ t ≤ ℓ − 1, since dG(u1 i , ut j) ̸= dG−ut jut−1 j (u1 i , ut j), it follows that ut jut−1 j ∈ EM(u1 i ), and hence EM(u1 i ) = {u1 i u1 i+1, u1 iu1 i−1} ∪ {cu1} ∪ {up−1 j up j|2 ≤ p ≤ ℓ − 1, 1 ≤ j ≤ n, j /∈ {i−2, i−1, i+1, i+2}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Without loss of generality, take u1 1 ∈ M, then, for any edge e ∈ E(Pi) (j /∈ 2, 3, k, k − 1), it is monitored by u1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge e ∈ E(Pi) (j ∈ {2, 3, k, k − 1}), if k is even, then k − 3 − 2 ≥ 3, since k ≥ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Obviously, u1 3, u1 k−1, u1 k−3 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If k is odd, then it is monitored by vertex set {u1 3, u1 k}, which is a subset of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any e ∈ E(Pi) (1 ≤ i ≤ k), it is monitored by Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Hence, we have dem(G) ≤ �⌊(k+2)/4⌋+1 i=1 ⌈k/(4i − 2)⌉ + 2(ℓ − Ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For ℓ ≤ Ak, it is similar to the case that ℓ ≥ Ak + 1, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' obtained the following result about distance-edge-monitoring numbers of Cn□Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' [12] Let ℓ and k be two integers with ℓ ≥ 3 and k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then dem (Ck□Pℓ) = \uf8f1 \uf8f2 \uf8f3 k if k ≥ 2ℓ + 1, 2ℓ if k < 2ℓ + 1 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any positive integer k ≥ 8, there exists a connected graph subgraph of G such that such that dem(G − v) − dem(G) = ⌊k/2⌋ − ⌈k/6⌉, where v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G = C(ℓ, k), where ℓ = 2 and k ≥ 2ℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that G−v0 = Ck□K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2, we have dem(G − v0) = dem(Ck□K2) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, we have dem(C(2, k)) = �2 i=1 ⌈k/(4i − 2)⌉ = ⌈k/2⌉ − ⌈k/6⌉, and hence dem(G − v) − dem(G) = k − ⌈k/2⌉ − ⌈k/6⌉ = ⌊k/2⌋ − ⌈k/6⌉, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 13 Let G = E(C(ℓ, k)) and H = Ck□Pℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2, if ℓ ≫ k, then dem(G) dem(H) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2, if k ≫ ℓ = 2, then dem(G) dem(H) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' There exist two graphs H and G such that dem(G) dem(H) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3, where H is an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='12: Let G = � K2k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='11, there exists a vertex u ∈ V (G) such that dem(� K2k+2) − dem(� K2k+2 − u) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that G − u = P2k+2 is connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In addition, let H = C(2, t), k = ⌊t/2⌋ − ⌈t/6⌉ and v ∈ V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3, we have dem(C(2, t) − v) − dem(C(2, t)) = k and C(2, t) − v = Ct□K2 is a connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In fact, G − v is subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='12, for any positive integer k ≥ 3, there exists a graph G such that dem(G − v) − dem(G) ≥ k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G = C(2, t)(t ≥ 8) and H1 = C(2, t) − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='12, we have dem(H1) ≥ dem(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that G is not tree, it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 that dem(G) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let H2 is a subtree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, we have dem(H2) = 1 ≤ 2 ≤ dem(G), and so we have Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let G be a connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then there exist two non-spanning subgraphs H1, H2 ⊂ G such that dem(H1) ≥ dem(G) and dem(H2) ≤ dem(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2: For any graph G with order n, and G−v contain at least one edge, we have dem(G) ≤ n−1 and 1 ≤ dem(G−v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Obviously, we obtained dem(G)−dem(G−v) ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, let G = K3, then dem(G) − dem(G − v) = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So the upper bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Conversely, since dem(G) − dem(G − v) = n − 2, it follows that dem(G) = n − 1 and dem(G − v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, G − v must be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If |V (G − v)| ≥ 3, and note that dem(G) = n − 1, it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 that G = Kn (n ≥ 4), which is impossi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then |V (G − v)| ≤ 2, and hence G = Kn (n ≤ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If G = K2, then G − v = K1, which contradicts to the fact that G−v contains at least one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So, we have G = K3, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Next, we consider the subgraph H of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If H ⊂ G such that dem(H) ≤ dem(G), then, what is the relationship between H and G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' A natural question is, what is the maximum number of edges we can delete from G without changing number of distance-edge monitoring?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We give a partial answer as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The base graph Gb of a graph G is the graph obtained from G by iteratively removing vertices of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We know that Gb is subgraph of G, but dem(G) = dem(Gb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So we can obtain below result, which give a lower bound for E such that dem(G) = dem(G − E), so we have Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 14 Observation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any graph G, let E1 = E(G) − E(Gb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If dem(G) = dem(G − E) and G−E is connect graph with order at least 2, then |E| ≥ |E1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, the bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='13: Since dem(G) = 2, without loss of generality, let M = {u, v} be a distance-edge-monitoring set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If uv ∈ E(G), then it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='4 that |E(G)| ≤ 2(n − 1) − 1, and hence dem(G) = dem(G − E) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, G − E must contain an cycle, and hence |E| ≤ 2(n − 1) − 1 − 3 = 2n − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If uv ̸∈ E(G), then M = {u, v} is a distance-edge-monitoring set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If |EM(u) ∩ EM(u)| ≥ 1, which is similar to the case u, v is adjacent to in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So, we suppose that EM(u) ∩ EM(u) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='4, |E(G)| ≤ 2(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, G must contain a cycle and |E| ≤ 2(n − 1) − 3 = 2n − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, we have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' |E| ≤ 2n − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Assume, to the contrary, that |E| = 2n−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then G−E must be an cycle of length 3, say G −E = K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Without loss of generality, let V (G −E) = {v1, v2, v3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In addition, the subgraph induced by the edge set EM(u) and EM(v) are spanning trees in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If u, v ∈ {v1, v2, v3}, then uv ∈ E(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any vertex w ∈ V (G) − {v1, v2, v3}, if dG(w) = 1, then w is in one of spanning tree of EM(u) or EM(v), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Hence, we have dG(w) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Especially for two vertices u and v, there exists a vertex u1 ∈ N(u) such that uu1 ∈ EM(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3, we have uu1 ∈ EM(u), and hence |EM(u)∩EM(u)| ̸= 0, which contradicts to the fact that EM(u) ∩ EM(u) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Claim 2, we have |E| ≤ 2n − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, let G = (n − 2)K1 ∨ K2 with ver- tex set V (G) = {vi | 1 ≤ i ≤ n} and edge set E(G) = {v1v2} ∪ {v1vi, v2vi|3 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let E = {v1vi, v2vi | 4 ≤ i ≤ n} ⊂ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, we have dem(G − E) = dem(K3) +(n − 1) dem(K1) = 2, and so there exists an edge set E such that dem(G − E) = 2 and |E| = 2n − 6, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For H ⊂ G, the distance-edge-monitoring set of subgraph H in G is a set M of vertices of a graph H such that E(H) ⊆ � x∈M EM(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The restrict-distance-edge-monitoring number dem(G|H) of a graph G|H is de- fined as the smallest size of a distance-edge-monitoring set of H in G, that is, dem(G|H) = min � |M| ���E(H) ⊆ � x∈M EM(x), M ⊆ V (H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In Figure 5, let G = K4 with V (G) = {v0, v1, v2, v3} and E(G) = {vivj | 0 ≤ i < j ≤ 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let E(T1) = {v0v1, v0v2, v0v3} and E(T2) = {v0v3, v3v1, v1v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Obviously, we have dem(K4|Ti) = i 15 v0 v1 v2 v3 v2 v1 v0 v3 T1 T2 Figure 5: T1 and T2 (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The distance-edge-monitoring set of subgraph Ti (i = 1, 2) in K4 is shown in Figure 5, where the blue vertices form the set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In the end of this section, we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='14 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='14: Let G = Kn with vertex set {vi | 1 ≤ i ≤ n} and edge set {vivj | 1 ≤ i < j ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let T be a spanning tree in Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge uv ∈ E(T) and vertex w ∈ (NG(u) ∪ NG(v)) ∩ V (T), we have dG(w, u) = dG(w, v) = 1 and dG\\uv(w, u) = dG\\uv(w, v) = 1, and hence uv /∈ EM(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, any edge uv ∈ E(T) is only monitored by u or v, and hence dem(Kn|T) ≥ β(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='5, we have dem(Kn|T) ≤ β(T), and hence dem(Kn|T) = β(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since T is tree with order n, it follows that there is no odd cycles, and hence T is a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Without loss of generality, let V (T) = U ∪ V (|U| ≤ |V |), which is a bipartite partition of V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From the pigeonhole principle, we have |U| ≤ ⌊n 2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any uv ∈ E(T), we have {u, v} ∩ U ̸= ∅, and hence β(T) ≤ ⌊n 2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In addition, T contains at least one edge, and hence 1 ≤ β(T), and so 1 ≤ β(T) ≤ ⌊n 2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Suppose that T = Sn with edge set E(Sn) = {v1vi | 2 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Without loss of generality, let dT(v1) = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then {v1} is the vertex cover set of Sn, and hence β(T) = 1, and so the lower bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Suppose that T = Pn with edge set E(Pn) = {vivi+1 | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Obviously, we have {vi | i ≖ 0 (mod 2), and 1 ≤ i ≤ n, vi ∈ V (Pn)} is vertex cover set of Pn, and hence β(T) = ⌊n 2⌋, and hence the upper bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Similar to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='14, we have below Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let H be a subgraph with order p of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Then we have 1 ≤ dem(G|H) ≤ p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, the left (right) equality hold if and only if H = Sp−1 (H = Kp−1, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If H is an induced subgraph of graph G, then we have dem(G) − dem(G|H) ≤ |V (G)| − |V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 16 Moreover, If both G and H are complete graph, then the bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any graph G and H, where H is induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let M1 ⊂ V (H) is the smallest size of a distance-edge-monitoring set of H in G, without loss of generality, suppose that the restrict distance-edge-monitoring number is r, that is, dem(G|H) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let M = (V (G) − V (H)) ∪ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We will prove that dem(G) ≤ |M|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge e = uv ∈ E(G), if u or v in V (G) − V (H), then it follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3 that e is monitored by V (G) − V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge e ∈ E(H), since M1 is the a restrict distance-edge-monitoring set, it follows that e is monitored by M1, and hence M is a distance-edge-monitoring set in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since |M| = |M1| + (|V (G)| − |V (H)|) = |V (G)| − |V (H)| + dem(G|H), it follows that dem(G) ≤ |M| = dem(G|H) + (|V (G)| − |V (H)|), and so dem(G) − dem(G|H) ≤ (|V (G)| − |V (H)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, we take G = Kn and H = Km (3 ≤ m ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2, we have dem(G) = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any uv ∈ E(H) and w ∈ N(u) ∪ N(v), we have dG(w, u) = dG(w, v) = 1 and dG\\uv(w, u) = dG\\uv(w, v) = 1, and hence uv /∈ EM(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, any edge uv ∈ E(H) is only monitored by u or v, and hence dem(Kn|Km) ≥ β(Km) = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='5, we have dem(Kn|Km) ≤ β(Km) = m − 1, and hence dem(Kn|Km) = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Obviously, dem(G) − dem(G|H) = (n − 1) − (m − 1) = |V (G)| − |V (H)|, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 5 Complexity issues Foucaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' proved in [10] that the problem DISTANCE-EDGE-MONITORING SET is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any graph G, a natural question is whether the original detection set can detect all edges if some edges or vertices in G are broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We design the algorithm and then give the results on the algorithm analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' DISTANCE-EDGE-MONITORING SET Instance: A graph G = (V, E), an edge e ∈ E(G) and a distance-edge-monitoring set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Question: Whether M is still a distance-edge-monitoring set for the graph G − e?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Given a graph G, a distance-edge-monitoring set M, and an edge e ∈ E(G), our goal is to determine whether the original distance-edge-monitoring set M is still valid in the resulting graph G \\ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The algorithm is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The algorithm of how to compute the edge set EM(x) from G was given by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' in [15], and is based on the breadth-first spanning tree algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The time complexity is polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 Deleting one edge or vertex from some known graphs In this section, we study the change for distance-edge-monitoring numbers of some well-known graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 17 Algorithm 1 The algorithm for determining M is or not monitor set for G \\ e Input: A graph G, M ⊂ V (G) and e ∈ E(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Output: E(G \\ e) ⊂ ∪x∈MEM(x) is TRUE or FALSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 1: M1 ← E(G \\ e) 2: for each vertex v ∈ M do 3: M1 ← M1 − EM(v) 4: if M1 = ∅ then return E(G \\ e) ⊂ ∪x∈MEM(x) is TRUE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 5: else return E(G \\ e) ⊂ ∪x∈MEM(x) is FALSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We denote NVi(G) = {v | dG(v) = i, v ∈ V (G)} and NEa,b(G) = {uv | uv ∈ E(G), dG(u) = a, dG(v) = b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that in Pn, we have E(Pn) = NE1,2(Pn) ∪ NE2,2(Pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If e ∈ NE1,2(Pn), then we have dem (Pn \\ e) = 1, and hence dem (Pn)−dem (Pn \\ e) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If e ∈ NE2,2(Pn), then dem (Pn \\ e) = 2, and hence dem (Pn) − dem (Pn \\ e) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Therefore these arguments imply the following Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, which imply that it is possible to find classes of graphs G and edge e such that dem(G \\ e) ≤ dem(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let n be an integer and Pn be a path with order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any e ∈ E(Pn), we have dem(Pn \\ e) = \uf8f1 \uf8f2 \uf8f3 dem(Pn), if e ∈ NE1,2(Pn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' dem(Pn) + 1, if e ∈ NE2,2(Pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The following corollary is immediate, which imply that there exists graphs G and edge e such that dem(G) > dem(G \\ e) and dem(G) = dem(G \\ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let n ≥ 3 be an integer and for any edge e ∈ E(G), we have (i) dem(Cn \\ e) = dem(Cn) − 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (ii) dem(Kn \\ e) = dem(Kn) − 1 = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (iii) dem(Kn,n \\ e) = dem(Kn,n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (i) From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, we have dem(Cn \\ e) = dem(Pn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Since dem(Cn) = 2, it follows that dem(Cn \\ e) = dem(Cn) − 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (ii) From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='2, we have dem(Kn) = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For e = uv ∈ E(Kn \\ e), if we take M = V (Kn) − {u, v}, then one end-vertex of e belongs to M for any edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3, we have dem(G \\ uv) ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any edge xy ∈ E(G \\ uv) and vertex w ∈ (NG(x) ∪ NG(y)) ∩ V (G − {u}), we have dG(w, x) = dG(w, y) = 1 and dG\\xy(w, x) = dG\\xy(w, y) = 1, and hence xy /∈ EM(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1, any edge xy ∈ E(G − {u}) is only monitored by x or y, and hence dem(G \\ uv) ≥ n − 2, and so dem(G \\ uv) = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' This implies that dem(Kn) − dem(Kn \\ e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (iii) Let G = Kn,n with vertices set V (Kn,n) = {ui | 1 ≤ i ≤ n} ∪{vi | 1 ≤ i ≤ n} and edge set E(Kn,n) = {uivj | 1 ≤ i, j ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Without loss of generality, let e = u1v1, For any vertex 18 set M ⊂ V (G) with |M| = n − 1, let U1 = M ∩ {ui | 1 ≤ i ≤ n}, V1 = M ∩ {vi | 1 ≤ i ≤ n}, and |U1| + |V1| = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If |U1| = n − 1 and |V1| = 0, then for any w ∈ M, there exist ut /∈ M and v2 ∈ V2 such that dG(ut, w) = dG\\utv2(ut, w) = 2 and dG\\e(v2, w) = dG\\utv2(v2, w) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So utv2 is not monitored by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If |U1| ≤ n−2 and |V1| ≤ n−2, then there exist p, q ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' , n} such that upvq ∈ E(Kn,n), upvq ̸= u1v1 and up, vq /∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If w ∈ U1, then dG(up, w) = dG\\upvq(up, w) = 2 and dG(vq, w) = dG\\upvq(vq, w) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If w ∈ V1, then dG(up, w) = dG\\upvq(up, w) = 1 and dG(vq, w) = dG\\upvq(vq, w) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' So upvq is not monitored by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Therefore, M is not an distance-edge-monitoring set, and hence dem(G) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='5, we have dem(G) ≤ β(G) = n, and hence dem(G) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' It is possible to find that there exists a graph G and v ∈ V (G) such that dem(G) > dem(G − v), dem(G) = dem(G − v), dem(G) < dem(G − v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' The following corollary shows that there exists a graph G such that dem(G) > dem(G − v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let n ≥ 3 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any v ∈ v(G), we have (i) dem(Cn − v) = dem(Cn) − 1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (ii) dem(Kn − v) = dem(Kn) − 1 = n − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (iii) dem(Kn,n − v) = dem(Kn,n) − 1 = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Let n ≥ 5 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For any v ∈ V (Pn), we have dem(Pn − v) = \uf8f1 \uf8f2 \uf8f3 dem(Pn), if v ∈ NV1(Pn) ∪ {v2, vn−1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' dem(Pn) + 1, if v ∈ NV2(Pn) \\ {v2, vn−1}, where Pn = v1v2 · · · vn−1vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Note that V (Pn) = NV1(Pn) ∪ NV2(Pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If v ∈ NV1(Pn), then dem (Pn − v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If v ∈ {v2, vn−1}, then dem (Pn − v) = dem (Pn−1) + dem (K1) = 1, where v(K1) = {v1} or v(K1) = {vn}, and hence dem (Pn) − dem (Pn − v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' If v ∈ NV2(Pn) \\ {v2, vn−1}, then dem (Pn − v) = 2, and hence dem (Pn) − dem (Pn − v) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content='1 shows that there exists a graph G and v ∈ V (G) such that dem(G) = dem(G − v) or dem(G) < dem(G − v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 6 Conclusion In this paper, we have studied the effect of deleting edges and vertices in a graph G on the distance-edge-monitoring number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' We obtained that for any graph G and edge e ∈ E(G), we have dem(G − e) − dem(G) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' Furthermore, the bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In addition, we can find a graph H and v ∈ V (H) such that dem(H − v) − dem(H) can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' This fact gives an answer to the monotonicity of the distance-edge-monitoring number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' This means that there exist two graphs H and G such that H ⊂ G and dem(H) ≥ dem(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' It is interesting to consider the following problems for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' 19 (1) It is interesting to characterize the graphs dem(H) = dem(G) if H ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (2) For a graph G and E ⊂ E(G), what is the maximum value of |E| such that dem(G) = dem(G − E)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' (3) For any ǫ > 0, whether the ratio dem(G) dem(H) ≤ ǫ holds, where H is an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' For further future work, it would be interesting to study distance-edge monitoring sets in further standard graph classes, including pyramids, Sierpi´nki-type graphs, circulant graphs, graph products, or line graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE0T4oBgHgl3EQfnAG2/content/2301.02507v1.pdf'} +page_content=' In addition, characterizing the graphs with dem(G) = n − 2 would be of interest, as well as clarifying further the relation of the parameter dem(G) to other standard graph parameters, such as 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+1,549 @@ +arXiv:2301.13167v1 [math.LO] 30 Jan 2023 +A complete classification of categoricity spectra of accessible +categories with directed colimits +Christian Esp´ındola +Abstract +We provide a complete classification of all the possible categoricity spectra, in terms of +internal size, that can appear in a large accessible category with directed colimits, assuming +the Singular Cardinal Hypothesis (SCH), and providing as well explicit threshold cardinals +for eventual categoricity. This includes as a particular case the first complete classification of +categoricity spectra of abstract elementary classes (AEC’s) entirely in ZFC. More specifically, +we have: +Theorem. Let K be a large κ-accessible category with directed colimits. Assume the Singular +Cardinal Hypothesis SCH (only if the restriction to monomorphisms is not an AEC). Then the +categoricity spectrum Cat(K) = {λ ≥ κ : K is λ-categorical} is one of the following: +1. Cat(K) = ∅. +2. Cat(K) = [α, β] for some α, β ∈ [κ, ℶω(κ)). +3. Cat(K) = [χ, ∞) for some χ ∈ [κ, ℶ(2κ)+). +This solves in particular Shelah categoricity conjecture for AEC’s. There are examples of +each of the three cases of the classification, showing that they indeed occur. +1 +Introduction +This short paper is a sequel to the work of the author in [Esp22] in which a generalization of +Shelah’s eventual categoricity conjecture (Conjecture 4.2 in the introduction of [She09]) is proven +in the more general context of accessible categories with directed colimits. When all morphisms are +monomorphisms in such categories of models, an analogous form of Shelah’s presentation theorem +exhibits them as a projective class of an infinite quantifier logic, for which even the Hanf number +for model existence has no known explicit bound in ZFC (the only known bound is a strongly +compact cardinal, and in fact in some models of ZFC the Hanf number for Lω1,ω1 exceeds the first +measurable cardinal). +In the special case of those accessible categories which have directed colimits, however, we will +prove that one can find ZFC bounds for the Hanf number of model existence. +Grossberg has +emphasized the importance of also having explicit threshold cardinals for the eventual categoricity +phenomenon, and we now intend to use the same setup and results of [Esp22], together with Morley’s +method, to provide the provably best possible explicit thresholds. Shelah categoricity conjecture +asks to prove, in ZFC, that the threshold for eventual categoricity in an AEC K is ℶ(2LS(K))+ (see +Conjecture 4.3 b) in the introduction of [She09]). We will prove this conjecture here. An example +from Shelah mentioned in [Vas19] shows that this threshold is best possible. +Assuming SCH, we are also going to provide a proof of a direct generalization of Shelah cate- +goricity conjecture to the more general context of accessible categories with directed colimits. If K +is such a category, we show that such that if K is categorical in some in some λ ≥ ℶ(2LS(K))+ (i.e., it +has only one object of some high enough internal size up to isomorphism), then K is λ′-categorical +for every λ′ ≥ ℶ(2LS(K))+. When considering cardinalities of models of infinitary theories T of Lκ,θ +1 + +that axiomatize K, the result implies, under SCH, the following infinitary version of Morley’s +categoricity theorem: +Theorem 1.1. (Morley’s categoricity theorem for infinitary theories) Let φ be a Lκ,θ sentence +whose category of models and Lκ,θ-elementary embeddings has directed colimits. Let S be the class +of cardinals λ which are of cofinality at least θ but are not successors of cardinals of cofinality +less than θ. Assume the weakening of the Singular Cardinal Hypothesis SCHθ,≥2<θ. Then, if φ +is λ-categorical for some λ ≥ ℶ(2κ)+ in S, then φ is λ′-categorical for every λ′ ≥ ℶ(2κ)+ in S. +Moreover: +1. if the directed colimits are concrete, we can spare the assumption SCHθ,≥2<θ and take S as +the class of all cardinals. +2. if φ is compact and the morphisms of our category are Lω,ω-elementary embeddings, we can +replace ℶ(2κ)+ with κ. +Here SCHθ,≥2<θ is defined as “for all µ ≥ 2<θ there is a set of cardinals λi ≤ µ unbounded below +µ such that, for each i, ν<θ ≤ λi for all ν < λi”, see Remark 2.3 of [LRV19]. Also, we know from +[Esp22] that there are examples showing that the exceptions in the class S are needed. +The case θ = ω in Theorem 1.1 is Shelah categoricity conjecture for Lκ,ω, since in this case +SCHω,≥2<ω is provable in ZFC. When the directed colimits are concrete, since we restrict to +monomorphisms, the result is Shelah categoricity conjecture for AEC’s, since SCHθ,≥2<θ can also be +removed by the methods of [Esp22]. The case when φ is compact (i.e., when it has the property that +φ is consistent with an arbitrary set of first-order finitary formulas if and only if it is consistent with +each of its finite subsets, see [Kei14]) is precisely a proper generalization of Morley’s categoricity +theorem (for countable first-order theories) and of Shelah’s categoricity theorem (for uncountable +first-order theories). Compact sentences of infinitary logic forms a much wider class than these two +particular cases, since they include (but are not limited to) all those conjunctive sentences (i.e., +sentences where only conjunctions are infinitary but disjunctions are finitary). Thus, Theorem 1.1 +is a vast generalization of those conjectures and results to the realm of infinite quantifier theories +and provides new proofs of the known theorems for finitary first-order theories. +As it turns out, the existence of directed colimits is what allows for a smooth classification theory. +The main tool for this will be Theorem 3.2, which at the same time extends work of Shelah for +Lω1,ω showing, under the Weak Generalized Continuum Hypothesis (WGCH) that categoricity +in the first ω cardinals implies categoricity everywhere (see [She09]). +We remove here the set- +theoretic hypothesis and generalize this result to AEC’s, at the price of asking for categoricity in +the first ℶω cardinals. By the example of Shelah and Villaveces in [SV04], this seems to be close to +optimal, since they showed that categoricity can fail above ℶn(λ) for any n ∈ ω while holding at the +first n cardinals above the L¨owenheim-Skolem number λ (though it is open whether categoricity +holds up to ℶn(λ) or the gap could be reduced further). Theorem 3.2 also uses higher dimensional +amalgamation properties, which are shown to be a consequence of categoricity by means of a simple +categorical proof, thereby simplifying the methods of [?]. +Finally, we state the classification of categoricity spectra in AEC’s, in ZFC, and assuming SCH +also in accessible categories with directed colimits (the set theoretic assumption is needed to guar- +antee that the existence spectrum contains an end tail of cardinals). This uses Lemma 2.1, some +weaker versions of which in the context of AEC’s have appeared in the literature. We give here a +categorical proof based, among other things, on a form of Lawvere’s duality for algebraic theories. +The proof extends the result to µ-AEC’s with directed colimits, and greatly simplifies the argu- +ments given for AEC’s, in such a way that it can be applied to deduce eventual categoricity without +needing to use amalgamation, using instead an observation on the double negation topology. +2 + +As a word of warning, we emphasize that all the methods, results and notation from the authors’ +previous paper [Esp22] are assumed here throughout, so the reader is advised to go through that +paper first before continuing with this sequel. +2 +Saturation and stability +We start by stating the following lemma of independent interest: +Lemma 2.1. Let K be a µ-AEC with directed colimits and amalgamation that is ρ-stable for each +ρ < κ. Then κ-saturated models are closed under directed colimits. +Proof. Consider the topos Set[TB +κ ]λ/[M, −] ∼= SetKB +≥κ,<λ/[M, −] ∼= SetM/KB +≥κ,<λ, (where KB +≥κ,<λ +consists of the models in K≥κ,<λ and all its κ-Boolean homomorphisms, and where TB +κ is Tκ +plus all those instances of excluded middle for κ-coherent formulas). +We have a stable sur- +jection SetM/KB +≥κ,<λ ։ SetM/K≥κ,<λ; this can be seen by considering first the stable surjection +SetKB +≥κ,<λ ∼= Set[TB +κ ]λ ։ Set[Tκ]λ ∼= SetK≥κ,<λ. Then we consider the pullback functor to the +slice SetK≥κ,<λ +� Set[Tκ]λ/[M, −] ∼= SetM/K≥κ,<λ, which is a geometric morphism along whose +direct image we take the following (pseudo-)pullback: +SetKB +≥κ,<λ +SetK≥κ,<λ +Set[TB +κ ]λ/[Mi, −] +Set[Tκ]λ/[Mi, −] +Set[TB +κ ]λ/[M, −] +Set[Tκ]λ/[M, −] +The (pseudo-)pullback is precisely Set[TB +κ ]λ/[M, −] ∼= SetM/KB +≥κ,<λ, as can be verified using +the universal property of the slice. +More generally, if we have now a sequence of embeddings +M0 +� M1 +� ... +� M with directed colimit M, then SetM/K≥κ,<λ will be the limit of the chain +formed by the SetMi/K≥κ,<λ and induced by those embeddings. Since pullbacks preserve limits, this +implies that SetM/KB +≥κ,<λ will be the limit of the chain formed by the SetMi/KB +≥κ,<λ; in particular +(considering functors from the presheaves to Set preserving limits and colimits), the Cauchy com- +pletion of the slice M/KB +≥κ,<λ is the (pseudo-)limit in Cat of the Cauchy completion of the slices +Mi/KB +≥κ,<λ (note, in turn, that the Cauchy completions of the slices are equivalent to the slices of +the Cauchy completion KB +≥κ,<λ). +Note that for any κ-small model P and κ-geometric theory Sκ of models of size at least κ contain- +ing P, the pullback, in the 2-category of κ+-toposes and κ+-geometric morphisms, of the double +negation subtopos S of Set[Sκ]κ+ along the surjection s : Set[SB +κ ]κ+ ։ Set[Sκ]κ+ must be S itself, +so S embeds into Set[SB +κ ]κ+. Moreover, the embedding is dense: take a nonzero subterminal object +A in Set[SB +κ ]κ; then it is nonzero in some model M which, by density, must embed into the model +of size κ. Then, if M is the category of models of S, the κ+-coherent sentence ψ = lim evφi in +SetMB +κ , where lim evφi ∼= [M, −] in SetMκ, is non zero in SetMsat +κ +and it implies A, which is thus +also non zero (and thus nonzero in S, as we claimed). Note also that, as a side consequence of this +proof, double negation commutes with κ+-small conjunctions in SetMB +κ . +Assume now that all Mi are κ-saturated. Without loss of generality we can also assume that +κ = δ+ is a successor, since for limit κ the saturated model is a directed colimit of smaller saturated +3 + +models. Let us now prove that M must be κ-closed (whence also κ-saturated). So consider an +embedding f : M +� N; since K is ρ-stable with respect to Galois types over some δ-saturated +submodel P, it is ρ-stable with respect to κ-Boolean types of the same kind, so that an application +of the omitting types theorem from [Esp22] to the κ-Boolean theory of models of size at least κ +containing P, SB +κ , shows that all subobject lattices of (x, ⊤) in Set[SB +κ ]κ, for x a nonempty finite +tuple, are atomic and thus Boolean. On the other hand, note that Set[Sκ]κ is two-valued since +it is a subtopos of Set[Sδ]κ and that this latter is equivalent to the slice SetKδ/φP , where φP +is the κ-geometric existential sentence corresponding to the diagram of P, which is two-valued +since φP is an atom, as every model from Kδ embeds in P. Therefore, the mentioned subobject +lattices in Set[SB +κ ]κ coincide with those in Set[Sκ]κ, and this entails, in particular, that Set[SB +κ ]κ +is two-valued. This readily implies that the colimit coprojections as well as their composition with +f are κ-Boolean, and so we have κ-Boolean embeddings Mi +� N, which induce a cone between +the slices N/KB +≥κ,<λ and Mi/KB +≥κ,<λ. By the universal property of the limit, there is an induced +functor N/KB +≥κ,<λ +� M/KB +≥κ,<λ, which provides a natural transformation [N, −] +� [M, −]. By +Yoneda, this must correspond to a morphism M +� N in KB +≥κ,<λ, and since this must be f, it +follows that f is κ-closed, as we wanted to show. +3 +Categoricity and tameness +We start by showing that in any µ-AEC with amalgamation and no maximal models, categoricity +in a high enough cardinal implies eventual tameness. We consider the same setup of section 8 in +[Esp22], which we reproduce for the sake of convenience. Given a µ-AEC K with L¨owenheim-Skolem +number κ and µ ≤ κ+, following Baldwin-Boney-Vasey, we add a κ+-small arity predicate P whose +interpretation in a model M consists of the image of the underlying structure of a model N of size +κ embedded in M through a morphism in the µ-AEC. This particular expansion, which gives rise +to an isomorphic AEC, has the property that morphisms coincide with substructure embeddings. +Moreover, its models of size at least κ can be axiomatized as follows, extending further the language +with the symbol ⊆: +⊤ ⊢x ∃y + + � +M0∈S +ψM0(y) ∧ x ⊆ y ∧ P(y) + + +⊤ ⊢xy (x ⊆ y ∧ P(x) ∧ P(y)) +� +� +(M0,M1)∈T +ψ(M0,M1)(x, y) +⊤ ⊢xy (P(y) ∧ ψ(M0,M1)(x, y)) +� P(x) +⊤ ⊢xy x ⊆ y ↔ +� +i∈I +� +j∈J +xi = yj +Here S is a skeleton of the subcategory of models of size κ, T is the set of pairs (M0, M1) with +a morphism in the µ-AEC and M0, M1 ∈ S, while ψM0, ψM0,M1 are conjunctions of atomic and +negated atomic formulas of the extended language such that ψM0(z) holds if and only if z is +isomorphic to M0, and ψM0,M1(z, w) holds if and only if (z, w) is isomorphic to (M0, M1). +Assuming now categoricity at κ, we can get an axiomatization of an isomorphic µ-AEC which +can be entirely rewritten through sequents in the (2κ)+-Reg¬ fragment. This is an intuitionis- +tic fragment of first-order logic which contains no disjunctions, obtained from the (2κ)+-regular +fragment by adding ⊥, together with the axioms ⊥ ⊢x φ and the axioms for ¬ that make it into +a negation operator. Indeed, in the first sequent above the disjunction reduces to a single dis- +junct since we have categoricity at κ, while the last three sequents above have the general form +4 + +of universal sentences ∀z � +i∈I +� +j∈J ψij, and each such sentence is equivalent to the set of sequents +{∃z � +i∈I ¬ψif(i) ⊢ ⊥}f∈JI. +The (2κ)+-Reg¬ fragment contains the (2κ)+-Reg⊥ subfragment, not containing the symbol ¬. +The syntactic category C of any (2κ)+-Reg¬ theory can be studied through the category Kr +≥(2κ)+ +of its (2κ)+-Reg⊥ models (models of the (2κ)+-Reg⊥ internal theory of C, also known as the (2κ)+- +Reg⊥ Morleyization of the (2κ)+-Reg¬ theory). These latter are in particular (2κ)+-regular models +for the extended signature in which there is an extra propositional symbol ⊥ and one predicate +symbol S for each negated atomic formula ¬R and where the axioms of the theory contain all +axioms obtained from formally replacing ¬R by S in each (2κ)+-Reg¬ axiom and, additionally, all +those axioms of the form ⊥ ⊢x φ and R ∧ S ⊢x ⊥. +If (C)r +λ+ is the syntactic category of the λ+-Reg⊥ theory with the same axioms as the (2κ)+-Reg⊥ +theory of C, then its λ+-classifying topos Sh((C)r +λ+, τ) (where τ is the λ+-Reg⊥ coverage) will be +precisely equivalent to the presheaf topos Set +Kr +≥(2κ)+,≤λ, as can be seen as a special case of Theorem +4.1 from [Esp22] when λ is big enough. In particular, the embedding (C)r +λ+ +� Set +Kr +≥(2κ)+,≤λ will +preserve ¬ since it can be identified with Yoneda embedding, which preserves any right adjoint to +pullback functors that might exist, see [BJ98]). +Using the compactness of (2κ)+-Reg⊥ logic, it is also easy to verify that the canonical functor +F : C +�(C)r +λ+ also preserves ¬. For if given a λ+-regular sentence ∃x � +i<λ φi we have ∃y � +i<λ φi ∧ +R ⊢x ⊥ in λ+-Reg⊥ logic, there must be a (2κ)+-regular sentence ∃i∈T yi +� +i∈T φi, for some subset +T ⊂ λ of size at most 2κ, such that ∃i∈T yi +� +i∈T φi ∧ R ⊢x ⊥ in (2κ)+-Reg⊥ logic, from which our +result follows. +It follows, in fact, that the evaluation functor ev : C +� Set +Kr +≥(2κ)+,≤λ, the composite of Yoneda +embedding with F, preserves ¬,1 which in particular means that the interpretation of S in the +presheaf topos will be precisely that of ¬R. Note that, if we add to the (2κ)+-Reg¬ axiomatiza- +tion above all instances of excluded middle for atomic formulas, we get an axiomatization of (an +isomorphic copy of) the µ-AEC, a fact which we will use in the following: +Theorem 3.1. Let K be a µ-AEC with directed colimits which is categorical in κ and in λ > 2κ. +Then K is (2κ, < λ)-tame. +Proof. Note first that we can take the model M of size λ as a monster model for K≥(2κ)+,<λ, since +by the arguments of [Esp22], we have amalgamation there. Now Galois types in M correspond +to λ+-geometric syntactic types, as shown in [Esp22] (indeed, Galois types over M0 correspond to +syntactic types containing the complete formula that realizes the type of the tuple given by the +underlying set of M0). Thus, it is enough to show that a λ-coherent existential sentence of the form +∃xφ(x, d, c), with constants c from the submodel M0, the set of parameters of the type, where +d is a finite tuple and where φ is a conjunction of atomic formulas, holds in M if (and only if) +every (2κ)+-small approximation ∃x′ψ(x′, d, c′) holds there. So suppose this latter condition holds. +Let N be a (2κ)+-pure submodel containing c and d, and consider the following theory: to the +(2κ)+-Reg⊥ Morleyization of the sequents in Reg¬ logic that axiomatize K, add the diagram of +N, sequents expressing those negated existential sentences with constants from N holding there, +and sequents expressing that the (2κ)+-small approximations ∃x′ψ(x′, d, c′) hold. Clearly, every +(2κ)+-small subset has a model (the obvious expansion of the monster model) and so the whole +theory has a (2κ)+-Reg⊥ model. This means that there is a (2κ)+-pure embedding of N +� S into +a (2κ)+-Reg⊥ model of ∃xφ(x, c). By a similar proof to that of Grossberg conjecture in [Esp22], +we can see that M is injective with respect to (2κ)+-pure embeddings in Kr +≥(2κ)+,<λ, and thus we +get that ∃xφ(x, d, c) must hold there, as we wanted to prove. +1It is also possible to give a direct proof of this fact, using the compactness of (2κ)+-Reg⊥ logic, with the same +arguments as in the proof of Joyal’s theorem, according to which ev : C +�SetMod(C) preserves universal quantification +when Mod(C) is the category of coherent models of the Heyting category C. This is worked out in the author PhD +thesis for the more general disjunction-free fragment. +5 + +Assuming categoricity in a sufficiently large initial segment, we can also derive tameness: +Theorem 3.2. Let K be a µ-AEC with directed colimits which is µ-categorical for κ ≤ µ < ℶω(κ). +Then K is 2κ-tame. In particular, categoricity in [κ, ℶω(κ)) implies categoricity everywhere above +κ. +Proof. Note first that our whole analysis before Theorem 3.1 could be upgraded to the case in +which we know that we have categoricity in µ for κ ≤ µ < χ := ℶω(κ). In this case, it is possible +to have an axiomatization in χ+-Reg¬ logic by adding a µ+-arity predicate Pµ for each µ < χ and +proceeding similarly to the axiomatization above. Let Tr be the χ+-Reg⊥ Morleyization of the the +following theory in the disjunction-free fragment: to the χ+-Reg¬ axiomatization of the models of +size at least χ, we add all instances of the axioms � +i<χ ¬¬φi ⊢x ¬¬ � +i<χ φi, where φi are < χ-Reg⊥ +formulas. Let also KB,r +≥χ,<λ be its category of χ+-Reg⊥ models of size at least χ and less than λ with +χ-Boolean homomorphisms. It is enough to prove that the subtopos SetK≥χ,<λ ֒→ SetKB,r +≥χ,<λ is +dense. As in the proof of Grossberg conjecture from [Esp22], we know similarly that this subtopos +is obtained by adding the λ-topology generated by instances of excluded middle, so we just need to +prove that the sequents � +i<λ ¬¬φi ⊢x ¬¬ � +i<λ φi hold in SetKB,r +≥χ,<λ, for which it is in turn enough +to prove that KB,r +≥χ,<λ has directed bounds. Since it has bounds of chains of cofinality at least χ+, +it is enough to consider chains of cofinality less than χ+. So let {Mi}i<α be such a chain; we need +to prove that there is a model containing the union of the diagrams in the chain. By χ+-Reg⊥ +compactness, we can assume without loss of generality that all models in the chain have size at most +χ, in which case it is enough to take a χ+-saturated model as a directed bound.2 To show such a +model exists, we will prove in the next paragraph that KB,r +χ +has amalgamation at χ. On the other +hand, note that the double negation subtopos of SetKB,r +χ +satisfies all sequents in Tr due to axioms +added to the axiomatization, and that it has a χ+-point since it χ+-classifies a χ+-Reg⊥ theory: +it is obtained as the quotient of Tr by all sequents expressing that morphisms [M, −] +� [M′, −] +are epimorphisms. Thus, if M is a χ+-point of the double negation subtopos, i.e., a χ+-saturated +model of Tr, then M is clearly a directed bound for the given chain, as we wanted. +It remains to prove our claim. +This can be seen through an argument using 3-amalgamation +of models of Tr +n := Tr ∩ Lℶn(κ)+ of size ℶn(κ)+ < χ and ℶn(κ)+-pure morphisms. +First, 3- +amalgamation follows by taking as the amalgam at each level ℶn(κ)+ a weakly initial model of +the pushout, in the doctrine of ℶn(κ)+-coherent categories, of the ℶn(κ)+ theories of the models +that constitute the amalgamation diagram, which must be consistent since amalgamation holds. +Since the ℶn(κ)+-theory of each pushout is axiomatized by ℶn(κ)+-Reg⊥ axioms, they have weakly +initial models Mn. +Let us see that we can choose the homomorphisms between them to form +a directed diagram whose colimit diagram D is consistent with Tr. Indeed, the diagram of the +smallest weak initial model is consistent with Tr and so it has a model M; it suffices then to add to +each ℶn(κ)+-theory of the pushouts also the diagrams of the weakly initial models of size ℶn−1(κ)+ +that it contains: this provides a canonical homomorphism and we can use again weak initiality of +these and successively embed them into M. Whence M is the desired amalgam. This concludes +the proof. +4 +Shelah categoricity conjecture for AEC’s +We now get to the following: +Theorem 4.1. (Shelah categoricity conjecture for AEC’s). Let K be an AEC. If K is categorical +in some λ ≥ ℶ(2LS(K))+, then K is λ′-categorical for every λ′ ≥ ℶ(2LS(K))+. +2If the chain has cofinality bigger than ω, at limit levels we use the existence of weakly initial models of the +χ+-Reg⊥ theory of the union of models below that level. +6 + +Proof. Assume that the AEC is λ-categorical for some λ ≥ ℶ(2LS(K))+. Consider the double negation +subtopos of SetKκ, corresponding to the dense topology, when κ = LS(K). The first observation +is that the dense topology is generated by those covers {M +� Ni}, where the Ni have the same +cardinality of M, such that any M +� N, for an arbitrarily large model N, can be amalgamated +with one M +� Nj. Indeed, this choice of morphisms is enough to contain all covers {M +� Ni}i∈I +where, for some partition I = J ∪ K, {[Nj, −] ֌ [M, −]}j∈J corresponds to existential ν-coherent +sentences whose union is φ and {[Nk, −] ֌ [M, −]}k∈K corresponds to those whose union is ¬φ. +This guarantees that the sheaf topos is Boolean (whence, it is the double negation subtopos). +Clearly the model of size λ must be one of its points, so there is a type in the Boolean algebra +of subterminal objects corresponding to the sentences which hold in the model of size λ. Since +it is also Boolean, it is atomic, and thus the type is isolated by a subobject S ֌ 1. It follows +that the slice over S is a two-valued Boolean topos all of whose κ+-points are L∞,κ-elementarily +equivalent; moreover, since there is a κ+-saturated model of size κ+ by stability (which holds from +categoricity at λ), and any such model must be a κ+-point of the two-valued Boolean subtopos, +any two such κ-points are isomorphic. This means that the Lκ++,κ+-theory κ+-classified by the +slice subtopos defines a κ+-AEC categorical at κ+ and λ. By Lemma 2.1 it must be closed under +directed colimits and is, thus, an AEC. Indeed, we do not need to use amalgamation, since κ+- +closed models are precisely the κ+-points of the double negation subtopos, and if they satisfy in +addition the same sentences of the κ+-saturated model they factor through the slice topos and are +thus κ+-saturated. As a consequence of Theorems 3.1 and 3.2, it is χ-tame for χ = 2κ and thus +it is categorical in a proper class of cardinals. By the same arguments from [Esp22], we conclude +that we have categoricity everywhere above κ+. By Morley’s omitting types theorem for AEC’s3, +we conclude that the every model of the AEC is κ+-saturated above some χ < ℶ(2LS(K))+. Our +theorem follows. +The threshold cardinal is best possible, since by one of Shelah’s example (see e.g. Fact 9.8 in +[Vas19]) for any α < (2LS(K))+ there are AEC’s with models of size up to ℶα(LS(K)) but no bigger +models. This allows to construct AEC’s that are eventually categorical but non categorical above +ℶα(LS(K)). The threshold in the special case of Lκ+,ω has been conjectured by Shelah and appears +also in his work from the nineties. +5 +An infinitary extension of Morley’s categoricity theorem +We can now proceed with the: +Proof of Theorem 1.1: Assume we have an accessible category with directed colimits K which is +categorical for some λ ≥ ℶ(2κ)+ in S. By Theorem 4.5 in [BR12], K is a reflective subcategory of a +finitely accessible category, K′, which must be equivalent to the category of models of a ω-geometric +theory; without loss of generality we can restrict our accessible category to monomorphisms and +so we can assume the morphisms of K′ are embeddings. Both categories are axiomatizable in µ- +coherent logic for µ = LS(K)+. Let λ be the first cardinal in S bigger than LS(K); it follows that +the inclusion i : K≥µ,<λ ֒→ K′ +≥µ,<λ has a left adjoint π. Note that π is computed by considering +the expression of M as a directed colimit of finitely presentable models and setting π(M) to be +the directed colimit in K of the same diagram. It further follows that i∗ : SetK′ +≥µ,<λ +� SetK≥µ,<λ +is the inverse image of a geometric morphism which is left adjoint to π∗.4. To see this, we need +to check that there is an isomorphism [i∗(F), G] ∼= [F, π∗(G)] natural in F and G, and since i∗ +preserves colimits, it is enough to take F ∼= [A, −] a representable. Then an element of [i∗(F), G] +is a compatible family of functions [A, i(N)] +� G(N) indexed by K-models N, or, what amounts +3This is an application of Morley’s method to the AEC to prove the analogous version of Proposition 3.3 in [SM90]. +4Incidentally, that i∗ is right adjoint to π∗ is also true since it follows from the adjunction π ⊣ i. +7 + +to the same thing, a compatible family [A, π(M)] +� G(π(M)) indexed by K′-models M. But by +naturality this latter family determines completely a compatible family [A, M] +� G(π(M)) (given +that π(π(M)) = π(M)), which is the same as an element in [F, π∗(G)]. Note as well that π∗ must be +an embedding which is clearly dense, and so both presheaf toposes will have the same λ-saturated +models. +Consider now the κ+-AEC K′′ consisting of κ+-saturated models. By Lemma 2.1, we can see that +K′′ inherits the concrete directed colimits of K′. Also, since Galois types over submodels of size +µ correspond to µ+-geometric types containing the complete formula satisfied by the underlying +set of the submodel, we can use the same proof idea of Proposition 3.5 from [SM90] to derive +stability from categoricity of K in λ, for which it is enough to note that the usual Ehrenfeucht- +Mostowski model M with λ indiscernibles, which is a directed colimit (in K) of those with finitely +many indiscernibles (which are as well models of K) omits a type over some submodel if and only +if π(M) omits such type. Indeed, the directed colimit (in K′′) must embed into each one of the +models which is a Skolem hull of the directed union of models, and thus a realization of the type +there would entail a realization of the type in all possible Skolem hulls, whence by the transfinite +transitivity rule up to double negation the type would be already realized in the directed union (i.e. +in M).5 Finally, by Theorem 4.1 applied to the AEC K′′ we get thus categoricity in every λ ≥ κ+, +and since by Morley’s omitting types theorem applied to K′ we have that all models of size at least +ν for some ν < ℶ(2κ)+ belong to K′′, we get in particular that K is categorical in those cardinals in +S above ℶ(2κ)+, as we wanted. +Finally, we consider the case in which the sentence defining the category of models is a compact +sentence, and we consider those embeddings that are first-order elementary. This is a µ-AEC with +directed colimits, amalgamation and no maximal models. The corresponding finitely accessible +category K′ will also have, then, amalgamation and no maximal models. As we have proven above, +K′ will be eventually categorical. Since by the considerations in the proof of Lemma 2.1 we have +a stable surjection SetM/KB +≥κ,<λ ։ SetM/K≥κ,<λ, each model M of K′ has a proper ω-pure (in +fact, ω-Boolean) extension, as otherwise SetM/KB +≥κ,<λ would be two-valued and Boolean, forcing +SetM/K≥κ,<λ to be two-valued and Boolean, which is impossible since M is not maximal. It follows +that each model has a ω-pure morphism into the model of size λ, and thus every such model is +ω-saturated; in particular, ω-coherent formulas are either 0 or 1. By the arguments in section 9 +of [Esp22], any sentence of the form ∀x(ϑ +� η), where ϑ and η are ω-coherent, which is valid in +the LS(K)+-saturated model is provable in the LS(K)+-classifying topos of models of size at least +LS(K)+ (note that ∀x(ϑ +� η) will also be equivalent to a ω-coherent formula, as can be seen by +compactness arguments using the definition of universal quantification in the syntactic category). +This readily implies, as can be seen by the proof in [Kei14] for conjunctive formulas in saturated +models, that any ω1-coherent formula is equivalent to the conjunction of their approximations (in +particular, ω1-coherent sentences are either 0 or 1). This allows thus to prove that now sentences +of the form ∀x(ϑ +� η), where ϑ and η are ω1-coherent, which are valid in the LS(K)+-saturated +model is provable in the LS(K)+-classifying topos. Continuing with this process, we finally get +that all LS(K)+-coherent sentences are either 0 or 1, which is enough to get categoricity at LS(K)+ +and hence everywhere above LS(K)+. This finishes the proof. +6 +Classification of categoricity spectra +We end with: +Theorem 6.1. Let K be a large κ-accessible category with directed colimits. Assume the Singular +Cardinal Hypothesis SCH (only if the restriction to monomorphisms is not an AEC). Then the +5Since our theories and types are λ-coherent, we can eliminate the double negation by conservativity of the classical +fragment over the coherent fragment +8 + +categoricity spectrum Cat(K) = {λ ≥ κ : K is λ-categorical} is one of the following: +1. Cat(K) = ∅. +2. Cat(K) = [α, β] for some α, β ∈ [κ, ℶω(κ)). +3. Cat(K) = [χ, ∞) for some χ ∈ [κ, ℶ(2κ)+). +Proof. Our proof idea shares the same guidelines as the one in [Vas19], except that the amalgama- +tion hypothesis is not used and WGCH is replaced with SCH or, in AEC’s, eliminated completely +(all this at the price of replacing κ+ω with ℶω(κ)). If case 1 does not occur, suppose first there +is some categoricity cardinal λ ≥ ℶω(κ), and proceed as in the proof of Theorem 4.1 to define +a κ+-AEC with directed colimits categorical in κ+ and λ. By Theorems 3.1 and 3.2, it must be +2κ+-tame and so must be the original category of models of φ, which allows us to conclude that +case 3 occurs. If categoricity occurs only below ℶω(κ), it must be a segment as proven in [Esp22]. +Indeed, we do not need amalgamation by using the same argument as in the proof of Theorem +4.1. On the other hand, the assumption of maximal models in the proof of Theorem 9.1 in [Esp22] +can be dropped by building the tree of theories so that those theories Γ that admit a maximal +model are not extended to Γ′ by adding a new constant and so they skip this modification of the +construction; in the end, in the tree of theories, branches reach to the level λ or they reach a node +in which the theory is classical (corresponding to a maximal model). In either case, instances of +excluded middle can be shown to hold at the topos Set[Tκ+]κ+, making it Boolean. To sum up, +categoricity cannot alternate, which leaves us in the case 2 and thus finishes the proof. +Examples culled from the literature on AEC’s show that each of the three cases in the classification +can indeed occur (see e.g. the examples of [Vas19]). The non-trivial case is 2, which occurs in the +Shelah-Villaveces example in [SV04]. +References +[BJ98] +Carsten Butz and Peter Johnstone, Classifying toposes for first-order theories, Annals of +Pure and Applied Logic 91 (1998), no. 1, 33–58. +[BR12] +Tibor Beke and Jir´ı Rosick`y, Abstract elementary classes and accessible categories, Annals +of Pure and Applied Logic 163 (2012), no. 12, 2008–2017. +[Esp22] +Christian Esp´ındola, A proof of Shelah’s eventual categoricity conjecture and an extension +to accessible categories with directed colimits, https://arxiv.org/abs/1906.09169 (2022). +[Kei14] +Jerome Keisler, Finite approximations of infinitely long formulas, The theory of models, +Elsevier, 2014, pp. 158–169. +[LRV19] Michael Lieberman, Jiˇr´ı Rosick´y, and Sebastien Vasey, Internal sizes in µ-abstract ele- +mentary classes, Journal of Pure and Applied Algebra 223 (2019), no. 10, 4560–4582. +[She09] +Saharon Shelah, Classification theory for abstract elementary classes, Studies in Logic: +Mathematical logic and foundations, vol. 18, College Publications, 2009. +[SM90] +Saharon Shelah and Michael Makkai, Categoricity of theories in Lκ,ω, with κ a compact +cardinal, Annals of Pure and Applied Logic 47 (1990), no. 1, 41–97. +[SV04] +Saharon Shelah and Andr´es Villaveces, Categoricity may fail late, Preprint. URL: +http://arxiv. org/abs/math/0404258v1 (2004). +[Vas19] +Sebastien Vasey, The categoricity spectrum of large abstract elementary classes, Selecta +Mathematica 25 (2019), 1–51. +9 + diff --git a/cNFPT4oBgHgl3EQfxTXG/content/tmp_files/load_file.txt b/cNFPT4oBgHgl3EQfxTXG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..79f6d783b3febc9c5676516dc8e6641aedca3c7c --- /dev/null +++ b/cNFPT4oBgHgl3EQfxTXG/content/tmp_files/load_file.txt @@ -0,0 +1,299 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf,len=298 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='13167v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='LO] 30 Jan 2023 A complete classification of categoricity spectra of accessible categories with directed colimits Christian Esp´ındola Abstract We provide a complete classification of all the possible categoricity spectra, in terms of internal size, that can appear in a large accessible category with directed colimits, assuming the Singular Cardinal Hypothesis (SCH), and providing as well explicit threshold cardinals for eventual categoricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This includes as a particular case the first complete classification of categoricity spectra of abstract elementary classes (AEC’s) entirely in ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' More specifically, we have: Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let K be a large κ-accessible category with directed colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Assume the Singular Cardinal Hypothesis SCH (only if the restriction to monomorphisms is not an AEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then the categoricity spectrum Cat(K) = {λ ≥ κ : K is λ-categorical} is one of the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Cat(K) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Cat(K) = [α, β] for some α, β ∈ [κ, ℶω(κ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Cat(K) = [χ, ∞) for some χ ∈ [κ, ℶ(2κ)+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This solves in particular Shelah categoricity conjecture for AEC’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' There are examples of each of the three cases of the classification, showing that they indeed occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 1 Introduction This short paper is a sequel to the work of the author in [Esp22] in which a generalization of Shelah’s eventual categoricity conjecture (Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='2 in the introduction of [She09]) is proven in the more general context of accessible categories with directed colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' When all morphisms are monomorphisms in such categories of models, an analogous form of Shelah’s presentation theorem exhibits them as a projective class of an infinite quantifier logic, for which even the Hanf number for model existence has no known explicit bound in ZFC (the only known bound is a strongly compact cardinal, and in fact in some models of ZFC the Hanf number for Lω1,ω1 exceeds the first measurable cardinal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' In the special case of those accessible categories which have directed colimits, however, we will prove that one can find ZFC bounds for the Hanf number of model existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Grossberg has emphasized the importance of also having explicit threshold cardinals for the eventual categoricity phenomenon, and we now intend to use the same setup and results of [Esp22], together with Morley’s method, to provide the provably best possible explicit thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Shelah categoricity conjecture asks to prove, in ZFC, that the threshold for eventual categoricity in an AEC K is ℶ(2LS(K))+ (see Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='3 b) in the introduction of [She09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' We will prove this conjecture here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' An example from Shelah mentioned in [Vas19] shows that this threshold is best possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Assuming SCH, we are also going to provide a proof of a direct generalization of Shelah cate- goricity conjecture to the more general context of accessible categories with directed colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' If K is such a category, we show that such that if K is categorical in some in some λ ≥ ℶ(2LS(K))+ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=', it has only one object of some high enough internal size up to isomorphism), then K is λ′-categorical for every λ′ ≥ ℶ(2LS(K))+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' When considering cardinalities of models of infinitary theories T of Lκ,θ 1 that axiomatize K, the result implies, under SCH, the following infinitary version of Morley’s categoricity theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' (Morley’s categoricity theorem for infinitary theories) Let φ be a Lκ,θ sentence whose category of models and Lκ,θ-elementary embeddings has directed colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let S be the class of cardinals λ which are of cofinality at least θ but are not successors of cardinals of cofinality less than θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Assume the weakening of the Singular Cardinal Hypothesis SCHθ,≥2<θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then, if φ is λ-categorical for some λ ≥ ℶ(2κ)+ in S, then φ is λ′-categorical for every λ′ ≥ ℶ(2κ)+ in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Moreover: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' if the directed colimits are concrete, we can spare the assumption SCHθ,≥2<θ and take S as the class of all cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' if φ is compact and the morphisms of our category are Lω,ω-elementary embeddings, we can replace ℶ(2κ)+ with κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Here SCHθ,≥2<θ is defined as “for all µ ≥ 2<θ there is a set of cardinals λi ≤ µ unbounded below µ such that, for each i, ν<θ ≤ λi for all ν < λi”, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='3 of [LRV19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Also, we know from [Esp22] that there are examples showing that the exceptions in the class S are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The case θ = ω in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 is Shelah categoricity conjecture for Lκ,ω, since in this case SCHω,≥2<ω is provable in ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' When the directed colimits are concrete, since we restrict to monomorphisms, the result is Shelah categoricity conjecture for AEC’s, since SCHθ,≥2<θ can also be removed by the methods of [Esp22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The case when φ is compact (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=', when it has the property that φ is consistent with an arbitrary set of first-order finitary formulas if and only if it is consistent with each of its finite subsets, see [Kei14]) is precisely a proper generalization of Morley’s categoricity theorem (for countable first-order theories) and of Shelah’s categoricity theorem (for uncountable first-order theories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Compact sentences of infinitary logic forms a much wider class than these two particular cases, since they include (but are not limited to) all those conjunctive sentences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=', sentences where only conjunctions are infinitary but disjunctions are finitary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Thus, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 is a vast generalization of those conjectures and results to the realm of infinite quantifier theories and provides new proofs of the known theorems for finitary first-order theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' As it turns out, the existence of directed colimits is what allows for a smooth classification theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The main tool for this will be Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='2, which at the same time extends work of Shelah for Lω1,ω showing, under the Weak Generalized Continuum Hypothesis (WGCH) that categoricity in the first ω cardinals implies categoricity everywhere (see [She09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' We remove here the set- theoretic hypothesis and generalize this result to AEC’s, at the price of asking for categoricity in the first ℶω cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By the example of Shelah and Villaveces in [SV04], this seems to be close to optimal, since they showed that categoricity can fail above ℶn(λ) for any n ∈ ω while holding at the first n cardinals above the L¨owenheim-Skolem number λ (though it is open whether categoricity holds up to ℶn(λ) or the gap could be reduced further).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='2 also uses higher dimensional amalgamation properties, which are shown to be a consequence of categoricity by means of a simple categorical proof, thereby simplifying the methods of [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Finally, we state the classification of categoricity spectra in AEC’s, in ZFC, and assuming SCH also in accessible categories with directed colimits (the set theoretic assumption is needed to guar- antee that the existence spectrum contains an end tail of cardinals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This uses Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1, some weaker versions of which in the context of AEC’s have appeared in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' We give here a categorical proof based, among other things, on a form of Lawvere’s duality for algebraic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The proof extends the result to µ-AEC’s with directed colimits, and greatly simplifies the argu- ments given for AEC’s, in such a way that it can be applied to deduce eventual categoricity without needing to use amalgamation, using instead an observation on the double negation topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 2 As a word of warning, we emphasize that all the methods, results and notation from the authors’ previous paper [Esp22] are assumed here throughout, so the reader is advised to go through that paper first before continuing with this sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 2 Saturation and stability We start by stating the following lemma of independent interest: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let K be a µ-AEC with directed colimits and amalgamation that is ρ-stable for each ρ < κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then κ-saturated models are closed under directed colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Consider the topos Set[TB κ ]λ/[M, −] ∼= SetKB ≥κ,<λ/[M, −] ∼= SetM/KB ≥κ,<λ, (where KB ≥κ,<λ consists of the models in K≥κ,<λ and all its κ-Boolean homomorphisms, and where TB κ is Tκ plus all those instances of excluded middle for κ-coherent formulas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' We have a stable sur- jection SetM/KB ≥κ,<λ ։ SetM/K≥κ,<λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' this can be seen by considering first the stable surjection SetKB ≥κ,<λ ∼= Set[TB κ ]λ ։ Set[Tκ]λ ∼= SetK≥κ,<λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then we consider the pullback functor to the slice SetK≥κ,<λ � Set[Tκ]λ/[M, −] ∼= SetM/K≥κ,<λ, which is a geometric morphism along whose direct image we take the following (pseudo-)pullback: SetKB ≥κ,<λ SetK≥κ,<λ Set[TB κ ]λ/[Mi, −] Set[Tκ]λ/[Mi, −] Set[TB κ ]λ/[M, −] Set[Tκ]λ/[M, −] The (pseudo-)pullback is precisely Set[TB κ ]λ/[M, −] ∼= SetM/KB ≥κ,<λ, as can be verified using the universal property of the slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' More generally, if we have now a sequence of embeddings M0 � M1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' � M with directed colimit M, then SetM/K≥κ,<λ will be the limit of the chain formed by the SetMi/K≥κ,<λ and induced by those embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Since pullbacks preserve limits, this implies that SetM/KB ≥κ,<λ will be the limit of the chain formed by the SetMi/KB ≥κ,<λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' in particular (considering functors from the presheaves to Set preserving limits and colimits), the Cauchy com- pletion of the slice M/KB ≥κ,<λ is the (pseudo-)limit in Cat of the Cauchy completion of the slices Mi/KB ≥κ,<λ (note, in turn, that the Cauchy completions of the slices are equivalent to the slices of the Cauchy completion KB ≥κ,<λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Note that for any κ-small model P and κ-geometric theory Sκ of models of size at least κ contain- ing P, the pullback, in the 2-category of κ+-toposes and κ+-geometric morphisms, of the double negation subtopos S of Set[Sκ]κ+ along the surjection s : Set[SB κ ]κ+ ։ Set[Sκ]κ+ must be S itself, so S embeds into Set[SB κ ]κ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Moreover, the embedding is dense: take a nonzero subterminal object A in Set[SB κ ]κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' then it is nonzero in some model M which, by density, must embed into the model of size κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then, if M is the category of models of S, the κ+-coherent sentence ψ = lim evφi in SetMB κ , where lim evφi ∼= [M, −] in SetMκ, is non zero in SetMsat κ and it implies A, which is thus also non zero (and thus nonzero in S, as we claimed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Note also that, as a side consequence of this proof, double negation commutes with κ+-small conjunctions in SetMB κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Assume now that all Mi are κ-saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Without loss of generality we can also assume that κ = δ+ is a successor, since for limit κ the saturated model is a directed colimit of smaller saturated 3 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let us now prove that M must be κ-closed (whence also κ-saturated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' So consider an embedding f : M � N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' since K is ρ-stable with respect to Galois types over some δ-saturated submodel P, it is ρ-stable with respect to κ-Boolean types of the same kind, so that an application of the omitting types theorem from [Esp22] to the κ-Boolean theory of models of size at least κ containing P, SB κ , shows that all subobject lattices of (x, ⊤) in Set[SB κ ]κ, for x a nonempty finite tuple, are atomic and thus Boolean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' On the other hand, note that Set[Sκ]κ is two-valued since it is a subtopos of Set[Sδ]κ and that this latter is equivalent to the slice SetKδ/φP , where φP is the κ-geometric existential sentence corresponding to the diagram of P, which is two-valued since φP is an atom, as every model from Kδ embeds in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Therefore, the mentioned subobject lattices in Set[SB κ ]κ coincide with those in Set[Sκ]κ, and this entails, in particular, that Set[SB κ ]κ is two-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This readily implies that the colimit coprojections as well as their composition with f are κ-Boolean, and so we have κ-Boolean embeddings Mi � N, which induce a cone between the slices N/KB ≥κ,<λ and Mi/KB ≥κ,<λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By the universal property of the limit, there is an induced functor N/KB ≥κ,<λ � M/KB ≥κ,<λ, which provides a natural transformation [N, −] � [M, −].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By Yoneda, this must correspond to a morphism M � N in KB ≥κ,<λ, and since this must be f, it follows that f is κ-closed, as we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 3 Categoricity and tameness We start by showing that in any µ-AEC with amalgamation and no maximal models, categoricity in a high enough cardinal implies eventual tameness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' We consider the same setup of section 8 in [Esp22], which we reproduce for the sake of convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Given a µ-AEC K with L¨owenheim-Skolem number κ and µ ≤ κ+, following Baldwin-Boney-Vasey, we add a κ+-small arity predicate P whose interpretation in a model M consists of the image of the underlying structure of a model N of size κ embedded in M through a morphism in the µ-AEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This particular expansion, which gives rise to an isomorphic AEC, has the property that morphisms coincide with substructure embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' its models of size at least κ can be axiomatized as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' extending further the language with the symbol ⊆: ⊤ ⊢x ∃y \uf8eb \uf8ed � M0∈S ψM0(y) ∧ x ⊆ y ∧ P(y) \uf8f6 \uf8f8 ⊤ ⊢xy (x ⊆ y ∧ P(x) ∧ P(y)) � � (M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='M1)∈T ψ(M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='M1)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' y) ⊤ ⊢xy (P(y) ∧ ψ(M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='M1)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' y)) � P(x) ⊤ ⊢xy x ⊆ y ↔ � i∈I � j∈J xi = yj Here S is a skeleton of the subcategory of models of size κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' T is the set of pairs (M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' M1) with a morphism in the µ-AEC and M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' M1 ∈ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' while ψM0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' ψM0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='M1 are conjunctions of atomic and negated atomic formulas of the extended language such that ψM0(z) holds if and only if z is isomorphic to M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' and ψM0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='M1(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' w) holds if and only if (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' w) is isomorphic to (M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Assuming now categoricity at κ, we can get an axiomatization of an isomorphic µ-AEC which can be entirely rewritten through sequents in the (2κ)+-Reg¬ fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This is an intuitionis- tic fragment of first-order logic which contains no disjunctions, obtained from the (2κ)+-regular fragment by adding ⊥, together with the axioms ⊥ ⊢x φ and the axioms for ¬ that make it into a negation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Indeed, in the first sequent above the disjunction reduces to a single dis- junct since we have categoricity at κ, while the last three sequents above have the general form 4 of universal sentences ∀z � i∈I � j∈J ψij, and each such sentence is equivalent to the set of sequents {∃z � i∈I ¬ψif(i) ⊢ ⊥}f∈JI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The (2κ)+-Reg¬ fragment contains the (2κ)+-Reg⊥ subfragment, not containing the symbol ¬.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The syntactic category C of any (2κ)+-Reg¬ theory can be studied through the category Kr ≥(2κ)+ of its (2κ)+-Reg⊥ models (models of the (2κ)+-Reg⊥ internal theory of C, also known as the (2κ)+- Reg⊥ Morleyization of the (2κ)+-Reg¬ theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' These latter are in particular (2κ)+-regular models for the extended signature in which there is an extra propositional symbol ⊥ and one predicate symbol S for each negated atomic formula ¬R and where the axioms of the theory contain all axioms obtained from formally replacing ¬R by S in each (2κ)+-Reg¬ axiom and, additionally, all those axioms of the form ⊥ ⊢x φ and R ∧ S ⊢x ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' If (C)r λ+ is the syntactic category of the λ+-Reg⊥ theory with the same axioms as the (2κ)+-Reg⊥ theory of C, then its λ+-classifying topos Sh((C)r λ+, τ) (where τ is the λ+-Reg⊥ coverage) will be precisely equivalent to the presheaf topos Set Kr ≥(2κ)+,≤λ, as can be seen as a special case of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 from [Esp22] when λ is big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' In particular, the embedding (C)r λ+ � Set Kr ≥(2κ)+,≤λ will preserve ¬ since it can be identified with Yoneda embedding, which preserves any right adjoint to pullback functors that might exist, see [BJ98]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Using the compactness of (2κ)+-Reg⊥ logic, it is also easy to verify that the canonical functor F : C �(C)r λ+ also preserves ¬.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' For if given a λ+-regular sentence ∃x � i<λ φi we have ∃y � i<λ φi ∧ R ⊢x ⊥ in λ+-Reg⊥ logic, there must be a (2κ)+-regular sentence ∃i∈T yi � i∈T φi, for some subset T ⊂ λ of size at most 2κ, such that ∃i∈T yi � i∈T φi ∧ R ⊢x ⊥ in (2κ)+-Reg⊥ logic, from which our result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' It follows, in fact, that the evaluation functor ev : C � Set Kr ≥(2κ)+,≤λ, the composite of Yoneda embedding with F, preserves ¬,1 which in particular means that the interpretation of S in the presheaf topos will be precisely that of ¬R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Note that, if we add to the (2κ)+-Reg¬ axiomatiza- tion above all instances of excluded middle for atomic formulas, we get an axiomatization of (an isomorphic copy of) the µ-AEC, a fact which we will use in the following: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let K be a µ-AEC with directed colimits which is categorical in κ and in λ > 2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then K is (2κ, < λ)-tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Note first that we can take the model M of size λ as a monster model for K≥(2κ)+,<λ, since by the arguments of [Esp22], we have amalgamation there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Now Galois types in M correspond to λ+-geometric syntactic types, as shown in [Esp22] (indeed, Galois types over M0 correspond to syntactic types containing the complete formula that realizes the type of the tuple given by the underlying set of M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Thus, it is enough to show that a λ-coherent existential sentence of the form ∃xφ(x, d, c), with constants c from the submodel M0, the set of parameters of the type, where d is a finite tuple and where φ is a conjunction of atomic formulas, holds in M if (and only if) every (2κ)+-small approximation ∃x′ψ(x′, d, c′) holds there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' So suppose this latter condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let N be a (2κ)+-pure submodel containing c and d, and consider the following theory: to the (2κ)+-Reg⊥ Morleyization of the sequents in Reg¬ logic that axiomatize K, add the diagram of N, sequents expressing those negated existential sentences with constants from N holding there, and sequents expressing that the (2κ)+-small approximations ∃x′ψ(x′, d, c′) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Clearly, every (2κ)+-small subset has a model (the obvious expansion of the monster model) and so the whole theory has a (2κ)+-Reg⊥ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This means that there is a (2κ)+-pure embedding of N � S into a (2κ)+-Reg⊥ model of ∃xφ(x, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By a similar proof to that of Grossberg conjecture in [Esp22], we can see that M is injective with respect to (2κ)+-pure embeddings in Kr ≥(2κ)+,<λ, and thus we get that ∃xφ(x, d, c) must hold there, as we wanted to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 1It is also possible to give a direct proof of this fact, using the compactness of (2κ)+-Reg⊥ logic, with the same arguments as in the proof of Joyal’s theorem, according to which ev : C �SetMod(C) preserves universal quantification when Mod(C) is the category of coherent models of the Heyting category C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This is worked out in the author PhD thesis for the more general disjunction-free fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 5 Assuming categoricity in a sufficiently large initial segment, we can also derive tameness: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let K be a µ-AEC with directed colimits which is µ-categorical for κ ≤ µ < ℶω(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then K is 2κ-tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' In particular, categoricity in [κ, ℶω(κ)) implies categoricity everywhere above κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Note first that our whole analysis before Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 could be upgraded to the case in which we know that we have categoricity in µ for κ ≤ µ < χ := ℶω(κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' In this case, it is possible to have an axiomatization in χ+-Reg¬ logic by adding a µ+-arity predicate Pµ for each µ < χ and proceeding similarly to the axiomatization above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let Tr be the χ+-Reg⊥ Morleyization of the the following theory in the disjunction-free fragment: to the χ+-Reg¬ axiomatization of the models of size at least χ, we add all instances of the axioms � i<χ ¬¬φi ⊢x ¬¬ � i<χ φi, where φi are < χ-Reg⊥ formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let also KB,r ≥χ,<λ be its category of χ+-Reg⊥ models of size at least χ and less than λ with χ-Boolean homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' It is enough to prove that the subtopos SetK≥χ,<λ ֒→ SetKB,r ≥χ,<λ is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' As in the proof of Grossberg conjecture from [Esp22], we know similarly that this subtopos is obtained by adding the λ-topology generated by instances of excluded middle, so we just need to prove that the sequents � i<λ ¬¬φi ⊢x ¬¬ � i<λ φi hold in SetKB,r ≥χ,<λ, for which it is in turn enough to prove that KB,r ≥χ,<λ has directed bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Since it has bounds of chains of cofinality at least χ+, it is enough to consider chains of cofinality less than χ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' So let {Mi}i<α be such a chain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' we need to prove that there is a model containing the union of the diagrams in the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By χ+-Reg⊥ compactness, we can assume without loss of generality that all models in the chain have size at most χ, in which case it is enough to take a χ+-saturated model as a directed bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='2 To show such a model exists, we will prove in the next paragraph that KB,r χ has amalgamation at χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' On the other hand, note that the double negation subtopos of SetKB,r χ satisfies all sequents in Tr due to axioms added to the axiomatization, and that it has a χ+-point since it χ+-classifies a χ+-Reg⊥ theory: it is obtained as the quotient of Tr by all sequents expressing that morphisms [M, −] � [M′, −] are epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Thus, if M is a χ+-point of the double negation subtopos, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=', a χ+-saturated model of Tr, then M is clearly a directed bound for the given chain, as we wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' It remains to prove our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This can be seen through an argument using 3-amalgamation of models of Tr n := Tr ∩ Lℶn(κ)+ of size ℶn(κ)+ < χ and ℶn(κ)+-pure morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' First, 3- amalgamation follows by taking as the amalgam at each level ℶn(κ)+ a weakly initial model of the pushout, in the doctrine of ℶn(κ)+-coherent categories, of the ℶn(κ)+ theories of the models that constitute the amalgamation diagram, which must be consistent since amalgamation holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Since the ℶn(κ)+-theory of each pushout is axiomatized by ℶn(κ)+-Reg⊥ axioms, they have weakly initial models Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let us see that we can choose the homomorphisms between them to form a directed diagram whose colimit diagram D is consistent with Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Indeed, the diagram of the smallest weak initial model is consistent with Tr and so it has a model M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' it suffices then to add to each ℶn(κ)+-theory of the pushouts also the diagrams of the weakly initial models of size ℶn−1(κ)+ that it contains: this provides a canonical homomorphism and we can use again weak initiality of these and successively embed them into M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Whence M is the desired amalgam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 4 Shelah categoricity conjecture for AEC’s We now get to the following: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' (Shelah categoricity conjecture for AEC’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let K be an AEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' If K is categorical in some λ ≥ ℶ(2LS(K))+, then K is λ′-categorical for every λ′ ≥ ℶ(2LS(K))+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 2If the chain has cofinality bigger than ω, at limit levels we use the existence of weakly initial models of the χ+-Reg⊥ theory of the union of models below that level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Assume that the AEC is λ-categorical for some λ ≥ ℶ(2LS(K))+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Consider the double negation subtopos of SetKκ, corresponding to the dense topology, when κ = LS(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The first observation is that the dense topology is generated by those covers {M � Ni}, where the Ni have the same cardinality of M, such that any M � N, for an arbitrarily large model N, can be amalgamated with one M � Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Indeed, this choice of morphisms is enough to contain all covers {M � Ni}i∈I where, for some partition I = J ∪ K, {[Nj, −] \u058c [M, −]}j∈J corresponds to existential ν-coherent sentences whose union is φ and {[Nk, −] \u058c [M, −]}k∈K corresponds to those whose union is ¬φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This guarantees that the sheaf topos is Boolean (whence, it is the double negation subtopos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Clearly the model of size λ must be one of its points, so there is a type in the Boolean algebra of subterminal objects corresponding to the sentences which hold in the model of size λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Since it is also Boolean, it is atomic, and thus the type is isolated by a subobject S \u058c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' It follows that the slice over S is a two-valued Boolean topos all of whose κ+-points are L∞,κ-elementarily equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' moreover, since there is a κ+-saturated model of size κ+ by stability (which holds from categoricity at λ), and any such model must be a κ+-point of the two-valued Boolean subtopos, any two such κ-points are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This means that the Lκ++,κ+-theory κ+-classified by the slice subtopos defines a κ+-AEC categorical at κ+ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 it must be closed under directed colimits and is, thus, an AEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Indeed, we do not need to use amalgamation, since κ+- closed models are precisely the κ+-points of the double negation subtopos, and if they satisfy in addition the same sentences of the κ+-saturated model they factor through the slice topos and are thus κ+-saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' As a consequence of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='2, it is χ-tame for χ = 2κ and thus it is categorical in a proper class of cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By the same arguments from [Esp22], we conclude that we have categoricity everywhere above κ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By Morley’s omitting types theorem for AEC’s3, we conclude that the every model of the AEC is κ+-saturated above some χ < ℶ(2LS(K))+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Our theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The threshold cardinal is best possible, since by one of Shelah’s example (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Fact 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='8 in [Vas19]) for any α < (2LS(K))+ there are AEC’s with models of size up to ℶα(LS(K)) but no bigger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This allows to construct AEC’s that are eventually categorical but non categorical above ℶα(LS(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The threshold in the special case of Lκ+,ω has been conjectured by Shelah and appears also in his work from the nineties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 5 An infinitary extension of Morley’s categoricity theorem We can now proceed with the: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1: Assume we have an accessible category with directed colimits K which is categorical for some λ ≥ ℶ(2κ)+ in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='5 in [BR12], K is a reflective subcategory of a finitely accessible category, K′, which must be equivalent to the category of models of a ω-geometric theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' without loss of generality we can restrict our accessible category to monomorphisms and so we can assume the morphisms of K′ are embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Both categories are axiomatizable in µ- coherent logic for µ = LS(K)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let λ be the first cardinal in S bigger than LS(K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' it follows that the inclusion i : K≥µ,<λ ֒→ K′ ≥µ,<λ has a left adjoint π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Note that π is computed by considering the expression of M as a directed colimit of finitely presentable models and setting π(M) to be the directed colimit in K of the same diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' It further follows that i∗ : SetK′ ≥µ,<λ � SetK≥µ,<λ is the inverse image of a geometric morphism which is left adjoint to π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' To see this, we need to check that there is an isomorphism [i∗(F), G] ∼= [F, π∗(G)] natural in F and G, and since i∗ preserves colimits, it is enough to take F ∼= [A, −] a representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then an element of [i∗(F), G] is a compatible family of functions [A, i(N)] � G(N) indexed by K-models N, or, what amounts 3This is an application of Morley’s method to the AEC to prove the analogous version of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='3 in [SM90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 4Incidentally, that i∗ is right adjoint to π∗ is also true since it follows from the adjunction π ⊣ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 7 to the same thing, a compatible family [A, π(M)] � G(π(M)) indexed by K′-models M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' But by naturality this latter family determines completely a compatible family [A, M] � G(π(M)) (given that π(π(M)) = π(M)), which is the same as an element in [F, π∗(G)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Note as well that π∗ must be an embedding which is clearly dense, and so both presheaf toposes will have the same λ-saturated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Consider now the κ+-AEC K′′ consisting of κ+-saturated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1, we can see that K′′ inherits the concrete directed colimits of K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Also, since Galois types over submodels of size µ correspond to µ+-geometric types containing the complete formula satisfied by the underlying set of the submodel, we can use the same proof idea of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='5 from [SM90] to derive stability from categoricity of K in λ, for which it is enough to note that the usual Ehrenfeucht- Mostowski model M with λ indiscernibles, which is a directed colimit (in K) of those with finitely many indiscernibles (which are as well models of K) omits a type over some submodel if and only if π(M) omits such type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Indeed, the directed colimit (in K′′) must embed into each one of the models which is a Skolem hull of the directed union of models, and thus a realization of the type there would entail a realization of the type in all possible Skolem hulls, whence by the transfinite transitivity rule up to double negation the type would be already realized in the directed union (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' in M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='5 Finally, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 applied to the AEC K′′ we get thus categoricity in every λ ≥ κ+, and since by Morley’s omitting types theorem applied to K′ we have that all models of size at least ν for some ν < ℶ(2κ)+ belong to K′′, we get in particular that K is categorical in those cardinals in S above ℶ(2κ)+, as we wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Finally, we consider the case in which the sentence defining the category of models is a compact sentence, and we consider those embeddings that are first-order elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This is a µ-AEC with directed colimits, amalgamation and no maximal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The corresponding finitely accessible category K′ will also have, then, amalgamation and no maximal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' As we have proven above, K′ will be eventually categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Since by the considerations in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 we have a stable surjection SetM/KB ≥κ,<λ ։ SetM/K≥κ,<λ, each model M of K′ has a proper ω-pure (in fact, ω-Boolean) extension, as otherwise SetM/KB ≥κ,<λ would be two-valued and Boolean, forcing SetM/K≥κ,<λ to be two-valued and Boolean, which is impossible since M is not maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' It follows that each model has a ω-pure morphism into the model of size λ, and thus every such model is ω-saturated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' in particular, ω-coherent formulas are either 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By the arguments in section 9 of [Esp22], any sentence of the form ∀x(ϑ � η), where ϑ and η are ω-coherent, which is valid in the LS(K)+-saturated model is provable in the LS(K)+-classifying topos of models of size at least LS(K)+ (note that ∀x(ϑ � η) will also be equivalent to a ω-coherent formula, as can be seen by compactness arguments using the definition of universal quantification in the syntactic category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This readily implies, as can be seen by the proof in [Kei14] for conjunctive formulas in saturated models, that any ω1-coherent formula is equivalent to the conjunction of their approximations (in particular, ω1-coherent sentences are either 0 or 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This allows thus to prove that now sentences of the form ∀x(ϑ � η), where ϑ and η are ω1-coherent, which are valid in the LS(K)+-saturated model is provable in the LS(K)+-classifying topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Continuing with this process, we finally get that all LS(K)+-coherent sentences are either 0 or 1, which is enough to get categoricity at LS(K)+ and hence everywhere above LS(K)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 6 Classification of categoricity spectra We end with: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Let K be a large κ-accessible category with directed colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Assume the Singular Cardinal Hypothesis SCH (only if the restriction to monomorphisms is not an AEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Then the 5Since our theories and types are λ-coherent, we can eliminate the double negation by conservativity of the classical fragment over the coherent fragment 8 categoricity spectrum Cat(K) = {λ ≥ κ : K is λ-categorical} is one of the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Cat(K) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Cat(K) = [α, β] for some α, β ∈ [κ, ℶω(κ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Cat(K) = [χ, ∞) for some χ ∈ [κ, ℶ(2κ)+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Our proof idea shares the same guidelines as the one in [Vas19], except that the amalgama- tion hypothesis is not used and WGCH is replaced with SCH or, in AEC’s, eliminated completely (all this at the price of replacing κ+ω with ℶω(κ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' If case 1 does not occur, suppose first there is some categoricity cardinal λ ≥ ℶω(κ), and proceed as in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 to define a κ+-AEC with directed colimits categorical in κ+ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' By Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='2, it must be 2κ+-tame and so must be the original category of models of φ, which allows us to conclude that case 3 occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' If categoricity occurs only below ℶω(κ), it must be a segment as proven in [Esp22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Indeed, we do not need amalgamation by using the same argument as in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' On the other hand, the assumption of maximal models in the proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='1 in [Esp22] can be dropped by building the tree of theories so that those theories Γ that admit a maximal model are not extended to Γ′ by adding a new constant and so they skip this modification of the construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' in the end, in the tree of theories, branches reach to the level λ or they reach a node in which the theory is classical (corresponding to a maximal model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' In either case, instances of excluded middle can be shown to hold at the topos Set[Tκ+]κ+, making it Boolean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' To sum up, categoricity cannot alternate, which leaves us in the case 2 and thus finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' Examples culled from the literature on AEC’s show that each of the three cases in the classification can indeed occur (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' the examples of [Vas19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' The non-trivial case is 2, which occurs in the Shelah-Villaveces example in [SV04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' References [BJ98] Carsten Butz and Peter Johnstone, Classifying toposes for first-order theories, Annals of Pure and Applied Logic 91 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 1, 33–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' [BR12] Tibor Beke and Jir´ı Rosick`y, Abstract elementary classes and accessible categories, Annals of Pure and Applied Logic 163 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 12, 2008–2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' [Esp22] Christian Esp´ındola, A proof of Shelah’s eventual categoricity conjecture and an extension to accessible categories with directed colimits, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='org/abs/1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content='09169 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' [Kei14] Jerome Keisler, Finite approximations of infinitely long formulas, The theory of models, Elsevier, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 158–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' [LRV19] Michael Lieberman, Jiˇr´ı Rosick´y, and Sebastien Vasey, Internal sizes in µ-abstract ele- mentary classes, Journal of Pure and Applied Algebra 223 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 10, 4560–4582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' [She09] Saharon Shelah, Classification theory for abstract elementary classes, Studies in Logic: Mathematical logic and foundations, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 18, College Publications, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' [SM90] Saharon Shelah and Michael Makkai, Categoricity of theories in Lκ,ω, with κ a compact cardinal, Annals of Pure and Applied Logic 47 (1990), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 1, 41–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' [SV04] Saharon Shelah and Andr´es Villaveces, Categoricity may fail late, Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' URL: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' org/abs/math/0404258v1 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' [Vas19] Sebastien Vasey, The categoricity spectrum of large abstract elementary classes, Selecta Mathematica 25 (2019), 1–51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFPT4oBgHgl3EQfxTXG/content/2301.13167v1.pdf'} diff --git a/ctFAT4oBgHgl3EQfYR3c/content/tmp_files/2301.08539v1.pdf.txt b/ctFAT4oBgHgl3EQfYR3c/content/tmp_files/2301.08539v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..20b60dbee201c22777c60f060325703bb1f3f566 --- /dev/null +++ b/ctFAT4oBgHgl3EQfYR3c/content/tmp_files/2301.08539v1.pdf.txt @@ -0,0 +1,1323 @@ +1 + +Temperature-dependent magnetic particle imaging with multi-harmonic lock-in detection +Thinh Q. Bui, Mark-Alexander Henn, Weston L. Tew, Megan A. Catterton, Solomon I. Woods +T.Q.B., W.L.T, M.A.C, S.I.W are with the National Institute of Standards and Technology, Gaithersburg, +MD 20895 USA (e-mail:thinh.bui@nist.gov). +M.A.H is an associate with the National Institute of Standards and Technology, Gaithersburg, MD 20895 +USA, and University of Maryland, College Park, MD, USA 20742. + +ABSTRACT +Advances in instrumentation and tracer materials are still required to enable sensitive and accurate 3D +temperature monitoring by magnetic particle imaging. We have developed a magnetic particle imaging +instrument to observe temperature variations using MPI, and discuss resolution dependence on +temperature and harmonic number. Furthermore, we present a low noise approach using lock-in +detection for temperature measurement, and discuss implications for a new detection modality of MPI. + +I. Introduction +Magnetic particle imaging relies on measurement of the magnetization of magnetic +nanoparticles (MNP) driven by a strong AC magnetic field [1]-[10]. At NIST, our efforts are directed at +establishing magnetic nanoparticles as an SI-traceable nano-thermometer standard for in situ thermal +imaging, and relying on magnetic particle imaging (MPI) as a core technology. Potential prospects for +theranostic applications of temperature sensing [11]-[20] and hyperthermia [16]-[20] based on MPI have +been proposed previously. The quantitative nature of thermal imaging with magnetic nanoparticles is +demanding and requires thorough investigation of technical noise and drifts in the measurement +system. Yet, a systematic evaluation of temperature on the sensitivity and spatial resolution of MPI +remains an unresolved problem. +For MPI, the magnetization response of MNPs driven into saturation by a sinusoidal excitation +field are detected by inductive coil sensors. Typical MPI systems record the magnetization response in +the time domain with a digitizer, and the harmonic components are obtained by digital Fourier +transform. While this procedure has the advantage of detecting a broadband harmonic signal, often +exceeding 1 MHz, the broad bandwidth carries along with it excess broadband noise. Previously, a +narrowband method using intermodulation frequency detection (a single harmonic and its +intermodualtion frequency) was introduced to improve MPI noise performance [21]. However, by +singling out individual harmonics, high harmonic information content that is responsible for high +imaging spatial resolution is not available, commonly described as a resolution-bandwidth trade-off +[22]. Instead of purposefully reducing bandwidth, it was later proposed that low noise performance can +be achieved by careful noise-matching inductive coil sensors to pre-amplifiers with a reported 11x gain +in noise performance [23],24], but this approach requires tailor-made transformer-like networks specific +to each MPI system. An additional problem with this strategy is that the complex impedance of the +inductive coil introduces a frequency-dependence of noise that complicates noise matching over too +wide of a bandwidth > 1 MHz. + +2 + +Here, we propose an alternative and more general approach by implementing a massively- +parallel lock-in detection scheme in which many harmonics are measured simultaneously while still +leveraging the narrowband detection of the lock-in technique, resulting in a reduction in measurement +noise by a factor of 100-1000, depending on the desired acquisition speed. In contrast to [21], we are +detecting many harmonics from the particle response in a narrowband fashion, thereby overcoming the +resolution-bandwidth trade-off. Lock-in detection can also reduce the data streaming requirements, +since it does not require saving the full time response. It is a general goal of magnetic imaging to reach +the body noise limit, which can potentially be achieved at a high frequency (MHz range) by detecting at +a high harmonic number and reaching coil resistance (thermal) noise. In this work, we present our multi- +harmonic lock-in detection approach, characterize noise performance, and compare performance of +parallel lock-in detection against the commonly used broadband detection with a digitizer. +In addition to concentration imaging, nanoparticle thermometry has been a longstanding goal of +MPI. Two key approaches for nanoparticle thermometry have been proposed: multi-color, system +matrix-based [13],[25],[26] and multi-harmonic-based measurements [11],[14],[15],27]. The system +matrix approach requires measurements of calibration system matrices to reconstruct both +concentration and temperature, and a linear interpolation step for intermediate temperatures [13]. In +this work we will focus on the multi-harmonic technique, where the temperature can be determined by +measurement of two harmonics and their ratios. Here, we will also present temperature-dependent +MPI, which relies on the multi-harmonic detection, and show that performance is improved by lock-in +detection. Finally, we discuss the observed spatial resolution dependence on harmonic number, which +will have important implications for image reconstruction when temperature is involved. + +II. MPI instrumentation +The MPI instrumentation developed at NIST was designed for thermal imaging and for +thermometric metrology using magnetic nanoparticles. Inspired by our previous temperature-tunable +AC magnetometers [28],[29], the MNP sample holder is machined out of thermally conductive Shapal +ceramic. The temperature of the sample is tuned by water-cooling with a recirculating chiller, and the +temperature is measured by two 100 ohm platinum resistance thermometers. An additional water- +cooling line is used to control the temperature of the excitation solenoid coil to ensure thermal stability +over long acquisition times. The selection field (gradient field) is generated from a pair of permanent +magnets (diameter = 50 mm, height = 50 mm, N52 grade NdFeB) separated by ≈30-40 mm to produce +the field-free-point (FFP). The FFP is fixed in our MPI imager, and spatial scanning is achieved by +mechanical translation of a 3D stage in which the sample holder and excitation coils are mounted. This +image acquisition scheme relies on spatial mechanical step-scanning at a maximum rate of ≈5 Hz, which +makes it more similar to the x-space method [8],[9],[30],[31] that uses low frequency (≈10's Hz) shift +coils rather than the original Lissajous pattern-based electromagnetic shifting with three orthogonal +fields near 25 kHz [1],[6],[32].The transmit chain is a resonant circuit at the excitation frequency 𝑓0 = +31.75 kHz with a second order low-pass filter to attenuate (>50 dB) the higher harmonic (feedthrough) +signals. Magnetization response from AC excitation (10 mT) is recorded with an inductive receive coil, +wound as a first-order gradiometer in series with an additional $\approx-30$ dB notch filter at the +excitation frequency. This output is directly connected to a SR560 pre-amplifier, subsequently +connected in series to an impedance matched, cascaded notch filter stage with ≈60 dB of attenuation at + +3 + +the f0 and ≈40 dB at 2f0. Output from the second notch filter is sent to a final low-pass filter stage (SRS +SIM965). Finally, the filtered magnetization response is directed to either a digitizer (GaGe, 16 bit, 1 +GSa/s) or lock-in amplifier (Zurich HF2LI) to measure the harmonic amplitudes for image reconstruction. +For the digitizer, the harmonic amplitudes are determined from the Fourier transform of the time +domain signal. The multi-frequency capability of the HF2LI lock-in amplifier allows the recording of six +independent harmonics simultaneously in a parallel fashion. For quantitative imaging, we use a glass +phantom with dimensions of 8 x 12 mm (diameter x height). The phantom has four channels (1 mm +diameter, 8 mm height) where MNP samples can be introduced and sealed by epoxy. To demonstrate +the 3D imaging capability of our MPI imager, Fig 1 shows the 3D image of this glass phantom in two, +rotated orientations, with four clearly resolved channels. For a more systematic evaluation of spatial +resolution, we used a 3D printed acrylic phantom shown in Fig. 2. Here, the sets of four channels have +different edge-to-edge hole spacing to evaluate the spatial resolution using Rayleigh criterion. The 2D +image in Fig. 2 reveal that we can resolve 0.25 mm in the x-direction, the direction of the highest +gradient field. These results, along with the measured point-spread-function (PSF) of Vivotrax, indicate +that the magnitude of the gradient field is ≈20 T/m for the x-axis and ≈10 T/m for the y-axis. These +results are consistent with our COMSOL simulations of the permanent magnet pair. +A. Parallel harmonic lock-in MPI +A voltage-based measurement system of MPI comprised of an inductive coil sensor connected +to a pre-amp has SNR given by [21],[23]: +𝑆𝑁𝑅  ∝   +𝑆 ⋅𝑑𝑚 +𝑑𝑡 +√(4𝑘𝐵𝑇𝑅𝐿 + 𝑒𝑛2 +̅̅̅ + 𝑖𝑛2̅ |𝑍𝐿| +2 )⋅ Δ𝑓 + (1) +The quantity in the numerator is the product of the coil sensitivity, $S$, and voltage induced by the +magnetic nanoparticles, dm/dt. The denominator denotes the noise sources from the sensor, an +inductive coil sensor with real (RL) and complex impedance (ZL), and the pre-amp. The pre-amplifier has +a voltage and current noise spectral density (NSD,𝑉/√𝐻𝑧), 𝑒𝑛 +̅̅̅ and 𝑖𝑛̅, respectively. Since Δf is the +detection bandwidth, the best SNR is obtained by minimizing the bandwidth. This ideal scenario is, +however, at odds with the MPI technique in that it is desirable to record all detectable information +content-carrying harmonics. Reducing the sampling rate (detection bandwidth) in the digitizer system +could improve SNR, but at the expense of spatial resolution [22]. Electronic noise matching in the +receive chain has been proposed, but the complex impedance of the inductive coil introduces a +frequency dependence that complicates noise matching strategies over broadband frequencies +[21],[23],[32]. NSD measurements for our signal chain (IC with 2nd order notch filter (Fig. 14, SI)) does +indeed show a complex frequency dependence with increasing noise density at higher (>1 MHz) +frequencies. To overcome the noise matching and bandwidth requirement incompatibility challenges, +we present a lock-in detection method with parallel multi-harmonic capabilities that can improve SNR +by permitting detection at high frequencies without the need for broadband noise matching. +Relying on measurement of only a few harmonics has been proposed for in-situ temperature +measurement [11],[15], and also as a general modality for imaging [9],[24]. Isolated harmonics are most +sensitively measured with lock-in detection, where the noise bandwidth (1 to 100 Hz range) is most ideal +and adjustable based on the desired acquisition speed. With a digitizer, a slower acquisition rate will +reduce broadband noise, but also reduce the detection bandwidth to observe the rich harmonic + +4 + +spectrum of saturated MNPs, which often exceed 1 MHz. For detection of a single harmonic, the +equivalent noise bandwidth Δ flock−in for lock-in detection is set by the lock-in time constant, T, which +typically in our experiment is 10 ms. For a first order RC filter, Δ𝑓𝑙𝑜𝑐𝑘−𝑖𝑛 = +1 +4𝑇, which gives a Δ𝑓𝑙𝑜𝑐𝑘−𝑖𝑛 ≈ +25 Hz. Compared with a 1 MHz bandwidth digitizer acquisition, the SNR enhancement of lock-in +detection relative to digitizer is √106/25 ≈ 200. This analysis shows that multi-harmonic lock-in +measurement can reduce broadband noise associated with the inductive coil detection. Improving MPI +noise performance by electronic noise matching in the receive chain [23],[24],[33],[34] can only further +improve overall performance. +B. Noise performance and SNR of lock-in detection +Despite the superior noise performance of lock-in detection, commercial lock-in amplifiers often +limit to one or two harmonics at a time. Yet, if many of the particle harmonics could be measured in a +parallel fashion, then SNR is optimized without sacrificing spatial information content by filtering higher +harmonics. These considerations motivated us to compare the noise and imaging performance of multi- +harmonic detection and time-domain detection with a digitizer. First, we compared the detection +background noise by analyzing the time domain signal when the receive chain is connected to either the +digitizer or the lock-in amplifier (Fig 13 (SI)). Here, the sampling conditions are set the same for both +instruments, which provides a direct comparison of the analog-to-digital conversion (ADC) noise of the +data acquisition devices. As presented in Fig 13 (SI), the RMS noise (VRMS) of the lock-in amplifier signal is +≈ 3x lower noise than the digitizer for the same acquisition rate +To resolve the different contributions to the total noise, the noise spectral densities were +measured for (1) the lock-in amplifier only and (2) the IC with the notch filter only, and (3) all +components in the receive chain including the pre-amp, as shown in Figure 14 (SI). The 1 MHz +bandwidth of the pre-amp suppresses noise at above 1 MHz, but increases the noise level by a factor of +100 in the range of interest of 30 kHz to 1 MHz. Fig. 14c shows a higher noise-baseline from broadband +detection of 1 MHz. These NSD measurements in Fig. 14b and c were used to calculate the total RMS +noise by taking the integral of NSD2 over the full 1 MHz bandwidth. This calculation agrees with the +measured RMS noise in Fig. 13 (SI). +To validate the enhanced noise performance of multi-harmonic lock-in detection with real +samples, we compared images of the Vivotrax glass phantom used in Fig. 1 acquired with the lock-in +amplifier and with the digitizer. Figure 3 show the 1D image slice across two channels of the glass +phantom. The four panels show a comparison of the 3rd to 9th harmonic signals from the lock-in and +digitizer. The SNR enhancement is most clearly observed in the 7th harmonic, with an improvement of ≈ +3x with the lock-in relative to the digitizer, in reasonable agreement with the predicted noise +improvement of ≈ 200x from broadband noise reduction from 1 MHz (digitizer) to 25 Hz (lock-in). +C. Temperature dependent MPI +Temperature impacts the equilibrium magnetization as well as the dynamics of magnetization, +which will be imprinted on the MPI signal. There have been a number of reported studies on the +temperature dependence of suspended magnetic nanoparticles [11]-[15],[27],[35]-[38], yet quantitative +effects of temperature on MPI signal and spatial resolution is not fully understood. Using the imaging + +5 + +phantom in Fig. 1, we measured the temperature dependence of Vivotrax to assess temperature's +impact on image reconstruction with MPI. +The temperature of the Vivotrax sample was tuned from 278 K to 305 K and allowed to +equilibrate to ± 50 mK prior to imaging. The 2D MPI images for 4 of the 7 temperature steps are plotted +in Fig. 4 as columns, with each 2D image being an average of 3 repeated measurements. It is evident by +visual inspection that the 9th harmonic has a stronger temperature dependence than the 7th harmonic. +The quantitative dependence is obtained from the 1D representation of Fig. 4 versus pixel number, and +is shown in the Fig. 15 (SI) top row plots. For each plot, the overlay of 7 temperatures is shown. To +obtain the magnitude, the integrated areas are calculated for the peaks labeled in Figs. 15a-b (SI) and +are presented in Figs. 15c-d (SI) . The slopes for each harmonic gives the temperature sensitivity [28]. +Since each peak gives a different spatial location of magnetic nanoparticles, the agreement in their +calculated sensitivities gives evidence for the stability of the system over long periods of scanning and +averaging (30 minutes of acquisition time at each temperature, 3 averages). To gauge the short and long +time stability, the Allan variance plots for a single image pixel are displayed in Fig. 16 (SI). In our +instrument, statistical noise can be improved by averaging up to 10 seconds before systematic drifts +degraded noise performance. +Previous temperature-dependent MPS measurement of Vivotrax samples in an 8 mm diameter +sapphire vial show a negative temperature dependence on harmonic amplitude predicted by the +Langevin function [28], which we reproduced with the current MPS instrument [29]. In contrast to the 8 +mm diameter glass vial used, our MPI results for Vivotrax using a glass phantom as a sample vessel +display the opposite (positive) temperature dependence. One possible explanation for this is that the +narrow (1 mm diameter) channel of the phantom induces MNP aggregation, which hinders Brownian +motion. If Brownian motion is blocked and the temperature dependence is dominated by Néel rotation, +the magnetization response displays a positive temperature dependence as reported in the literature +from MPS [37] and MPI [38] studies. We independently verified with MPS that the phantom vessel +displays a positive temperature dependence consistent with our MPI results, while the larger glass vial +shows the opposite trend. Conversely, measurement of Vivotrax in the larger glass vial using both the +MPS and MPI instrumentation showed the same (negative) temperature dependence (Fig. 17, SI). +The imaging spatial resolution dependence on harmonic number and temperature were also +measured for Vivotrax with the glass phantom. 1-D scans across the two channels of the phantom are +presented in Fig. 5 for both the 7th and 9th harmonics. Only a modest change in the FWHM of the peaks +was observed. A more quantitative measurement of the temperature dependence is determined from +the PSF of Vivotrax (Fig. 6). In contrast to measurements in Fig. 5, the PSF in Fig. 6 includes all harmonics +and therefore exhibits significantly higher SNR. By fitting the PSF curves to a Voigt function, the +temperature dependence of the FWHM was obtained (Fig. 6b). A modest slope of 0.041(6) mT/K is +consistent with a small temperature dependence shown in Fig. 5. +For thermal imaging applications using magnetic nanoparticles, measurement of harmonic +ratios, rather than individual harmonic amplitudes, has been proposed to remove the concentration +dependence [11]. We recorded 2D images using individual harmonics and also their ratios, and the data +is presented in Fig. 7. In the case of harmonic ratios, 5th/3rd and 7th/3rd, the higher harmonic +dominates the spatial resolution of the ratio image, which mostly resembles the image of the higher +harmonic number. In general, images obtained with either individual harmonics or harmonic ratios + +6 + +clearly show a harmonic dependence of spatial resolution, and suggests a corresponding point-spread- +function (PSF) dependence on harmonic number. This concept of a harmonic PSF is crucial for this +imaging modality, and will be discussed in detail in the following sections. + +III. Harmonic dependence of spatial resolution +Previous MPI implementations typically utilize the full time signal for image reconstruction, +which obscures the impact of individual harmonics on spatial resolution. The effect of harmonics on the +reconstructed image has been introduced previously [2],[31], mainly in the context of feedthrough (1st +harmonic) suppression. More importantly, these studies reveal a direct mapping of harmonic frequency +space to spatial frequency in the image space for sinusoidal excitation. Our work extends upon these +analyses by experimentally determining the impact of individual harmonics on image reconstruction, a +new concept denoted as the harmonic PSF. To demonstrate the existence of a harmonic PSF, we +acquired MPI images reconstructed using individual harmonics by means of (1) the spectrum of the +digitized time signal and (2) the lock-in amplifier demodulated signal. Using both approaches, the +apparent dependence of resolution on harmonic number was observed for two different MNP samples. +Here, a glass phantom with four channels was used as a sample holder for 70 nm Synomag (Fig. 8) and +for Vivotrax (Fig. 9). Figure 8a shows a strong resolution dependence on harmonic number, as the four +wells become more spatially resolved with increasing harmonic number. To quantitatively analyze the +resolution dependence, an acrylic phantom with two sample wells spaced by 0.5 mm was machined and +imaged in 1D (Fig. 8b). The spatial resolution was determined by fitting the FWHM of the measured +profile, and the result is plotted in Fig. 8c. The resolution enhancement experiences an exponential +improvement at low harmonic number and saturates at higher harmonics. A similar analysis was done +for Vivotrax tracers (Fig. 9) with an observed exponential-like dependence of resolution on harmonic +number. +A. PSF measurement +The spatial resolution of MPI is dependent on the PSF of the magnetic nanoparticles and the +gradient of the selection field [39]. By applying a slow varying (10 Hz) bias field at 80 mTpp superimposed +on the excitation field, the PSF can be determined in a similar fashion to [40]. Different from this PSF +reconstruction procedure that includes all harmonics similar to previous reports, we extended this +concept to measure the PSF of individual harmonics. The PSF of the individual harmonics were resolved +by two different methods: (1) short time Fourier transform (STFT) of the digitized time response signal +and (2) time-swept lock-in detection. For these measurements, the bias field is driven by a triangle +waveform with 𝑓𝑏𝑖𝑎𝑠 = 10 Hz with an amplitude of 40 mTp. +For method (1) using the time signal $f(t)$ from the digitizer, the short time Fourier transform is +implemented by first binning the digitized data in defined window segments 𝑤(𝑡). At each time shift 𝑡, a +new window 𝑤(𝑡′ − 𝑡) is applied and the FT is calculated of 𝑓(𝑡′)𝑤(𝑡′ − 𝑡): +𝑓̂(𝑡, 𝑢) = ∫ +𝑓(𝑡′)𝑤(𝑡′ − 𝑡)𝑒−𝑖2π𝑡′𝑢 +∞ +−∞ +𝑑𝑡′ (2) +This procedure produces a spectrogram, or a plot of the Fourier spectrum at each time segment +window. With the bias field turned on, the harmonic spectrum will only appear when the bias field +amplitude is near zero, that is when the MNP are not fully saturated. For method (2), the harmonic PSF + +7 + +is obtained with the lock-in by implementing a time-domain acquisition of the output at the specified +demodulation frequency (f0, 3f0, 5f0, etc.) to resolve the time-swept PSF at each harmonic, similar to +data in Fig. 6a. +Figure 10 is a step-by-step layout of the STFT procedure. Harmonic PSF data taken with the +digitizer (method 1) and lock-in amplifier (method 2) are plotted in Figs. 11a-b. The general shape of the +PSF obtained using the two independent approaches display qualitative agreement for each harmonic, +and provides experimental justification for the origin of the harmonic dependence of PSF. By inspection, +it can be seen that the harmonic PSF displays a decreasing FWHM (higher spatial resolution) with +increasing harmonic number, and the FWHM also converges to an asymptotic value. This trend explains +our imaging results for nanoparticle samples in Figs. 8 and 9 and the exponential dependence in Figure +8c. +B. Simulation of harmonic dependence of resolution +To obtain further validation and insight into the origin of the observed spatial resolution +dependence on harmonic number, we performed the following simulation study. The magnetization of a +single magnetic nanoparticle is modeled using the Langevin function, i.e., +𝑀⃗⃗ (𝐻⃗⃗ ) = ℒ(𝑘𝐻⃗⃗ ) +𝐻⃗⃗ +||𝐻⃗⃗ || , with 𝑘 = +μ0𝑚 +𝑘𝐵𝑇, (3) +and μ0 being the vacuum permeability, 𝑚 the particle magnetic moment. The moment is a function of +the saturation magnetization 𝑀𝑠, the Boltzmann constant 𝑘𝐵, and the temperature 𝑇. The PSF 𝑆 (𝑡) is +derived by calculating the time-derivative of the Langevin function, Eq. (3) [30]. Finally, the harmonic +dependence of PSF is calculated in the same fashion as the STFT and lock-in approaches. Measured and +simulated harmonic PSF presented in Figs. 11a-b and Figs. 11c-d, respectively, show that the PSF FWHM, +defined by the broad envelope at each harmonic, is narrower with increasing harmonic number, with +nearly an exponential-like drop off in resolution [41]. The simulations also reveal the origin of the +narrower PSF for higher harmonics: the amplitude decreases faster for higher harmonics relative to +lower harmonics for nanoparticles that are increasingly farther away from the FFP region. + +We also simulated the temperature dependence of the PSF to validate the observations in Fig. 6. Note +that due to the explicit temperature dependence of 𝑘 in Eq. 3, the magnetization of a single particle +shows a corresponding temperature dependence, even if the saturation magnetization was set to be +constant. Figure 12 shows the simulated temperature dependence of PSF over the range 250 K to 300 K, +in which only a modest change in the PSF FWHM with temperature (0.043 mT/K) was observed, in good +agreement with the Vivotrax data from Fig. 6. + +Conclusion +Towards the goal of advancing temperature measurement and thermal imaging with magnetic +nanoparticles, this work comprised experimental and theoretical studies on the harmonic dependence +of MPI and measurements of noise performance based on a digitizer and multi-channel lock-in amplifier. +The study shows the advantage of using parallel multi-harmonic lock-in detection as a general data + +8 + +acquisition method with exceptional noise performance and straightforward implementation. However, +scaling up lock-in amplifiers to multiple independent channels is expensive with commercial solutions. A +more cost-effective solution to increase the number of independent lock-in detection channels is to +implement high frequency lock-in with FPGA [42]. This will require developments of lock-in electronics +that are both noise-matched to the receive chain and operate at high frequencies. + +Appendix +A. Numerical Details +Note, that for this study we assumed all quantities from Eq. 3 to be scalars. With the time-dependent +PSF 𝑆(𝑡) evaluated at 𝑁 + 1 different times 𝑡−𝑁/2, 𝑡−𝑁/2−1, … , 𝑡0, … , 𝑡𝑁/2+1, 𝑡𝑁/2, with center point 𝑡0, +that fall within the window segment 𝑤(𝑡0) = [𝑡−𝑁/2, 𝑡𝑁/2], and a triangular shaped window function +𝑊(𝑡) that has its maximum at 𝑡0, we can calculate the PSF's 𝑘-th harmonic through the discrete STFT: +𝑐𝑘 = +1 +𝑁+1 ∑ +𝑊(𝑡𝑖)𝑆(𝑡𝑖) exp(−𝑖2π𝑘𝑓0𝑡𝑖). +𝑁/2 +𝑖=−𝑁/2 + (4) +The lock-in measurement for the 𝑘-th harmonic is simulated by first multiplying the time-dependent PSF +𝑆(𝑡) with a reference signal such that: +𝑆̂(𝑡) = 𝑆(𝑡) exp(−𝑖2π𝑘𝑓0𝑡), (5) +and then taking the discrete Fourier transform of Eq. 5 over a time window 𝑤(𝑡). +The time resolution in all our simulations was set to 10−9 s to find a balance between necessary +accuracy and memory constraints. + +Acknowledgment +We acknowledge Jo Wu for assistance with the glass imaging phantom. We thank Angela Hight Walker, +Cindi Dennis, Michael Donahue, Adam Biacchi, and Frank Abel for helpful discussions. 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An open-source high-frequency +lock-in amplifier. Review of Scientific Instruments, 90(9), 094701. + + + + + +12 + +FIGURES + +Figure 1: Magnetic particle imaging: a) 4-channel glass phantom containing undiluted (5 mg Fe/ml) +Vivotrax tracer at two rotated orientations shown from the top view. The four white circles (diameter = +1 mm, height = 8 cm) are the sample channels. b) 3D-MPI measurement of the phantom, measured by +mechanically scanning voxels in steps of 0.25 x 0.25 x 1 mm to map out the 3D phantom image at the +two orientations. + +(a) +rotate +(b) +10 +10 +8 +8 +6 +6 +ww] +ww] +N +4 +N +4 +2 +2 +2 +2 +2 +0 +0 +0 +0 +2 +-2 +-4-4 +-2 +-2 +-4-4 +-2 +y [mm] +x[mm] +y [mm] +x[mm]13 + + +Figure 2:(a) 3D-printed acrylic spatial resolution phantom with 16 channels. The diameter of each +channel is 0.75 mm. The group of 4 channels have edge-to-edge separations indicated by the labels on +the top view cartoon. (b) 2D MPI image. For clarity, the dashed red circle denotes the image +corresponding to the circled portion in the top view cartoon (0.5 mm separation). + +(a) +(b) +-6 +88 +88 +-7 +0 +-8 +-9 +-10 +[ww] +-11 +0.25 mm +>-12 +88 +-13 +00 +88 +-14 +0.1 mm +0.5 mm +-15 +O +-16 +1 mm +-17 +40 +41 +42 +43 +44 +45 +46 +47 +48 +49 +x [mm]14 + + +Figure 3: 1D scan of Vivotrax tracer in the glass phantom from Fig. 1. The two peaks correspond to two +of the four sample channels. The pre-amplifier gain is set at 10x to avoid the quantization noise floor of +the digitizer. + + +3rd harmonic +5th harmonic +50 +200 +20 +70 +digitizer +?digitizer +A +日lock-in +O-lock-in +60 +40 +150 +15 +50 +[uM] +30 +40 +100 +10 +30 +20 +50 +5 +10 +7 +710 +0 +0 +0 +40 +42 +44 +46 +48 +50 +40 +42 +44 +46 +48 +x [mm] +x[mm] +7th harmonic +gth harmonic +8 +25 +4.5 +digitizer +digitizer +—lock-in +4 +△lock-in +8 +20 +6 +3.5 +amplitude [μuV] +6 +15 +3 +2.5 +4 +10 +2 +2 +1.5 +5 +2 +1 +0 +0 +0.5 +0 +40 +42 +44 +46 +48 +50 +42 +44 +46 +48 +x[mm] +x [mm]15 + + +Figure 4: Thermal imaging of Vivotrax in a glass phantom as a function of harmonic number. Each +column is a different temperature and each row a different harmonic number. + +Figure 5: 1D images of Vivotrax as a function of temperature for the 7th (a) and 9th (b) harmonics. + +278 K +288 K +298 K +305 K +-6 +7th +7th +7th +7th +-8 +y [mm] +-10 +-12 +14 +-6 +9th +9t +9th +9th +-8 +y [mm] +10 +-12 +-14 +42 +44 +46 +48 +42 +44 +46 +48 +42 +44 +46 +48 +42 +44 +46 +48 +x [mm] +x [mm] +x [mm] +x[mm](a) +(b) +6 +1.2 +273K +7th +273 K +gth +283K +5 +283 K +e—293K +e—293 K +303K +303 K +0.8 +0.6 +2 +0.4 +0.2 +0 +40 +42 +44 +46 +48 +50 +40 +42 +44 +46 +48 +50 +x [mm] +x [mm]16 + + +Figure 6: (a) Vivotrax PSF and (b) fitted FWHM as a function of temperature (slope = 0.041(6) mT/K). This +PSF includes information from all harmonics. + +Figure 7: Spatial resolution dependence on (a) harmonic number and (b) harmonic ratio for 70 nm +Synomag nanoparticles. + +(a) +(b) +15 +278K +288K +298K +14.5 +0.8 +308K +FWHM [mT] +14 +.0.6 +13.5 +0.4 +P +0.2 +13 +0 +12.5 +-15 +-10 +-5 +0 +5 +10 +15 +275 +280 +285 +290 +295 +300 +305 +310 +μoH [mT] +Temperature [K]a +-6 +3 +-7 +-8 +[mm] +-9 +V-10 +-11 +-12 +-13 +16 +18 +20 +22 +246 +18 +20 +22 +246 +18 +20 +22 +24 +x [mm] +x [mm] +x[mm] +(b) +5th/3rd +7th/3rd +-6 +-6 +-7 +8 +-8 +[mm] +y [mm] +-9 +-9 +> +-10 +-10 +-11 +-11 +-12 +-12 +-13 +-13 +16 +18 +20 +22 +24 +16 +18 +20 +22 +24 +x[mm] +x[mm]17 + + +Figure 8: (a) 2D-MPI images of undiluted (15 mg/ml) 70 nm Synomag nanoparticles in a glass phantom. +The apparent spatial resolution improves with increasing harmonic number. (b) An acrylic phantom +(inset) is used to obtain the 1D image for quantifying the spatial resolution dependence on harmonic +number. (c) Fitted FWHM from (b) along with a bi-exponential fit (dashed blue curve) to the FWHM. + +3rd harmonic +5th harmonic +7th harmonic +-6 +-8 +-12 +-6 +-8 +[ww] +7.10 +-12 +16 +18 +20 +22 +24 16 +18 +20 +22 +24 16 +18 +20 +22 +24 +x [mm] +x [mm] +x [mm]18 + + +Figure 9: 2D-MPI images of undiluted (5 mg Fe/ml) of Vivotrax tracers in a glass phantom. Vivotrax +shows a more gradual harmonic dependence of resolution than 70 nm Synomag. + +-8 +-10 +[mm] +-12 +-14 +-16 +-10 +[mm] +-12 +-14 +-16 +41 +42 +43 +44 +45 +46 +47 +4841 +42 +43 +44 +45 +46 +47 +48 +x[mm] +x [mm]19 + + +Figure 10: Short time Fourier transform (STFT) procedure for obtaining the harmonic PSF at excitation +and bias field frequencies 𝑓0 = 31.75 kHz and 𝑓𝑏𝑖𝑎𝑠 = 10 Hz, respectively. (a) The applied field (sum of +excitation and bias) as a function of time. The red line i) denotes the time when the bias field is zero, and +the black line ii) denotes the time when the bias field is at its maximum of 40 𝑚𝑇𝑝, (b) zoomed-in view +of position i) at the local maximum (thick red line) with the integration time window (thin red lines) and +ii) at the the local minimum (thick black line) with the integration time window (thin black lines). (c) +Spectrogram of nanoparticle response signal using a triangular shaped window function that results in a +50% overlap in the integration domain. The red and black lines correspond to the time positions i) and +ii), respectively. (d) Fourier spectrum for the two different time position i) and ii). The blue vertical lines +denote the odd harmonics. + +50 +(a) +0.01 +(b) +TTETTT +amp. [a.u.] +Lw +(i) ! +normalizedamplitude +magnetic field [mT] +0.5 +0014 +0.075 +0.0751 +0.0752 +-0.1 +0 +0.1 +time [s] +field [mT] +0 +0 +(ii): +(ii) +0.5 +-50 +-1 +0 +0.02 +0.04 +0.06 +0.08 +-40 +-20 +0 +20 +40 +time [s] +magneticfield[mT] +C +40 +(d) +600 +()0mTbias +40 +.-.-(i)40mTbias +500 +20 +[kHz] +amplitude [a.u.] +20 +400 +frequency +0 +300 +20 +1 +200 +-20 +100 +-40 +0 +-40 +-40 +-20 +0 +20 +40 +0 +100 +200 +300 +400 +500 +600 +frequency[kHz] +magneticfield[mT]20 + + +Figure 11: Harmonic PSF measurement of Vivotrax using a digitizer (a,c) and a lock-in amplifier (b, d) at +excitation frequency 𝑓0 = 31.75 kHz. (a,b) and (c,d) are the measurements and simulations, respectively. + +Lock-in(experiment) +STFT(experiment) +10-2 +102 +(a) +(b) +10-3 +A.U. +amplitude [A.U.] +10-4 +amplitude +10~5 +100 +10~6 +10-7 +10-1 +Lock-in (simulation) +STFT（simulation） +60 +50 +(c) +(p) +40 +amplitude in dB [A.U.] +40 +amplitude in dB [A.U.] +30 +20 +20 +10 +-20 +-10 +-40 +-20 +0 +20 +40 +-40 +-20 +0 +20 +40 +magneticfield[mT] +magneticfield[mT]21 + + +Figure 12: Simulated PSF for Vivotrax at three different temperatures, along with the fits using a Voigt +function. The inset shows the fitted FWHM with a slope of 0.043 mT/K. + +0.35 +250 K +[FWHM [mT] +13 +0 +275 K +0.3 +300 K +12 +fit (250 K) +0.25 +11 +fit (275 K) +fit (300 K) +amplitude [A.U.] +240 +260 +280 +300 +temperature [K] +0.2 +0.15 +0.1 +0.05 +0 +-20 +-10 +0 +10 +20 +magneticfield[mT]22 + + +Figure 13 (SI): Background noise measurement of receive chain recorded with a digitizer (blue) and lock- +in amplifier (red). The black trace is the lock-in measurement with 50 ohm termination. + +2.5 +X10-3 +digtizer,IC input +lock-in, IC input +lock-in50ohmterminatorinput +1.5 +amplitude [V] +0.5 +0.5 +-1.5 +2 +-2.5 +0 +0.005 +0.01 +0.015 +0.02 +time [s]23 + + +Figure 14 (SI): Noise spectral density of receive chain measured with a lock-in amplifier. + +lock-in only +-IC + notch filter +.-IC + notch filter + pre-amp +10~6 +10 +10~8 +a) +AM +10-9 +102 +103 +104 +105 +106 +107 +freguency[Hz]24 + + +Figure 15 (SI): 1D representation of the 2D MPI image of Vivotrax in the glass phantom using the a) 7th +and b) 9th harmonic. The two peaks indicated by red arrows were used to determine the temperature +dependence of the c) 7th and d) 9th harmonic. + +(a) +(b) +3.5 +X104 +1.2 +X104 +305K +305K +3 +301.5K +peak #1 +peak #2 +301.5K +298K +1 +298K +293K +2.5 +293K +M +288K +288K +283K +283K +amplitude +2 +278K +278K +0.6 +1.5 +0.4 +1 +0.5 +0.2 +0 +0 +200 +400 +600 +800 +100 +0 +200 +400 +600 +800 +1000 +scannumber +scannumber +(c) +(d) +24 +8 +peak#1 +牛 +peak#1 +slope(3.8(2)μV/K) +22 +7 +slope (1.9(1)μV/K +更peak#2 +本 +peak#2 +-slope(3.7(1)μV/K) +-slope(1.9(1)μV/K +木 +3: +Amplitude [A.L +20 +5 +18 +4 +16 +3 +14 +由 +2 +12 +280 +285 +290 +295 +300 +305 +280 +285 +290 +295 +300 +305 +Temperature [K] +Temperature [K]25 + + +Figure 16 (SI): Allan deviation for harmonic MPI at a single pixel with Vivotrax tracers. The red dashed +line shows the Allan deviation for a statistical (random) noise source. + + +10-5 +0 +5thharmonic +口 +7thharmonic +9th harmonic +-1/2dependence +10 +6 +10-7 +10-1 +100 +101 +102 +Integration time [s]26 + + +Figure 17 (SI): Temperature dependence of Vivotrax in a large glass vial using the (a) MPS and (b) MPI +instrument. + + +(a) +(b) +10~5 +278K +278K +283K +283K +10-4 +—288K +288K +293K +amplitude [M] +amplitude [M] +中 +293K +298K +298K +303K +303K +308K +10-6 +313K +10-5 +10-7 +1 +2 +3 +4 +5 +6 +8 +9 +10 +1213 +2 +4 +6 +8 +10 +12 +harmonicnumber +harmonicnumber \ No newline at end of file diff --git a/ctFAT4oBgHgl3EQfYR3c/content/tmp_files/load_file.txt b/ctFAT4oBgHgl3EQfYR3c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..43c69b846fa2f3d492da8d82473a669d8b0d0014 --- /dev/null +++ b/ctFAT4oBgHgl3EQfYR3c/content/tmp_files/load_file.txt @@ -0,0 +1,718 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf,len=717 +page_content='1 Temperature-dependent magnetic particle imaging with multi-harmonic lock-in detection Thinh Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Bui, Mark-Alexander Henn, Weston L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Tew, Megan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Catterton, Solomon I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Woods T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='T, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='C, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='W are with the National Institute of Standards and Technology, Gaithersburg, MD 20895 USA (e-mail:thinh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='bui@nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='gov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='H is an associate with the National Institute of Standards and Technology, Gaithersburg, MD 20895 USA, and University of Maryland, College Park, MD, USA 20742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' ABSTRACT Advances in instrumentation and tracer materials are still required to enable sensitive and accurate 3D temperature monitoring by magnetic particle imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' We have developed a magnetic particle imaging instrument to observe temperature variations using MPI, and discuss resolution dependence on temperature and harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Furthermore, we present a low noise approach using lock-in detection for temperature measurement, and discuss implications for a new detection modality of MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Introduction Magnetic particle imaging relies on measurement of the magnetization of magnetic nanoparticles (MNP) driven by a strong AC magnetic field [1]-[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' At NIST, our efforts are directed at establishing magnetic nanoparticles as an SI-traceable nano-thermometer standard for in situ thermal imaging, and relying on magnetic particle imaging (MPI) as a core technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Potential prospects for theranostic applications of temperature sensing [11]-[20] and hyperthermia [16]-[20] based on MPI have been proposed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The quantitative nature of thermal imaging with magnetic nanoparticles is demanding and requires thorough investigation of technical noise and drifts in the measurement system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Yet, a systematic evaluation of temperature on the sensitivity and spatial resolution of MPI remains an unresolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For MPI, the magnetization response of MNPs driven into saturation by a sinusoidal excitation field are detected by inductive coil sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Typical MPI systems record the magnetization response in the time domain with a digitizer, and the harmonic components are obtained by digital Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' While this procedure has the advantage of detecting a broadband harmonic signal, often exceeding 1 MHz, the broad bandwidth carries along with it excess broadband noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Previously, a narrowband method using intermodulation frequency detection (a single harmonic and its intermodualtion frequency) was introduced to improve MPI noise performance [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' However, by singling out individual harmonics, high harmonic information content that is responsible for high imaging spatial resolution is not available, commonly described as a resolution-bandwidth trade-off [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Instead of purposefully reducing bandwidth, it was later proposed that low noise performance can be achieved by careful noise-matching inductive coil sensors to pre-amplifiers with a reported 11x gain in noise performance [23],24], but this approach requires tailor-made transformer-like networks specific to each MPI system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' An additional problem with this strategy is that the complex impedance of the inductive coil introduces a frequency-dependence of noise that complicates noise matching over too wide of a bandwidth > 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 2 Here, we propose an alternative and more general approach by implementing a massively- parallel lock-in detection scheme in which many harmonics are measured simultaneously while still leveraging the narrowband detection of the lock-in technique, resulting in a reduction in measurement noise by a factor of 100-1000, depending on the desired acquisition speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In contrast to [21], we are detecting many harmonics from the particle response in a narrowband fashion, thereby overcoming the resolution-bandwidth trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Lock-in detection can also reduce the data streaming requirements, since it does not require saving the full time response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' It is a general goal of magnetic imaging to reach the body noise limit, which can potentially be achieved at a high frequency (MHz range) by detecting at a high harmonic number and reaching coil resistance (thermal) noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In this work, we present our multi- harmonic lock-in detection approach, characterize noise performance, and compare performance of parallel lock-in detection against the commonly used broadband detection with a digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In addition to concentration imaging, nanoparticle thermometry has been a longstanding goal of MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Two key approaches for nanoparticle thermometry have been proposed: multi-color, system matrix-based [13],[25],[26] and multi-harmonic-based measurements [11],[14],[15],27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The system matrix approach requires measurements of calibration system matrices to reconstruct both concentration and temperature, and a linear interpolation step for intermediate temperatures [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In this work we will focus on the multi-harmonic technique, where the temperature can be determined by measurement of two harmonics and their ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Here, we will also present temperature-dependent MPI, which relies on the multi-harmonic detection, and show that performance is improved by lock-in detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Finally, we discuss the observed spatial resolution dependence on harmonic number, which will have important implications for image reconstruction when temperature is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' MPI instrumentation The MPI instrumentation developed at NIST was designed for thermal imaging and for thermometric metrology using magnetic nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Inspired by our previous temperature-tunable AC magnetometers [28],[29], the MNP sample holder is machined out of thermally conductive Shapal ceramic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The temperature of the sample is tuned by water-cooling with a recirculating chiller, and the temperature is measured by two 100 ohm platinum resistance thermometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' An additional water- cooling line is used to control the temperature of the excitation solenoid coil to ensure thermal stability over long acquisition times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The selection field (gradient field) is generated from a pair of permanent magnets (diameter = 50 mm, height = 50 mm, N52 grade NdFeB) separated by ≈30-40 mm to produce the field-free-point (FFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The FFP is fixed in our MPI imager, and spatial scanning is achieved by mechanical translation of a 3D stage in which the sample holder and excitation coils are mounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=" This image acquisition scheme relies on spatial mechanical step-scanning at a maximum rate of ≈5 Hz, which makes it more similar to the x-space method [8],[9],[30],[31] that uses low frequency (≈10's Hz) shift coils rather than the original Lissajous pattern-based electromagnetic shifting with three orthogonal fields near 25 kHz [1],[6],[32]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='The transmit chain is a resonant circuit at the excitation frequency 𝑓0 = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='75 kHz with a second order low-pass filter to attenuate (>50 dB) the higher harmonic (feedthrough) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Magnetization response from AC excitation (10 mT) is recorded with an inductive receive coil, wound as a first-order gradiometer in series with an additional $\\approx-30$ dB notch filter at the excitation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' This output is directly connected to a SR560 pre-amplifier, subsequently connected in series to an impedance matched, cascaded notch filter stage with ≈60 dB of attenuation at 3 the f0 and ≈40 dB at 2f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Output from the second notch filter is sent to a final low-pass filter stage (SRS SIM965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Finally, the filtered magnetization response is directed to either a digitizer (GaGe, 16 bit, 1 GSa/s) or lock-in amplifier (Zurich HF2LI) to measure the harmonic amplitudes for image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For the digitizer, the harmonic amplitudes are determined from the Fourier transform of the time domain signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The multi-frequency capability of the HF2LI lock-in amplifier allows the recording of six independent harmonics simultaneously in a parallel fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For quantitative imaging, we use a glass phantom with dimensions of 8 x 12 mm (diameter x height).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The phantom has four channels (1 mm diameter, 8 mm height) where MNP samples can be introduced and sealed by epoxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' To demonstrate the 3D imaging capability of our MPI imager, Fig 1 shows the 3D image of this glass phantom in two, rotated orientations, with four clearly resolved channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For a more systematic evaluation of spatial resolution, we used a 3D printed acrylic phantom shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Here, the sets of four channels have different edge-to-edge hole spacing to evaluate the spatial resolution using Rayleigh criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The 2D image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 2 reveal that we can resolve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='25 mm in the x-direction, the direction of the highest gradient field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' These results, along with the measured point-spread-function (PSF) of Vivotrax, indicate that the magnitude of the gradient field is ≈20 T/m for the x-axis and ≈10 T/m for the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' These results are consistent with our COMSOL simulations of the permanent magnet pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Parallel harmonic lock-in MPI A voltage-based measurement system of MPI comprised of an inductive coil sensor connected to a pre-amp has SNR given by [21],[23]: 𝑆𝑁𝑅 ∝ 𝑆 ⋅𝑑𝑚 𝑑𝑡 √(4𝑘𝐵𝑇𝑅𝐿 + 𝑒𝑛2 ̅̅̅ + 𝑖𝑛2̅ |𝑍𝐿| 2 )⋅ Δ𝑓 (1) The quantity in the numerator is the product of the coil sensitivity, $S$, and voltage induced by the magnetic nanoparticles, dm/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The denominator denotes the noise sources from the sensor, an inductive coil sensor with real (RL) and complex impedance (ZL), and the pre-amp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The pre-amplifier has a voltage and current noise spectral density (NSD,𝑉/√𝐻𝑧), 𝑒𝑛 ̅̅̅ and 𝑖𝑛̅, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Since Δf is the detection bandwidth, the best SNR is obtained by minimizing the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' This ideal scenario is, however, at odds with the MPI technique in that it is desirable to record all detectable information content-carrying harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Reducing the sampling rate (detection bandwidth) in the digitizer system could improve SNR, but at the expense of spatial resolution [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Electronic noise matching in the receive chain has been proposed, but the complex impedance of the inductive coil introduces a frequency dependence that complicates noise matching strategies over broadband frequencies [21],[23],[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' NSD measurements for our signal chain (IC with 2nd order notch filter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 14, SI)) does indeed show a complex frequency dependence with increasing noise density at higher (>1 MHz) frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' To overcome the noise matching and bandwidth requirement incompatibility challenges, we present a lock-in detection method with parallel multi-harmonic capabilities that can improve SNR by permitting detection at high frequencies without the need for broadband noise matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Relying on measurement of only a few harmonics has been proposed for in-situ temperature measurement [11],[15], and also as a general modality for imaging [9],[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Isolated harmonics are most sensitively measured with lock-in detection, where the noise bandwidth (1 to 100 Hz range) is most ideal and adjustable based on the desired acquisition speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' With a digitizer, a slower acquisition rate will reduce broadband noise, but also reduce the detection bandwidth to observe the rich harmonic 4 spectrum of saturated MNPs, which often exceed 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For detection of a single harmonic, the equivalent noise bandwidth Δ flock−in for lock-in detection is set by the lock-in time constant, T, which typically in our experiment is 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For a first order RC filter, Δ𝑓𝑙𝑜𝑐𝑘−𝑖𝑛 = 1 4𝑇, which gives a Δ𝑓𝑙𝑜𝑐𝑘−𝑖𝑛 ≈ 25 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Compared with a 1 MHz bandwidth digitizer acquisition, the SNR enhancement of lock-in detection relative to digitizer is √106/25 ≈ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' This analysis shows that multi-harmonic lock-in measurement can reduce broadband noise associated with the inductive coil detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Improving MPI noise performance by electronic noise matching in the receive chain [23],[24],[33],[34] can only further improve overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Noise performance and SNR of lock-in detection Despite the superior noise performance of lock-in detection, commercial lock-in amplifiers often limit to one or two harmonics at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Yet, if many of the particle harmonics could be measured in a parallel fashion, then SNR is optimized without sacrificing spatial information content by filtering higher harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' These considerations motivated us to compare the noise and imaging performance of multi- harmonic detection and time-domain detection with a digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' First, we compared the detection background noise by analyzing the time domain signal when the receive chain is connected to either the digitizer or the lock-in amplifier (Fig 13 (SI)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Here, the sampling conditions are set the same for both instruments, which provides a direct comparison of the analog-to-digital conversion (ADC) noise of the data acquisition devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' As presented in Fig 13 (SI), the RMS noise (VRMS) of the lock-in amplifier signal is ≈ 3x lower noise than the digitizer for the same acquisition rate To resolve the different contributions to the total noise, the noise spectral densities were measured for (1) the lock-in amplifier only and (2) the IC with the notch filter only, and (3) all components in the receive chain including the pre-amp, as shown in Figure 14 (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The 1 MHz bandwidth of the pre-amp suppresses noise at above 1 MHz, but increases the noise level by a factor of 100 in the range of interest of 30 kHz to 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 14c shows a higher noise-baseline from broadband detection of 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' These NSD measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 14b and c were used to calculate the total RMS noise by taking the integral of NSD2 over the full 1 MHz bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' This calculation agrees with the measured RMS noise in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 13 (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' To validate the enhanced noise performance of multi-harmonic lock-in detection with real samples, we compared images of the Vivotrax glass phantom used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 1 acquired with the lock-in amplifier and with the digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Figure 3 show the 1D image slice across two channels of the glass phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The four panels show a comparison of the 3rd to 9th harmonic signals from the lock-in and digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The SNR enhancement is most clearly observed in the 7th harmonic, with an improvement of ≈ 3x with the lock-in relative to the digitizer, in reasonable agreement with the predicted noise improvement of ≈ 200x from broadband noise reduction from 1 MHz (digitizer) to 25 Hz (lock-in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Temperature dependent MPI Temperature impacts the equilibrium magnetization as well as the dynamics of magnetization, which will be imprinted on the MPI signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' There have been a number of reported studies on the temperature dependence of suspended magnetic nanoparticles [11]-[15],[27],[35]-[38], yet quantitative effects of temperature on MPI signal and spatial resolution is not fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Using the imaging 5 phantom in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=" 1, we measured the temperature dependence of Vivotrax to assess temperature's impact on image reconstruction with MPI." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The temperature of the Vivotrax sample was tuned from 278 K to 305 K and allowed to equilibrate to ± 50 mK prior to imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The 2D MPI images for 4 of the 7 temperature steps are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 4 as columns, with each 2D image being an average of 3 repeated measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' It is evident by visual inspection that the 9th harmonic has a stronger temperature dependence than the 7th harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The quantitative dependence is obtained from the 1D representation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 4 versus pixel number, and is shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 15 (SI) top row plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For each plot, the overlay of 7 temperatures is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' To obtain the magnitude, the integrated areas are calculated for the peaks labeled in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 15a-b (SI) and are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 15c-d (SI) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The slopes for each harmonic gives the temperature sensitivity [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Since each peak gives a different spatial location of magnetic nanoparticles, the agreement in their calculated sensitivities gives evidence for the stability of the system over long periods of scanning and averaging (30 minutes of acquisition time at each temperature, 3 averages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' To gauge the short and long time stability, the Allan variance plots for a single image pixel are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 16 (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In our instrument, statistical noise can be improved by averaging up to 10 seconds before systematic drifts degraded noise performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Previous temperature-dependent MPS measurement of Vivotrax samples in an 8 mm diameter sapphire vial show a negative temperature dependence on harmonic amplitude predicted by the Langevin function [28], which we reproduced with the current MPS instrument [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In contrast to the 8 mm diameter glass vial used, our MPI results for Vivotrax using a glass phantom as a sample vessel display the opposite (positive) temperature dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' One possible explanation for this is that the narrow (1 mm diameter) channel of the phantom induces MNP aggregation, which hinders Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' If Brownian motion is blocked and the temperature dependence is dominated by Néel rotation, the magnetization response displays a positive temperature dependence as reported in the literature from MPS [37] and MPI [38] studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' We independently verified with MPS that the phantom vessel displays a positive temperature dependence consistent with our MPI results, while the larger glass vial shows the opposite trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Conversely, measurement of Vivotrax in the larger glass vial using both the MPS and MPI instrumentation showed the same (negative) temperature dependence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 17, SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The imaging spatial resolution dependence on harmonic number and temperature were also measured for Vivotrax with the glass phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 1-D scans across the two channels of the phantom are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 5 for both the 7th and 9th harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Only a modest change in the FWHM of the peaks was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' A more quantitative measurement of the temperature dependence is determined from the PSF of Vivotrax (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In contrast to measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 5, the PSF in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 6 includes all harmonics and therefore exhibits significantly higher SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' By fitting the PSF curves to a Voigt function, the temperature dependence of the FWHM was obtained (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' A modest slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='041(6) mT/K is consistent with a small temperature dependence shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For thermal imaging applications using magnetic nanoparticles, measurement of harmonic ratios, rather than individual harmonic amplitudes, has been proposed to remove the concentration dependence [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' We recorded 2D images using individual harmonics and also their ratios, and the data is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In the case of harmonic ratios, 5th/3rd and 7th/3rd, the higher harmonic dominates the spatial resolution of the ratio image, which mostly resembles the image of the higher harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' In general, images obtained with either individual harmonics or harmonic ratios 6 clearly show a harmonic dependence of spatial resolution, and suggests a corresponding point-spread- function (PSF) dependence on harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' This concept of a harmonic PSF is crucial for this imaging modality, and will be discussed in detail in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Harmonic dependence of spatial resolution Previous MPI implementations typically utilize the full time signal for image reconstruction, which obscures the impact of individual harmonics on spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The effect of harmonics on the reconstructed image has been introduced previously [2],[31], mainly in the context of feedthrough (1st harmonic) suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' More importantly, these studies reveal a direct mapping of harmonic frequency space to spatial frequency in the image space for sinusoidal excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Our work extends upon these analyses by experimentally determining the impact of individual harmonics on image reconstruction, a new concept denoted as the harmonic PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' To demonstrate the existence of a harmonic PSF, we acquired MPI images reconstructed using individual harmonics by means of (1) the spectrum of the digitized time signal and (2) the lock-in amplifier demodulated signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Using both approaches, the apparent dependence of resolution on harmonic number was observed for two different MNP samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Here, a glass phantom with four channels was used as a sample holder for 70 nm Synomag (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 8) and for Vivotrax (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Figure 8a shows a strong resolution dependence on harmonic number, as the four wells become more spatially resolved with increasing harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' To quantitatively analyze the resolution dependence, an acrylic phantom with two sample wells spaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 mm was machined and imaged in 1D (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The spatial resolution was determined by fitting the FWHM of the measured profile, and the result is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The resolution enhancement experiences an exponential improvement at low harmonic number and saturates at higher harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' A similar analysis was done for Vivotrax tracers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 9) with an observed exponential-like dependence of resolution on harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' PSF measurement The spatial resolution of MPI is dependent on the PSF of the magnetic nanoparticles and the gradient of the selection field [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' By applying a slow varying (10 Hz) bias field at 80 mTpp superimposed on the excitation field, the PSF can be determined in a similar fashion to [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Different from this PSF reconstruction procedure that includes all harmonics similar to previous reports, we extended this concept to measure the PSF of individual harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The PSF of the individual harmonics were resolved by two different methods: (1) short time Fourier transform (STFT) of the digitized time response signal and (2) time-swept lock-in detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For these measurements, the bias field is driven by a triangle waveform with 𝑓𝑏𝑖𝑎𝑠 = 10 Hz with an amplitude of 40 mTp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For method (1) using the time signal $f(t)$ from the digitizer, the short time Fourier transform is implemented by first binning the digitized data in defined window segments 𝑤(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' At each time shift 𝑡, a new window 𝑤(𝑡′ − 𝑡) is applied and the FT is calculated of 𝑓(𝑡′)𝑤(𝑡′ − 𝑡): 𝑓̂(𝑡, 𝑢) = ∫ 𝑓(𝑡′)𝑤(𝑡′ − 𝑡)𝑒−𝑖2π𝑡′𝑢 ∞ −∞ 𝑑𝑡′ (2) This procedure produces a spectrogram, or a plot of the Fourier spectrum at each time segment window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' With the bias field turned on, the harmonic spectrum will only appear when the bias field amplitude is near zero, that is when the MNP are not fully saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For method (2), the harmonic PSF 7 is obtained with the lock-in by implementing a time-domain acquisition of the output at the specified demodulation frequency (f0, 3f0, 5f0, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=') to resolve the time-swept PSF at each harmonic, similar to data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Figure 10 is a step-by-step layout of the STFT procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Harmonic PSF data taken with the digitizer (method 1) and lock-in amplifier (method 2) are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 11a-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The general shape of the PSF obtained using the two independent approaches display qualitative agreement for each harmonic, and provides experimental justification for the origin of the harmonic dependence of PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' By inspection, it can be seen that the harmonic PSF displays a decreasing FWHM (higher spatial resolution) with increasing harmonic number, and the FWHM also converges to an asymptotic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' This trend explains our imaging results for nanoparticle samples in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 8 and 9 and the exponential dependence in Figure 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Simulation of harmonic dependence of resolution To obtain further validation and insight into the origin of the observed spatial resolution dependence on harmonic number, we performed the following simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The magnetization of a single magnetic nanoparticle is modeled using the Langevin function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', 𝑀⃗⃗ (𝐻⃗⃗ ) = ℒ(𝑘𝐻⃗⃗ ) 𝐻⃗⃗ ||𝐻⃗⃗ || , with 𝑘 = μ0𝑚 𝑘𝐵𝑇, (3) and μ0 being the vacuum permeability, 𝑚 the particle magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The moment is a function of the saturation magnetization 𝑀𝑠, the Boltzmann constant 𝑘𝐵, and the temperature 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The PSF 𝑆 (𝑡) is derived by calculating the time-derivative of the Langevin function, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (3) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Finally, the harmonic dependence of PSF is calculated in the same fashion as the STFT and lock-in approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Measured and simulated harmonic PSF presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 11a-b and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 11c-d, respectively, show that the PSF FWHM, defined by the broad envelope at each harmonic, is narrower with increasing harmonic number, with nearly an exponential-like drop off in resolution [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The simulations also reveal the origin of the narrower PSF for higher harmonics: the amplitude decreases faster for higher harmonics relative to lower harmonics for nanoparticles that are increasingly farther away from the FFP region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' We also simulated the temperature dependence of the PSF to validate the observations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Note that due to the explicit temperature dependence of 𝑘 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 3, the magnetization of a single particle shows a corresponding temperature dependence, even if the saturation magnetization was set to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Figure 12 shows the simulated temperature dependence of PSF over the range 250 K to 300 K, in which only a modest change in the PSF FWHM with temperature (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='043 mT/K) was observed, in good agreement with the Vivotrax data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Conclusion Towards the goal of advancing temperature measurement and thermal imaging with magnetic nanoparticles, this work comprised experimental and theoretical studies on the harmonic dependence of MPI and measurements of noise performance based on a digitizer and multi-channel lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The study shows the advantage of using parallel multi-harmonic lock-in detection as a general data 8 acquisition method with exceptional noise performance and straightforward implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' However, scaling up lock-in amplifiers to multiple independent channels is expensive with commercial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' A more cost-effective solution to increase the number of independent lock-in detection channels is to implement high frequency lock-in with FPGA [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' This will require developments of lock-in electronics that are both noise-matched to the receive chain and operate at high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Numerical Details Note, that for this study we assumed all quantities from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 3 to be scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=" With the time-dependent PSF 𝑆(𝑡) evaluated at 𝑁 + 1 different times 𝑡−𝑁/2, 𝑡−𝑁/2−1, … , 𝑡0, … , 𝑡𝑁/2+1, 𝑡𝑁/2, with center point 𝑡0, that fall within the window segment 𝑤(𝑡0) = [𝑡−𝑁/2, 𝑡𝑁/2], and a triangular shaped window function 𝑊(𝑡) that has its maximum at 𝑡0, we can calculate the PSF's 𝑘-th harmonic through the discrete STFT: 𝑐𝑘 = 1 𝑁+1 ∑ 𝑊(𝑡𝑖)𝑆(𝑡𝑖) exp(−𝑖2π𝑘𝑓0𝑡𝑖)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 𝑁/2 𝑖=−𝑁/2 (4) The lock-in measurement for the 𝑘-th harmonic is simulated by first multiplying the time-dependent PSF 𝑆(𝑡) with a reference signal such that: 𝑆̂(𝑡) = 𝑆(𝑡) exp(−𝑖2π𝑘𝑓0𝑡), (5) and then taking the discrete Fourier transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 5 over a time window 𝑤(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The time resolution in all our simulations was set to 10−9 s to find a balance between necessary accuracy and memory constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Acknowledgment We acknowledge Jo Wu for assistance with the glass imaging phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' We thank Angela Hight Walker, Cindi Dennis, Michael Donahue, Adam Biacchi, and Frank Abel for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' We acknowledge financial support from the NIST Innovation in Measurement Science (IMS) program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Reference is made to commercial products to adequately specify the experimental procedures involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that these products are the best for the purpose specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' REFERENCES [1] Gleich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Tomographic imaging using the nonlinear response of magnetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Nature, 435(7046), 1214-1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [2] Rahmer, J.' metadata={'source': 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Projection X-Space Magnetic Particle Imaging Patrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' IEEE Trans Med Imaging, 31(5), 1076–1085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 9 [4] Saritas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Goodwill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Croft, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Konkle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Lu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Zheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Conolly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Magnetic Particle Imaging (MPI) for NMR and MRI Researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Journal of Magnetic Resonance, 23(1), 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [5] Vogel, P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 2D Images Recorded With a Single-Sided Magnetic Particle Imaging Scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' IEEE Transactions on Medical Imaging, 35(4), 1056- 1065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [7] Vogel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Rückert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Kemp, S.' metadata={'source': 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K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Behr, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Micro-Traveling Wave Magnetic Particle Imaging - Sub-Millimeter Resolution with Optimized Tracer LS-008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' IEEE Transactions on Magnetics, 55(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [8] Kurt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Muslu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Saritas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Partial FOV Center Imaging (PCI): A Robust X-Space Image Reconstruction for Magnetic Particle Imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' IEEE transactions on medical imaging, 39(11), 3441–3450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [9] Mason, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Excitation frequency dependence of temperature resolution in magnetic nanoparticle temperature imaging with a scanning magnetic particle spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Journal of Magnetism and Magnetic Materials, 471, 340-345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [16] Franke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Heinen, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Lehr, H.' metadata={'source': 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Characterization of a Highly Integrated Preclinical Hybrid MPI-MRI Scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' IEEE Transactions on Medical Imaging, 35(9), 1993-2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [17] Bauer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Situ, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Griswold, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Samia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' High-performance iron oxide nanoparticles for magnetic particle imaging – guided hyperthermia (hMPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Nanoscale, 8, 12162-12169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 10 [18] Hensley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Tay, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Dhavalikar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Zheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Goodwill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Rinaldi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Conolly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Combining magnetic particle imaging and magnetic fluid hyperthermia in a theranostic platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Physics in Medicine and Biology, 62(9), 3483–3500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [19] Tay, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Chandrasekharan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Chiu-Lam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Hensley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Dhavalikar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Yu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Goodwill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Zheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Rinaldi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Conolly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Magnetic Particle Imaging-Guided Heating in Vivo Using Gradient Fields for Arbitrary Localization of Magnetic Hyperthermia Therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' ACS Nano, 12(4), 3699– 3713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [20] Chandrasekharan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Tay, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Hensley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Fung, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Colson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Fellows, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Huynh, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Saayujya, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Yu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Orendorff, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Zheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Goodwill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Rinaldi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Conolly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Using 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (Doctoral dissertation, Massachusetts Institute of Technology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [25] Rahmer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Halkola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Gleich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Schmale, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Borgert, J.' metadata={'source': 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Rauwerdink, Eric W Hansen, & John B Weaver (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Nanoparticle temperature estimation in combined ac and dc magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Physics in Medicine & Biology, 54(19), L51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [28] Bui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Tew, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Woods, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Tew, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Hight Walker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Woods, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Advanced characterization of magnetization dynamics in iron oxide magnetic nanoparticle tracers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Applied Physics Letters, 120(1), 012407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 11 [30] Goodwill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Saritas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Croft, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Kim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Krishnan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Schaffer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Conolly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' X-space MPI: magnetic nanoparticles for safe medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Advanced materials, 24(28), 3870–3877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [31] Lu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Goodwill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Saritas, E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Physics in Medicine and Biology, 54(5), L1-L10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [33] Graeser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Knopp, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Grüttner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Sattel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Buzug, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Schilling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Ludwig, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Determination of dominating relaxation mechanisms from temperature-dependent Magnetic Particle Spectroscopy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Journal of Magnetism and Magnetic Materials, 474(September 2018), 570–573.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Chandrasekharan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Zheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Conolly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Optimization of Drive Parameters for Resolution, Sensitivity and Safety in Magnetic Particle Imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' IEEE Transactions on Medical Imaging, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Bui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Woods, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (submitted for publication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Investigating the Harmonic Dependence of MPI Resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' IJMPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [42] Stimpson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Skilbeck, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Patel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', Green, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=', & Morley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' An open-source high-frequency lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Review of Scientific Instruments, 90(9), 094701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 12 FIGURES Figure 1: Magnetic particle imaging: a) 4-channel glass phantom containing undiluted (5 mg Fe/ml) Vivotrax tracer at two rotated orientations shown from the top view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The four white circles (diameter = 1 mm, height = 8 cm) are the sample channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' b) 3D-MPI measurement of the phantom, measured by mechanically scanning voxels in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='25 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='25 x 1 mm to map out the 3D phantom image at the two orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (a) rotate (b) 10 10 8 8 6 6 ww] ww] N 4 N 4 2 2 2 2 2 0 0 0 0 2 2 4-4 2 2 4-4 2 y [mm] x[mm] y [mm] x[mm]13 Figure 2:(a) 3D-printed acrylic spatial resolution phantom with 16 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The diameter of each channel is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='75 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The group of 4 channels have edge-to-edge separations indicated by the labels on the top view cartoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (b) 2D MPI image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' For clarity, the dashed red circle denotes the image corresponding to the circled portion in the top view cartoon (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 mm separation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (a) (b) 6 88 88 7 0 8 9 10 [ww] 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='25 mm >-12 88 13 00 88 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='1 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 mm 15 O 16 1 mm 17 40 41 42 43 44 45 46 47 48 49 x [mm]14 Figure 3: 1D scan of Vivotrax tracer in the glass phantom from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The two peaks correspond to two of the four sample channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The pre-amplifier gain is set at 10x to avoid the quantization noise floor of the digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 3rd harmonic 5th harmonic 50 200 20 70 digitizer ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='digitizer A 日lock-in O-lock-in 60 40 150 15 50 [uM] 30 40 100 10 30 20 50 5 10 7 710 0 0 0 40 42 44 46 48 50 40 42 44 46 48 x [mm] x[mm] 7th harmonic gth harmonic 8 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 digitizer digitizer —lock-in 4 △lock-in 8 20 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 amplitude [μuV] 6 15 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 4 10 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 5 2 1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 0 40 42 44 46 48 50 42 44 46 48 x[mm] x [mm]15 Figure 4: Thermal imaging of Vivotrax in a glass phantom as a function of harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Each column is a different temperature and each row a different harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Figure 5: 1D images of Vivotrax as a function of temperature for the 7th (a) and 9th (b) harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 278 K 288 K 298 K 305 K 6 7th 7th 7th 7th 8 y [mm] 10 12 14 6 9th 9t 9th 9th 8 y [mm] 10 12 14 42 44 46 48 42 44 46 48 42 44 46 48 42 44 46 48 x [mm] x [mm] x [mm] x[mm](a) (b) 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='2 273K 7th 273 K gth 283K 5 283 K e—293K e—293 K 303K 303 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='6 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='2 0 40 42 44 46 48 50 40 42 44 46 48 50 x [mm] x [mm]16 Figure 6: (a) Vivotrax PSF and (b) fitted FWHM as a function of temperature (slope = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='041(6) mT/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' This PSF includes information from all harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Figure 7: Spatial resolution dependence on (a) harmonic number and (b) harmonic ratio for 70 nm Synomag nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (a) (b) 15 278K 288K 298K 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='8 308K FWHM [mT] 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='4 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='2 13 0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 15 10 5 0 5 10 15 275 280 285 290 295 300 305 310 μoH [mT] Temperature [K]a 6 3 7 8 [mm] 9 V-10 11 12 13 16 18 20 22 246 18 20 22 246 18 20 22 24 x [mm] x [mm] x[mm] (b) 5th/3rd 7th/3rd 6 6 7 8 8 [mm] y [mm] 9 9 > 10 10 11 11 12 12 13 13 16 18 20 22 24 16 18 20 22 24 x[mm] x[mm]17 Figure 8: (a) 2D-MPI images of undiluted (15 mg/ml) 70 nm Synomag nanoparticles in a glass phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The apparent spatial resolution improves with increasing harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (b) An acrylic phantom (inset) is used to obtain the 1D image for quantifying the spatial resolution dependence on harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (c) Fitted FWHM from (b) along with a bi-exponential fit (dashed blue curve) to the FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 3rd harmonic 5th harmonic 7th harmonic 6 8 12 6 8 [ww] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='10 12 16 18 20 22 24 16 18 20 22 24 16 18 20 22 24 x [mm] x [mm] x [mm]18 Figure 9: 2D-MPI images of undiluted (5 mg Fe/ml) of Vivotrax tracers in a glass phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Vivotrax shows a more gradual harmonic dependence of resolution than 70 nm Synomag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 8 10 [mm] 12 14 16 10 [mm] 12 14 16 41 42 43 44 45 46 47 4841 42 43 44 45 46 47 48 x[mm] x [mm]19 Figure 10: Short time Fourier transform (STFT) procedure for obtaining the harmonic PSF at excitation and bias field frequencies 𝑓0 = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='75 kHz and 𝑓𝑏𝑖𝑎𝑠 = 10 Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (a) The applied field (sum of excitation and bias) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The red line i) denotes the time when the bias field is zero, and the black line ii) denotes the time when the bias field is at its maximum of 40 𝑚𝑇𝑝, (b) zoomed-in view of position i) at the local maximum (thick red line) with the integration time window (thin red lines) and ii) at the the local minimum (thick black line) with the integration time window (thin black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (c) Spectrogram of nanoparticle response signal using a triangular shaped window function that results in a 50% overlap in the integration domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The red and black lines correspond to the time positions i) and ii), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (d) Fourier spectrum for the two different time position i) and ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The blue vertical lines denote the odd harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 50 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='01 (b) TTETTT amp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='] Lw (i) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' normalizedamplitude magnetic field [mT] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 0014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='0751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='0752 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='1 time [s] field [mT] 0 0 (ii): (ii) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 50 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='08 40 20 0 20 40 time [s] magneticfield[mT] C 40 (d) 600 ()0mTbias 40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='-(i)40mTbias 500 20 [kHz] amplitude [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='] 20 400 frequency 0 300 20 1 200 20 100 40 0 40 40 20 0 20 40 0 100 200 300 400 500 600 frequency[kHz] magneticfield[mT]20 Figure 11: Harmonic PSF measurement of Vivotrax using a digitizer (a,c) and a lock-in amplifier (b, d) at excitation frequency 𝑓0 = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='75 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (a,b) and (c,d) are the measurements and simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' Lock-in(experiment) STFT(experiment) 10-2 102 (a) (b) 10-3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' amplitude [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='] 10-4 amplitude 10~5 100 10~6 10-7 10-1 Lock-in (simulation) STFT（simulation） 60 50 (c) (p) 40 amplitude in dB [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='] 40 amplitude in dB [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='] 30 20 20 10 20 10 40 20 0 20 40 40 20 0 20 40 magneticfield[mT] magneticfield[mT]21 Figure 12: Simulated PSF for Vivotrax at three different temperatures, along with the fits using a Voigt function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The inset shows the fitted FWHM with a slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='043 mT/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='35 250 K [FWHM [mT] 13 0 275 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='3 300 K 12 fit (250 K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='25 11 fit (275 K) fit (300 K) amplitude [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='] 240 260 280 300 temperature [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='05 0 20 10 0 10 20 magneticfield[mT]22 Figure 13 (SI): Background noise measurement of receive chain recorded with a digitizer (blue) and lock- in amplifier (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The black trace is the lock-in measurement with 50 ohm termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 X10-3 digtizer,IC input lock-in, IC input lock-in50ohmterminatorinput 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 amplitude [V] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='02 time [s]23 Figure 14 (SI): Noise spectral density of receive chain measured with a lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' lock-in only IC + notch filter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='-IC + notch filter + pre-amp 10~6 10 10~8 a) AM 10-9 102 103 104 105 106 107 freguency[Hz]24 Figure 15 (SI): 1D representation of the 2D MPI image of Vivotrax in the glass phantom using the a) 7th and b) 9th harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The two peaks indicated by red arrows were used to determine the temperature dependence of the c) 7th and d) 9th harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' (a) (b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 X104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='2 X104 305K 305K 3 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5K peak #1 peak #2 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5K 298K 1 298K 293K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 293K M 288K 288K 283K 283K amplitude 2 278K 278K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='4 1 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content='L 20 5 18 4 16 3 14 由 2 12 280 285 290 295 300 305 280 285 290 295 300 305 Temperature [K] Temperature [K]25 Figure 16 (SI): Allan deviation for harmonic MPI at a single pixel with Vivotrax tracers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' The red dashed line shows the Allan deviation for a statistical (random) noise source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFAT4oBgHgl3EQfYR3c/content/2301.08539v1.pdf'} +page_content=' 10-5 0 5thharmonic 口 7thharmonic 9th harmonic 1/2dependence 10 6 10-7 10-1 100 101 102 Integration time [s]26 Figure 17 (SI): Temperature dependence of Vivotrax in a large glass vial using the (a) MPS and (b) MPI instrument.' metadata={'source': 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b/hdFJT4oBgHgl3EQfVyxJ/content/tmp_files/2301.11514v1.pdf.txt @@ -0,0 +1,3602 @@ +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial +Manufacturing: A Survey +Jiaqi Liu1†, Guoyang Xie1,2†, Jingbao Wang1†, Shangnian +Li1, Chengjie Wang3, Feng Zheng1 and Yaochu Jin2,4 +1Research Institute of Trustworthy Autonomous Systems, +Southern University of Science and Technology, Shenzhen 518055, +China. +2NICE Group, University of Surrey, Guildford GU2 7YX, United +Kingdom. +3Youtu Lab, Tencent, Shenzhen 10000, China. +4NICE Group, Bielefeld University, Bielefeld 33619, Germany. +†These authors contributed equally to this work. +Abstract +The recent rapid development of deep learning has laid a milestone in +visual anomaly detection (VAD). In this paper, we provide a comprehen- +sive review of deep learning-based visual anomaly detection techniques, +from the perspectives of neural network architectures, levels of supervi- +sion, loss functions, metrics and datasets. In addition, we extract the +new setting from industrial manufacturing and review the current VAD +approaches under our proposed our new setting. Moreover, we highlight +several opening challenges for visual anomaly detection. The merits +and downsides of representative network architectures under varying +supervision are discussed. Finally, we summarize the research findings +and point out future research directions. More resources are available at +https://github.com/M-3LAB/awesome-industrial-anomaly-detection. +Keywords: Visual anomaly detection, Industrial manufacturing, Deep +learning +1 +arXiv:2301.11514v1 [cs.CV] 27 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +1 Introduction +We review the recent advances of deep learning based visual anomaly detec- +tion since the rapid development of deep learning can bring the capabilities +of visual anomaly detection into the factory floor. In modern manufacturing, +VAD is always performed at the end of the manufacturing process and tries to +identify product defects. The price of a product is significantly affected by the +defect’s severity. In addition, if the flaw reaches a certain threshold, the prod- +uct will be discarded. Historically, the majority of anomaly detection tasks are +performed by humans, which suffers from the following many disadvantages: +• It is impossible to avoid human fatigue, resulting in a false positive phe- +nomenon (i.e.,the ground truth is abnormal, while the human’s judgment is +normal). +• Long and intensive working on anomaly detection may cause health prob- +lems, such as visual impairment. +• Locating anomalies requires a significant number of employees, raising +operational costs. +Thus, the goal of VAD algorithms is to reduce human labour and improve pro- +ductivity and product quality. In the pre deep learning era, the performance +of VAD cannot fulfill the demands of industrial manufacturing. But nowadays +most of deep learning-based visual anomaly detection is approaching 97% accu- +racy. So more careful analysis of VAD applied on industrial manufacturing is +called for. +Table 1 demonstrates clearly the merits of our survey in terms of dataset, +metric, neural network architecture, levels of supervision and new setting for +industrial manufacturing. Firstly, our study uses twice as many VAD datasets +as Tao et al. [1]. Secondly, we analyze the performance of VAD using the most +comprehensive image level and pixel level metrics. Nevertheless, Cui et al. [2] +and Tao et al. [1] only employ image level metrics, neglecting the anomalies +localization performance of VAD. Thirdly, our study develops a taxonomy +based on the design of neural network architecture with varying degrees of +supervision. Finally, to bridge the gap between academic research and real- +world industry needs, we review the current VAD algorithms under industrial +manufacturing settings. +Table 1: Our survey and existing surveys for VAD. +Content +[3] +[4] +[1] +[2] +Ours +VAD mataset +- +5 +9 +7 +20 +VAD metric +- +- +3 +1 +6 +Neural network architecture +� +� +� +� +� +Levels of supervision +� +� +� +� +� +Industrial manufacturing setting +� +� +� +� +� + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +3 +As an emerging field, research on VAD must fully consider industrial man- +ufacturing requirements. The following is a summary of the challenging issues +that need to be investigated: +• VAD dataset should be gathered from actual manufacturing lines, not labs. +The public cannot access the real-world anomalous dataset due to privacy +concerns. The majority of open-source VAD datasets generate anomalies +from anomaly-free products. In other words, the abnormalities from open- +source VAD datasets may not occur in actual production lines, which makes +deploying VADs in industrial manufacturing very challenging. +• It is challenging to enable the creation of a unified VAD model in the absence +of multiple domain VAD datasets. Recently, You et al. [5] propose a unified +VAD model for multiple class objects. However, they disregard the notion +that commodities produced in the same plant should be of the same sort. For +example, an automaker manufactures several types of workpieces but does +not produce fruit. Current popular VAD datasets, like MVTec AD [6] and +MVTec LOCO [7], consist of numerous classes but not multiple domains. +To simulate a realistic manufacturing process, we must create a new VAD +dataset collected from multiple domains. +• It is urgent to set up a uniform assessment for the image-level and pixel +level of VAD performance. The majority of VAD metrics shrink the anoma- +lous mask (ground truth) into the size of feature map for evaluation, which +inevitably reduces the precision of assessment. Moreover, we discover that +certain VAD methods perform well on image AUROC but poorly on pixel +AP, or vice versa. Therefore, it is essential to develop a uniform metric for +assessment VAD performance at both image and pixel level. +• We should design a more efficient loss function that can leverage both the +guidance of labelled data and the exploration of unlabelled data. In realistic +manufacturing scenario, limited number of anomalous samples are available. +However, most of unsupervised VAD methods outperform semi-supervised +VAD methods. By observing the failure of semi-supervised VAD, we would +call for more attention to the feature extraction and loss function, which can +leverage both the guidance from labels efficiently and the exploration from +the unlabeled data. Regarding the key problem mentioned above, improving +feature extraction from abnormal samples and redesigning deviation loss +function can fully use labelled anomalies and diverge the feature space of +abnormal samples from those of normal samples. +1.1 Contribution +The main contributions of this survey can be summarized as following: +• We provide an in-depth review of visual anomaly detection by consid- +ering the design of neural network architecture with varying degrees of +supervision. + +Springer Nature 2021 LATEX template +4 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +Visual Anomaly Detection (VAD) + in Industrial Manufacturing +Feature Embedding +Based Methods +§2.1 +Reconstruction +Based Methods +§2.2 +Unsupervised +Anomaly Detection +§2 +Supervised +Anomaly Detection +§3 +Different Settings +in Anomaly Detection +§4 +Few-shot +Anomaly Detection +§4.1 +Noisy +Anomaly Detection +§4.2 +3D +Anomaly Detection +§4.3 +Datasets and Metrics +§5 +Performance Analysis +and Experiments +§6 +Future Directions +§7 +Fig. 1: Framework of this review. +• It provides a comprehensive review of the current VAD algorithms in +the industrial manufacturing setting to bridge the gap between academic +research and real-world industrial manufacturing. +• It summarizes the main issues and potential challenges in VAD, which +outlines the underlying research directions for future works. +The rest of this paper is organized as Figure 1. In Section 2 and Section 3, +we review VAD on the basis of the neural network architecture with different +levels of supervision. Next, we review the recent advances of VAD under our +proposed setting from industrial manufacturing in Section 4. We describe the +popular dataset in Section 5 and take a retrospective view of the metrics +function in Section 5. Then, we provide an analysis of the performance of +current VAD methods on various datasets in Section 6. Finally, we provide +future research directions for VAD in Section 7. +2 Unsupervised Anomaly Detection +The majority of current research focuses on unsupervised anomaly detec- +tion, based on the assumption that the collection of abnormal samples incurs +massive human and financial costs. This indicates that only normal samples +are included in the training set, whereas both abnormal and normal samples +are included in the test set. Anomaly detection in industrial images is a subset +of problems with out-of-distribution (OOD). Before the rise of deep learning, +differential detection and filtering were frequently used to detect anomalies in +industrial images. Following the release of the MVTec AD [6], methods for +anomaly detection in industrial images can be divided into two categories: +feature-embedding and reconstructed-based. Currently, more AD techniques +are based on feature embedding. + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +5 +2.1 Feature Embedding based Methods +2.1.1 Teacher-Student Architecture +The performance of these methods is outstanding, but they depend on pre- +trained models such as ResNet [8] VGG [9] and EfficientNet [10]. The selection +of the ideal teacher model is crucial. This type of instructional strategy is sum- +marized in Table 2. The structure of the network and the method of distillation +are the primary distinctions between various techniques. +Table 2: A summary of teacher-student methods regarding loss function, +pre-trained model, and highlights. +Method +Loss Function +Pre-trained +Highlights +Uninformed +Students [11] +L2, Compactness +ResNet +The paper designs a basic approach to anomaly detection problems using a +teacher-student model. +MKD [12] +L2 +VGG +The paper uses multi-scale features and lighter networks for distillation. +STPM [13] +L2 +ResNet +The paper uses multi-scale features under different network layers for +distillation. +STFPM [14] +L2 +ResNet +The paper adds another teacher-student pair to get different feature recon- +struction results. +RD4AD [15] +Cosine Similarity +ResNet +The paper designs the teacher-student model of reverse distillation in a similar +way to reconstruction. +IKD [16] +Context Similarity +ResNet +The paper adds context similarity loss and adaptive hard sample mining +module to prevent overfitting. +AST [17] +L2, Log-Likelihood +EfficientNet +The paper uses a asymmetric teacher-student network to make the represen- +tation of anomaly more different. +The teacher-student network architecture depicted in Figure 2 is the most +standard technique for detecting industrial image anomalies. This method typ- +ically selects a partial layer of a backbone network pre-trained on a large-scale +dataset as a fixed-parameter teacher model. During training, the teacher model +imparts to the student model the knowledge of extracting normal sample fea- +tures. During inference, the characteristics of normal images extracted from +the test set by the teacher network and the student network are comparable, +whereas the characteristics of abnormal images extracted from the test set are +quite distinct. By comparing the feature maps generated by the two networks, +it is possible to generate anomaly score maps with the same size. Then, by +enlarging the anomaly score map to the same proportion as the input image, +we can obtain the anomaly scores of various input image locations. On the +justification of this model, it is possible to determine whether the test image +is abnormal. +Bergmann et al. [11] is the first to use teacher-student architecture for +anomaly detection. The model is straightforward and effective, significantly +outperforming other benchmark methods. While STPM [13] and MKD [12] +both use multi-scale features under different network layers for distillation, +they do so in different ways. In this instance, the normal sample features +extracted by the student network are more similar to those extracted by the +teacher network, whereas the abnormal sample features are more dissimilar. +In addition, MKD finds that the lighter student network structure performs +better than the student network structure identical to that of the teacher +network. Based on STPM, RSTPM [14, 18] adds a pair of teacher-student + +Springer Nature 2021 LATEX template +6 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +Teacher Network +Student Network +Distillation +Anomaly +Map +Normal feature from teacher network +Normal feature from student network +Abnormal feature from teacher network +Abnormal feature from teacher network +Distance of image feature from two networks +Fig. 2: Teacher-student model architecture. +networks. During reasoning, the new teacher network is placed behind the +original teacher-student network and is responsible for recreating the features. +When anomalous images are presented, the student network typically recon- +structs normal features that can be distinguished from those of the teacher +network. RSTPM also includes a mechanism for transferring features from the +teacher network to the student network in order to facilitate feature recon- +struction. RD4AD [15] and RSTPM share certain similarities in their learning. +RSTPM employs two pairs of teacher-student networks for feature reconstruc- +tion, whereas RD4AD only employs one pair of teacher-student networks. +RD4AD proposes a Multi-scale Feature Fusion (MFF) block and One-Class +Bottleneck (OCB) to form an embedding, which is used to eliminate redundant +features at multiple scales so that a single pair of teacher-student networks +can perform feature reconstruction effectively. The abnormal image features +extracted by the teacher-student network of RD4AD differ significantly dur- +ing inference. AST [17] concludes that the abnormal image features extracted +by the teacher-student model with the same structure are significantly sim- +ilar, so they propose an asymmetric teacher-student architecture to address +this issue. AST also introduces a normalized flow to avoid this problem and +prevent estimation bias caused by the inconsistency of the two network struc- +tures. Previous teacher-student architecture anomaly detection methods suffer +from overfitting as a result of inconsistency between neural network capacity +and knowledge amount. By incorporating the Context Similarity Loss (CSL) +and Adaptive Hard Sample Mining (AHSM) modules, Informative Knowledge +Distillation (IKD) [16] hopes to reduce overfitting. CSL can assist the stu- +dent network in comprehending the structure of a context-containing data +manifold. The AHSM can concentrate on difficult samples containing a lot of +information. + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +7 +2.1.2 One-Class Classification +One-class classification techniques rely more heavily on abnormal samples. +If the generated abnormal samples are of poor quality, the method’s perfor- +mance will be severely compromised. As demonstrated in Table 3, with the +exception of MemSeg [19], the training of other methods relies on SVDD +and Cross-Entropy loss; consequently, the performance of the vast majority of +methods is marginally inadequate. +Table 3: A summary of one-class classification methods regarding loss +function, pre-trained model, and highlights. +Method +Loss Function +Pre-trained +Highlights +Patch SVDD [20] +Cross-Entropy, SVDD +- +The paper divides image into patches and sends them to SVDD for training. +DDSPSVDD [21] +L2, SVDD +VGG +DSPSVDD takes reconstruction error into model training. +SE-SVDD [22] +SVDD +ResNet +The paper proposes a Semantic Correlation module (SCB) to represent +abnormal semantics information. +MOCCA [23] +L2, SVDD +- +The paper extends a single boundary to a hard boundary and a soft boundary, +it also trains AE as feature exactor. +[24] +Cross-Entropy +Xception [25] +The paper uses Xception to train a classification network. +PANDA [26] +SVDD, Log-Likelihood +DN2 [27] +The paper introduces a method to avoid combating collapse in model +adaption. +[28] +Cross-Entropy, +Contrastive +ResNet +The paper presents a novel distribution-augmented contrastive learning to +enhance the representing ability of network. +CutPaste [29] +Cross-Entropy +EfficientNet +The paper applies “cut and paste” augmentation into binary anomaly +classification. +[30] +Cosine Similarity, +Contrastive +- +The paper applies some dynamic local augmentation to generate negative +samples. +CPC-AD [31] +InfoNCE +- +The paper applies Contrastive Predictive Coding (CPC) model to AD and +get an anomaly score through pixel-wise loss. +MemSeg [19] +L1, Focal +ResNet +The paper artificially creates anomalies in the foreground of products and +makes detecting artificial anomalies a segmentation task. +Anomaly detection can also be viewed as a One-Class Classification (OCC) +problem, which has inspired some research. As depicted in Figure 3, the +method finds a hypersphere to distinguish normal sample features from abnor- +mal sample features during training. During inference, the method determines +whether the sample is abnormal based on the relative position of the test +sample’s features and the hypersphere. Since the training set does not con- +tain abnormal samples, some methods create abnormal samples artificially to +improve the accuracy of the hypersphere. +SVDD [32] is a classic algorithm in the OCC problem, PatchSVDD [20] +DSPSVDD [21] and SE-SVDD [22] improve it for industrial image AD. +PatchSVDD [20] divides the image into uniform patches and sends them to the +model for training, which significantly enhances the model’s ability to detect +anomalies. DSPSVDD [21] designs an improved comprehensive optimization +objective for the deep SVDD model that simultaneously considers hypersphere +volume minimization and network reconstruction error minimization to extract +deep data features more effectively. SE-SVDD proposes a Semantic Correla- +tion module (SCB) to improve the representation of abnormal semantics and +the accuracy of anomaly localization by extracting multi-level features. +MOCCA [23] employs multi-layer features for anomaly detection. MOCCA, +unlike SE-SVDD, uses an autoencoder to extract features and locates the +boundary position of normal features at each layer. And Sauter et al. [24] + +Springer Nature 2021 LATEX template +8 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +Normal Sample +Artificial Abnormal Sample +Abnormal Sample +Normal Sample +Training +Testing +Fig. 3: One-class classification model architecture. +attempt to use the Xception network for classification and obtained results +comparable to SVDD. FCDD [33] employs a fully convolutional neural net- +work for OCC. Since the relative positions of the features of each image layer +do not change during the convolution process, FCDD yields more interpretable +results than alternative methods. +PANDA [26] examines the migration method of pre-trained features and +introduces the early stopping mechanism to the OCC problem. In addition, +Reiss et al. [34] investigate the issue of catastrophic forgetting in PANDA. +They propose a new loss function capable of overcoming the failure modes of +both center-loss and contrastive-loss methods and replacing Euclidean distance +with a confidence-invariant angular center loss for prediction. +DisAug CLR [28] proposes a two-stage anomaly detection framework, in +which the first stage hinders the uniformity of contrastive representations by +means of a novel distribution-enhanced contrastive learning. After comparative +learning, abnormal and normal sample representations are easier to distinguish. +While the second stage builds a one-class classifier using the representations +learned in the first stage. Yoa et al. [30] presents a novel dynamic local aug- +mentation to generate negative image pairs from a normal training dataset, +which is effective for anomaly detection. Contrastive Predictive Coding (CPC) +[35] model is utilized by De et al. [31] for anomaly detection and segmentation, +which uses patch-wise contrastive loss as anomaly score to localize anomalies. +CutPaste [29] is a representative example of an OCC method for data +augmentation. It generates abnormal images by cutting and pasting por- +tions of normal images, allowing the network to distinguish abnormal images. +Additionally, segmentation-based methods are useful. This method puts more +emphasis on pixel-level anomaly localization. When the flow is known, Ique- +bal et al. [36] demonstrate that the maximum a posterior estimation of image +labels can be formulated as a continuous max-flow problem. Then, anomaly +segmentation is accomplished by obtaining flows iteratively using a novel +Markov random field on the image domain. The technique shows its adaptabil- +ity using a dataset for metal additive manufacturing anomaly detection [37]. +MemSeg [19] stores the features of normal images in a memory bank in order to + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +9 +improve the segmentation network’s ability to distinguish abnormal regions. In +order to prevent the influence of background factors, MemSeg only introduces +anomalies in external data sets in the foreground of items, which is another +reason for its excellent performance. +2.1.3 Distribution Map +Distribution-map based methods necessitate a suitable mapping objective +for training, and the choice of mapping method impacts model performance. +As shown in Table 4, Normalizing Flows (NF)-based methods predominate. +As a generative model, NF has a strong mapping ability, and it has also +demonstrated good performance in AD tasks. +Table 4: A summary of distribution-map based methods regarding loss +function, pre-trained model, and highlights. +Method +Loss Function +Pre-trained +Highlights +[38] +PCA +ResNet +The paper uses PCA and ResNet to extract features and count their +distribution. +[39] +Cross-Entropy +ResNet, +EfficientNet +The paper establishes a model of normality by fitting a multivariate Gaussian +to feature representations of a pre-trained network. +[40] +Mahalanobis Distance +EfficientNet +The paper generates a multi-variate Gaussian distribution for the normal class +and mitigates the catastrophic forgetting in past research. +PEDENet [41] +Log-Likelihood, +Cross-Entropy, +Regularization +- +The model can predict the location of the patch and compare it with the +actual location to judge the abnormality. +PFM [42] +L2 +ResNet +The paper proposes the bidirectional and multi-hierarchical bidirectional pre- +trained feature mapping based on the vanilla feature mapping. +PEFM [43] +L2 +ResNet +The paper introduces position encoding into PFM. +FYD [44] +L2 +ResNet +The paper aligns samples at image and feature levels to detect anomalies. +DifferNet [45] +Log-Likelihood +ResNet +The paper is the first one to introduce normalizing flow into anomaly +detection. +CS-Flow [46] +Log-Likelihood +ResNet +The paper uses information of multi-scale feature maps and improve Differ- +Net. +CFlow-AD [47] +Log-Likelihood +ResNet +The paper introduces positional encoding into the conditional normalizing +flow framework. +CAINNFlow [48] +Log-Likelihood +ViT [49] +The paper uses VIT to replace ResNet and achieve better result. +FastFlow [50] +Log-Likelihood +ResNet +The paper introduces an alternate stacking of large and small convolution +kernels in the NF module to model global and local distribution. +AltUB [51] +Log-Likelihood +ResNet +The paper designs a module for normalizing flow based methods and imporve +their performance. +Distribution-map based methods are very similar to OCC-based methods, +with the exception that OCC-based methods concentrate on finding feature +boundaries, whereas mapping-based methods attempt to map features into +desired distributions. A common framework for those methods is shown in +Figure 4. This expected distribution is typically a MultiVariate Gaussian +(MVG) distribution. This type of method first employs a strong pre-trained +network to extract the features of normal images, and then maps the extracted +features to the Gaussian distribution using a mapping module. This distribu- +tion will be deviated from by the features of abnormal images that appear +during the evaluation. The abnormal probability can be calculated based on +the level of deviation. +Tailanian et al. [38] propose a contrario framework that applies statistical +analysis to feature maps produced by patch PCA and ResNet, which per- +forms well on leather samples, to detect anomalies in images. By fitting a +multivariate Gaussian to the feature representations of a pre-trained network, + +Springer Nature 2021 LATEX template +10 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +. +. +. +Normal Samples +Pre-trained +Network +Origin Features +Mapping +Module +Well-distributed Features +Normal Feature +Abnormal Feature +Abnormal Sample +Origin Distribution +Mapped Distribution +Fig. 4: Distribution-map based method architecture. +Rippel et al. [39] establish a model of normality. Nonetheless, the issue of +catastrophic forgetting remains unresolved. Based on the relationship between +generative and discriminative modeling, Rippel et al. [40] generate a multi- +variable Gaussian distribution for the normal class and prove the efficacy of +this concept on Deep SVDD and FCDD, which mitigates the catastrophic for- +getting observed in previous research. PEDENet [41] framework consists of a +Patch Embedding (PE) network, a Density Estimation (DE) network, and a +Location Prediction (LP) network. At first, the PE module is used to reduce +the size of the features that the pre-trained network has extracted. Then, using +the DE module, which was inspired by the Gaussian mixture model, and the +LP module, the model can predict the relative position of the patch embedding +and, based on the difference between the predicted result and the actual result +during inference, decide if the image is abnormal. Pre-trained Feature Map- +ping (PFM) [42] proposes bidirectional and multi-hierarchical bidirectional +pre-trained feature mapping to enhance the performance of vanilla feature +mapping. In addition, Wan et al. [43] add position encoding to the PFM +framework and propose a novel Position Encoding enhanced Feature Map- +ping (PEFM) [43] to further enhance PFM. FYD [44] introduce registration +to industrial image AD for the first time. FYD suggests a coarse-to-fine align- +ment method that starts with aligning the foreground of objects at the image +level. Next, in the refinement alignment stage, non-contrastive learning is used +to increase the similarity of features between all corresponding positions in a +batch. +Normalizing Flows (NF) [52] is a technique for constructing complex dis- +tributions by transforming a probability density via a series of invertible +mappings. NF methods extract features from normal images from a pre-trained +model, such as ResNet [53] or Swin Transformer [54], and transform the fea- +ture distribution as a Gaussian distribution during the training phase. In the +test phase, after passing through NF, the features of abnormal images will + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +11 +deviate from the Gaussian distribution of the training phase, which is the +most important principle for classifying anomalies. DifferNet [45] is the first +research to use NF to address the industrial image AD issue. By incorporating +cross-convolution blocks within the normalizing flow to assign probabilities, +CS-Flow [46] makes use of the context within and between multi-scale fea- +ture maps to improve DifferNet. CFlow-AD [47] adds positional encoding to +the framework for conditional normalizing flow to achieve superior results. +In addition, CFlow-AD [47] analyzes in depth why the multivariate Gaussian +assumption is a reasonable prior in earlier models and why the more gen- +eral NF framework aims to converge to similar results with less computation. +FastFlow [50] introduces an alternate stacking of large and small convolution +kernels in the NF module to model global and local distribution efficiently. +CAINNFlow [48] enhances the performance of the model by introducing the +attention mechanism CBAM [55] to the NF module. In techniques such as +FastFlow and CFlow-AD, the feature distribution center is not 0 and their per- +formance is unstable. Kim et al. [56] propose a simple solution AltUB [51] that +uses alternating training to update the base distribution of normalizing flow +for anomaly detection in order to solve the problem. AltUB verifies the effect +of CFlow-AD and FastFlow using AltUB. +2.1.4 Memory Bank +As illustrated in Table 5, memory-based methods regularly do not require +the loss function for training, and models are constructed quickly. Their per- +formance is ensured by a robust pre-training network and additional memory +space, and this type of method is currently the most effective in VAD tasks. +Table 5: A summary of memory bank based methods regarding loss +function, pre-trained model, and highlights. +Method +Loss Function +Pre-trained +Highlights +SPADE [57] +- +ResNet +The paper uses multi-resolution feature to detect anomalies based on KNN. +[56] +- +ResNet +The paper reduces the computational cost for the inverse of multi-dimensional +covariance tensor so that bigger resolution image can be applied. +SOMAD [58] +- +ResNet +The paper maintains normal characteristics by using topological memory +based on multi-scale features. +GCPF [59] +- +ResNet +The paper processes normal features into multiple independent multivariate +Gaussian clustering. +MSPB [60] +Kmeans, Cosine +Similarity, SVDD +VGG +The paper enhances network representation capabilities by learning patch +position relationships. +SPD [61] +Focal, InfoNCE, SPD, +Cosine Similarity +- +Design a contrastive learning method to retrain ResNet to enhance the ability +of defect representation. +PatchCore [62] +- +ResNet +The paper introduces a core-set sampling method to build a memory bank. +CFA [63] +SVDD +ResNet +The paper improves PatchCore so that image features are distributed on a +hypersphere. +FAPM [64] +- +ResNet +The paper puts different position features of the image into different memory +banks to speed up retrieval. +N-pad [65] +Mahalanobis Distance, +Log-Likelihood +ResNet +The paper allows for possible edge misalignment by estimating a nominal +distribution for each pixel using the pixel’s neighborhood features. +The primary distinction between memory bank-based methods and OCC- +based methods, is that memory-based methods, such as SVDD, require +additional memory space to store image features. As shown in Figure 5, +these methods require minimal network training and only require sampling or + +Springer Nature 2021 LATEX template +12 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +. +. +. +Normal Samples +Pre-trained +Network +Abnormal Samples +Sampling or Mapping +Basic Memory Bank +Memory Bank and Testing Samples +Normal Samples +Training set normal feature +Testing set abnormal feature +Testing set normal feature +Distance from neighbor +Fig. 5: Memory bank based method architecture. +mapping the collected normal image features for inference. During inference, +features of the test image are compared to features in the memory bank. The +abnormal probability of the test image is equal to the spatial distance from +the normal features in the memory bank. +K Nearest Neighbors (KNN) [66] is a widely used algorithm for unsuper- +vised anomaly detection, but it operates only at the sample level. Semantic +Pyramid Anomaly Detection (SPADE) [57] is inspired by KNN and utilizes +correspondences based on a multi-resolution feature pyramid to obtain pixel- +level anomaly segmentation results. PaDim [67] employs multivariate Gaussian +distributions to construct a probabilistic representation of the normal class. +Consequently, the memory bank size is determined solely by the image resolu- +tion and not by the size of the training set. PaDiM requires the batch-inverse +of the multidimensional covariance tensor, which makes it challenging to scale +up to larger CNNs due to the increased feature size. To reduce the computa- +tional cost of the inverse by a factor of three, Kim et al. [56] generalize random +feature selection into semi-orthogonal embedding. +Meanwhile, Self-organizing Map for Anomaly Detection (SOMAD) [58] and +GCPF [59] enhance the storage of normal features. SOMAD preserves normal +characteristics by employing topological memory based on multi-scale features. +While GCPF transforms standard characteristics into multiple independent +multivariate Gaussian clustering +PatchCore [62] is a significant advancement in industrial image AD that +significantly raises the performance for MVTec AD. PatchCore contains two +special points. First, the memory bank of PatchCore is coreset-subsampled to +ensure a low inference cost while maximizing performance. PatchCore then +determines whether the test sample is abnormal based on the distance between +the test sample’s nearest neighbor feature in its memory bank and other +features. This process of reweighting makes PatchCore more robust. Since +PatchCore was proposed, numerous improved methods have been developed + +atttSpringer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +13 +on its foundation. Coupled-hypersphere-based Feature Adaptation (CFA) is +proposed by Lee et al. [63] to obtain target-oriented features. The center and +surface of the hypersphere in the memory bank are obtained through trans- +fer learning, and the positional relationship between the test feature and the +coupled-hypersphere can be used to determine whether it is abnormal or not. +FAPM [64] is comprised of numerous patch-wise and layer-wise memory banks +located in various places. FAPM calculates the features in different memory +banks independently during inference, which significantly accelerates inference +speed. N-pad [68] allows for the possibility of marginal misalignment by esti- +mating a per-pixel nominal distribution using neighboring and target pixel +features. In addition, anomaly scores are deduced using both Mahalanobis +and Euclidean distances between target pixels and the estimated distribution. +Similarly, Bae et al. [65] model the cumulative histogram using location infor- +mation as conditional probabilities, and neighborhood information was used to +establish the normal feature distribution. Furthermore, this work introduces +the first refinement approach in the anomaly detection and localization prob- +lem, using synthetic anomalous images to improve the anomaly map based on +the input image, as well as using neighborhood and location information to +estimate the distribution. +By learning the embedding position information and comparing the +extracted features with the normal embedding during inference, Tsai et +al. [60] propose a method to improve the network’s ability to represent +data. It is also based on the concept of self-supervised learning. Zou et +al. [61] use contrastive learning to train the backbone network and propose a +new data augmentation method called SPD to push the network to differenti- +ate between two images with slight differences. In addition, they demonstrate +the representation capability of the backbone network using PatchCore [62]. +2.2 Reconstruction based methods +Reconstruction-based methods primarily self-train encoders and decoders +to reconstruct images for anomaly detection, which makes them less reliant on +the pre-trained model and increases their ability to detect anomalies. However, +its image classification capability is poor due to its inability to extract high- +level semantic features. As shown in Table 6, the loss functions of various +methods are comparable; however, their performance varies due to different +reconstruction model paradigms and abnormal sample construction methods. +The structure of the reconstruction-based technique is depicted in Figure 6. +During the training process, normal or abnormal images are sent to the recon- +struction network, and the reconstruction loss function is used to guide the +training of the reconstruction network. Finally, the reconstruction network can +restore the reconstruction image in a manner similar to the original normal +image. In the inference stage, the comparison model compares the original +image to the reconstructed image to generate a prediction. In contrast to +the variety of methods for feature embedding, the majority of reconstruction- +based methods only differ in the construction of the reconstruction network. + +Springer Nature 2021 LATEX template +14 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +Table 6: A summary of reconstruction based methods. +Method +Loss Function +Pre-trained +Highlights +(1) Autoencoder Model +[69] +L2, SSIM +- +The paper firstly takes SSIM as a loss to reconstruct image and detect +anomalies. +[70] +L2, SSIM +- +The paper proposes two AEs and reduces style change during image recon- +struction. +UTAD [71] +L1, Adversarial +VGG +The paper uses two-stage reconstruction to generate high-fidelity images to +avoid reconstruction errors. +DFR [72] +L2 +VGG +The paper proposes to reconstruct and compare at the feature level to detect +anomalies. +ALT [73] +L1, Perceptual, +Adversarial +VGG +The paper proposes an adaptive attention-level transition strategy and uses +perceptual loss to improve reconstruction quality. +P-Net [74] +L1, Adversarial +- +The paper designs a new architecture for anomaly detection. +[75] +L1, L2 +- +The paper adds skip-connection in reconstruction network and adds noise +during training to improve reconstruction sharpness. +[76] +L2 +VGG +The paper proposes a dense feature fusion module to assist reconstruction. +[77] +L2, Adversarial +- +The paper uses memory to help reconstructing images. +EdgRec [78] +L2, SSIM +- +The paper reconstructs from the gray value edge and preserves the high- +frequency information with skip-connection. +PAE [79] +L2, Cross-Entropy +- +The paper gradually increases the resolution of the input image during +training. +SMAI [80] +L2, SSIM +- +The paper masks and inpaintings image by superpixel. +RIAD [81] +L2, MSGMS, SSIM +- +The paper proposes to inpaint and reconstruct images by patch. +I3AD [82] +L1, Adversarial +- +The paper gradually masks the high anomaly probability areas and recon- +structs them. +[83] +L2 +- +The paper proposes to reconstruct the anomalous area differently from the +original image. +[84] +L2, SSIM, GMS +- +Similar to I3AD, but the paper adds skip connections to reconstruction +network. +DREAM [85] +L2, SSIM, Focal +- +The paper designs a method to generate abnormal images and uses U-Net to +distinguish anomalies after reconstruction. +DSR [86] +L2, Focal +- +The paper generates abnormal samples in feature level and perform better +than DRAEM. +NSA [87] +L2, Cross-Entropy +- +The paper generates abnormal samples by pasting parts of other normal +samples, which is the SOTA method without extra data. +SSPCAB [88] +L2 +- +The paper designs a “plug and play” self-supervised block to improve the +reconstruction ability of many methods. +SSMCTB [89] +L2 +- +This paper replaces the SE-layer in SSPCAB with transformer architecture. +[90] +Cross-Entropy +- +The paper guides reconstruction using gradient descent with VAE. +[91] +Attention +Disentanglement +- +The paper proposes to use disentanglement VAE to detect anomalies. +DGM [92] +L2, Log-Likelihood +- +The paper proposes to use non-regularized objective functions for training +VAE under heterogeneous datasets. +FAVAE [93] +Log-Likelihood +VGG +The paper uses VAE to model the distribution of features extracted by its +pre-trained model. +[94] +L2, Cross-Entropy +- +The paper uses VQ-VAE to construct a discrete latent space and reconstructs +images based on the latent space. +(2) GAN Model +SCADN [95] +L2, Adversarial +- +The paper masks part of image and reconstruct image with GAN during +training. +AnoSeg [96] +L1, L2, Adversarial +- +The paper generates abnormal samples through a GAN and detects anomalies +with the discriminator. +OCR-GAN [97] +L1, L2, Adversarial +- +The paper uses the Frequency Decoupling module to decouple and reconstruct +images. +(3) Transformer Model +VT-ADL [98] +L2, SSIM, +Log-Likelihood +- +The paper proposes a transformer-based framework to reconstruct images and +detects anomalies. +ADTR [99] +L2, Cross-Entropy +EfficientNet +The paper makes it simple to identify anomalies when reconstruction fails by +reconstructing features from pre-trained network. +AnoViT [100] +L2 +ViT +The paper uses a pre-trained ViT to extract features and reconstruct images. +HaloAE [101] +L2, Cross-Entropy, +SSIM +VGG +The paper introduces an auto-encoder architecture based on a transformer +with HaloNet. +InTra [102] +L2, GMS, SSIM +- +The paper leverages more global information to repair images with trans- +former. +MSTUnet [103] +L2, SSIM, Focal +- +The paper uses swin transformer for inpainting masked images and detects +anomalies. +MeTAL [104] +L1, SSIM +- +The paper uses information from neighbor patches to inpainting images, +better accounting for local structural information. +UniAD [5] +L2 +EfficientNet +The paper trains all categories of products in one model. +(4) Diffusion Model +AnoDDPM [105] +L2, Log-Likelihood +- +The paper is the first to apply diffusion model for industrial image anomaly +detection. +[106] +L2, Log-Likelihood +- +The paper significantly speeds up the inference process of anomaly detection +using diffusion model. + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +15 +Reconstruction +Network +Reconstruction +Network +Comparison +Model +Reconstruction +Loss +Training +Testing +Predict +Fig. 6: Reconstruction based architecture. +Reconstruction-based methods outperform feature-embedding methods at the +pixel level due to their ability to identify anomalies through pixel-level com- +parison. In addition, the majority of reconstruction-based methods are trained +from scratch without employing robust pre-trained models, which results in +inferior performance compared to image-level feature embedding. +2.2.1 Autoencoder +Autoencoder (AE) is the most prevalent reconstruction network for AD. +Numerous other reconstruction networks also consist of encoder and decoder +components. Bergmann et al. [69] investigate the influence of Structure Simi- +larity Index Measure (SSIM) and L2 loss on AE reconstruction and anomaly +segmentation, providing numerous suggestions for future research. +How to resolve the difference between the reconstructed image and the orig- +inal image is the most foundational principle. There are regularly differences +in style between the reconstructed image and the original image, resulting +in over-detection. Chung et al. [70] present an Outlier-Exposed Style Dis- +tillation Network (OE-SDN) to preserve the style translation and suppress +the content translation of the AE in order to avoid over-detection. As the +anomaly prediction, Chung et al. replaced the difference between the original +image and the reconstruction image of AE with the difference between the +reconstruction image of OE-SDN and the reconstruction image of AE. Unsu- +pervised Two-stage Anomaly Detection (UTAD) [71] brings an IE-Net and +Expert-Net to extract and utilize impressions for anomaly-free and high-fidelity +reconstructions, thereby offering the framework interpretable. +Reconstruction-based methods are nearly effective as feature embedding +methods when utilizing features at different scales. Similar to teacher-student + +1Springer Nature 2021 LATEX template +16 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +architecture, Deep Feature Reconstruction (DFR) [72] method detects anoma- +lous through reconstruction at the level of features. DFR obtains multiple spa- +tial context-aware representations from a network that has been pre-trained. +Then, DFR reconstructs features using a deep yet efficient convolutional AE +and detects anomalous regions by comparing the original features to the +reconstruction features. Yan et al. [73] propose a novel Multi-Level Image +Reconstruction (MLIR) framework that forms the reconstruction process as +an image denoising task at different resolutions. Thus, MLIR accounts for the +detection of both global structure anomalies and detail anomalies. +Modifying the structure of AE can also improve its capacity for recon- +struction. Zhou et al. [74] introduce P-Net to compare the difference in +structure between the original and reconstruction images. Collin et al. [75] +include skip-connections between encoder and decoder to improve the recon- +struction’s sharpness. In addition, they propose corrupting them with a +synthetic noise model to prevent the network from convergently mapping +identities, and they introduce the innovative Stain noise model for this pur- +pose. Tao et al. [76] also operate at the feature level; they employ a dense +feature fusion module to obtain a dense feature representation of double +input in order to help reconstruction in the dual-Siamese framework. Hou et +al. [77] also use skip-connections to enhance the quality of reconstruction. In +addition to achieving expected results, they add a memory module to skip- +connections. Liu et al. [78] reconstruct the original RGB image from its gray +value edges, with the skip-connections in the model preserving the image’s +high-frequency information to better guide the reconstruction. Progressive +Autoencoder (PAE) [79] improves autoencoder reconstruction performance +through progressive learning and modified CutPaste augmentation. During +training, PAE achieves progressive learning by gradually increasing the input +image’s resolution. +Masking and repainting is an effective method for self-supervised learning. +The Superpixel Masking And Inpainting (SMAI) technique was developed by +Li et al [80]. SMAI divides the image into multiple blocks of superpixels and +trains the inpainting module to reconstruct a superpixel within a mask. SMAI +performs masking and inpainting superpixel-by-superpixel on the test image +during inference, and then compares the reconstruction image to the test image +to distinguish abnormal regions. Iterative Image Inpainting Anomaly Detection +(I3AD) is a method proposed by Nakanishi et al. [82] that reconstructs partial +regions based on the anomaly map. I3AD improves reconstruction quality by +only reconstructing inpainting masks over images, and only masking regions +with a high probability of abnormality. SSM [84] is conceptually similar to +I3AD. SSM adds skip-connections to the reconstruction network and predicts +the mask region as the training target. RIAD [81] randomly masks a portion +of the training set image at the patch level and reconstructs it using a U- +Net encoder-decoder network [107]. During inference, RIAD combines multiple +random masks and reconstruction patches to generate a reconstructed image, +which is then compared to the original image. Multi-Scale Gradient Magnitude + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +17 +Similarity (MSGMS) outperforms SSIM as an anomaly score, according to +RIAD. +DRAEM [85] is representative of reconstruction-based techniques. DRAEM +synthesizes abnormal images and reconstructs them as normal by introduc- +ing external datasets, which greatly improves the reconstruction network’s +generalization capacity. In addition, DRAEM feeds the original image and +the reconstructed image into the segmentation network to predict abnor- +mal regions, significantly enhancing the model’s ability to segment anomalous +regions. Nevertheless, DRAEM is susceptible to failure when synthesizing +near-in-distribution anomalies. DSR [86] proposes an architecture based on +quantized feature space representation and dual decoders to circumvent the +requirement for image-level anomaly generation. By sampling the learned +quantized feature space at the feature level, the near-in-distribution anomalies +are generated in a controlled way. NSA [87] does not use external data for data +data augmentation and adopts more data augmentation methods, allowing it +to outperform all previous methods that learned without utilizing additional +datasets. In contrast to other methods that attempt to reconstruct abnormal +images into normal images, Bauer [83] proposes reconstructing the abnormal +areas of the image so that they deviate from the original image’s appearance. +This approach produces comparable results to other methods. +In contrast to classical reconstruction-based methods, Ristea et al. [88] pro- +pose integrating reconstruction-based functionality into a Self-Supervised Pre- +dictive Architectural Building Block (SSPCAB). SSPCAB can be incorporated +into models such as DRAEM and CutPaste to enhance those models. Self- +Supervised Masked Convolutional Transformer Block (SSMCTB) [89] trans- +forms the SE-layer [108] in SSPCAB into a channel-wise transformer block +and achieves superior results. +VAE is a variant of AE, with the difference that the intermediate variables +of VAE are data from a normal distribution. Naturally, VAE has superior inter- +pretability. Dehaene et al. [90] iteratively guide reconstruction using gradient +descent with energy defined by the reconstruction loss, thereby overcoming the +tendency of VAE to produce blurry reconstructions and preserving the nor- +mal high-frequency structure. The variational autoencoder is trained with an +attention disentanglement loss by Liu et al [91]. Anomaly inputs in this VAE +will result in Gaussian-deviating latent variables during gradient backpropaga- +tion and attention generation. This deviation can be used to locate anomalies. +According to Matsubara et al. [92], datasets are commonly heterogeneous +rather than regularized, and non-regularized objective functions are more suit- +able for training VAE models on heterogeneous datasets. FAVAE [93] employs +VAE to model the distribution of features extracted by the pre-trained model, +implicitly simulating richer anomalies and enhancing the model’s generaliza- +tion. Wang et al. [94] use VQ-VAE to create a discrete latent space, resample +the discrete latent code deviate from the normal distribution, and reconstruct +the image using the resampled latent code. VQ-VAE reconstructs images that +are closer to the training set’s normal images. + +Springer Nature 2021 LATEX template +18 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +2.2.2 Generative Adversarial Networks +The stability of the reconstruction model based on Generative Adversarial +Networks (GANs) is not as good as that of AE, but the discriminant network +has a better effect on some scenes described as follows. +During training, Semantic Context based Anomaly Detection Network +(SCADN) [95] masks a portion of the image and reconstructs it with GAN. +SCADN detects anomalies for inference by comparing the input image to +the reconstruction image. In addition to masking images, AnoSeg [96] utilizes +hard augmentation, adversarial learning, and channel concatenation to gener- +ate abnormal samples. AnoSeg then trains GAN to generate normal samples. +AnoSeg differs from the AE reconstruction model in that its objective func- +tion incorporates both reconstruction loss and adversarial loss. OCR-GAN [97] +utilizes the Frequency Decoupling (FD) module to decouple the image into +information combinations of different frequencies, and then reconstructs and +combines the information of these different frequencies to yield reconstructed +images. During inference, the model can identify a statistically significant +difference between the frequency distributions of normal and abnormal images. +2.2.3 Transformer +Transformer has a higher capacity to represent global information, which +gives it the potential to surpass AE and become a new reconstruction network +foundation for anomaly detection. Mishra et al. [98] propose a transformer- +based framework to reconstruct images at the patch level and employ a +gaussian mixture density network to localize anomalous regions. You et al. [99] +propose ADTR for reconstructing pre-trained features. According to them, +the use of transformers prevents well-reconstructed anomalies, making it +easy to identify anomalies when reconstruction fails. Lee et al. [100] intro- +duce a vision transformer-based encoder-decoder model (AnoViT) and assert +that AnoViT is superior to the CNN-based l2-CAE in the issue of anomaly +detection. HaloAE [101] implements transformer into HaloNet [109] and facil- +itates image reconstruction by reconstructing features to achieve competitive +results on the MVTec AD dataset. A common self-supervised learning method +for reconstruction-based anomaly detection is the reconstruction of masked +images. However, traditional CNNs find it difficult to extract global context +information. In order to accomplish this, Pirnay et al. [102] propose Inpaint- +ing Transformer (InTra), which integrates information from larger regions +of the input image. InTra is representative of trained-from-scratch methods. +Masked Swin Transformer Unet (MSTUnet) [103] is comparable to InTra, but +MSTUnet employs additional enhancements [110] when simulating anomalies, +thereby achieving superior results. De et al. [104] used the neighbor patch +to reconstruct the masked patch and also achieved a powerful reconstruction +ability. + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +19 +2.2.4 Diffusion Model +Diffusion model [111] is a recently popular generative model that can also +be utilized for reconstruction-based anomaly detection. AnoDDPM [105] is, to +the best of our knowledge, the first to apply the diffusion model to industrial +image anomaly detection. In comparison to GAN-based methods, AnoDDPM +with simplex noise can also capture large anomaly regions without the need +for large datasets. When applying the diffusion model to anomaly detection, +Teng et al. [106] primarily make two improvements. As a replacement metric +for reconstruction loss, a time-dependent gradient value of normal data dis- +tribution is used to measure the defects. In addition, they develop a novel +T-scales method to reduce the required number of iterations and accelerate +the inference process. +3 Supervised Anomaly Detection +Despite the fact that abnormal data is diverse and difficult to collect, it +is still possible to collect abnormal samples in real-world scenarios. Therefore, +some research focuses on how to train models for anomaly detection using a +small number of abnormal samples and a large number of normal samples. +Chu et al. [112] propose a semi-supervised framework for detecting anoma- +lies in the presence of significant data imbalance. They assume that changes in +loss values during training can be used to identify abnormal data as features. To +achieve this, they train a reinforcement learning-based aneural batch sampler +to amplify the difference in loss curves between anomalous and non-anomalous +regions. FCDD [33] is an unsupervised method that synthesizes abnormal sam- +ples for training the OCC model. FCDD [33] is an unsupervised method that +synthesizes abnormal samples for training the OCC model. This concept is +transferable to other OCC methods. Venkataramanan et al. [113] propose a +Convolutional Adversarial Variational Autoencoder (CAVGA) with Guided +Attention that can be applied equally to cases with and without abnormal +images. In an unsupervised setting, CAVGA is guided to focus on all normal +regions of an image by an attention expansion loss. CAVGA uses a comple- +mentary guided attention loss in the weakly supervised setting to minimize the +attention map corresponding to abnormal regions of the image while focusing +on normal regions. Bovzivc et al. [114] examine the influence of image-level +supervision information, mixed supervision information, and pixel-level super- +vision information on surface defect detection tasks within the same deep +learning framework. Bovzivc et al. find that a small number of pixel-level anno- +tations can help the model achieve performance comparable to full supervision. +DevNet [115] uses a small number of abnormal samples to realize fine-grained +end-to-end differentiable learning. Wan et al. [116] propose a Logit Inducing +Loss (LIS) for training with imbalanced data distribution and an Abnormality +Capturing Module (ACM) for characterizing anomalous features in order to +effectively utilize a small amount of anomalous information. DRA [117] pro- +poses a framework for learning disentangled representations of seen, pseudo, + +Springer Nature 2021 LATEX template +20 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +and latent residual anomalies in order to detect both visible and invisible +anomalies. +Besides, a number of studies fail to account for the unbalanced distribution +of normal and abnormal samples and rely primarily on abnormal samples for +supervised training. Sindagi et al. [118] investigate the domain transfer prob- +lem of datasets for anomaly detection in various settings. Dual Weighted PCA +(DWPCA) is an algorithm proposed by Qiu et al. [119] for image registration +and surface defect detection. An interleaved Deep Artifacts-aware Attention +Mechanism (iDAAM) is proposed by Bhattacharya et al. [120] propose to +classify multi-object and multi-class defects in abnormal images. Zeng et +al. [121] view anomaly detection as a subset of target detection and designed a +Reference-based Defect Detection Network (RDDN) to detect anomalies using +template reference and context reference. Long et al. [122] investigate defect +detection in a tactile image, which has obvious benefits for fabric structure +defect detection in RGB images. In addition, there are methods that refer to +the concept of semantic segmentation. To detect defects in infrared thermal +volumetric data, Hu et al. [123] propose a hybrid multi-dimensional space and +temporal segmentation model. Ferguson et al. [124] use Mask Region-based +CNN architecture to detect and segment defects in X-ray images simultane- +ously. There are also numerous modified models on anomaly detection based +on the object detection and semantic segmentation model of natural images +under full supervision [125–127]. Here we will not discuss them one by one. +4 Industrial Manufacturing Setting +This section introduces the classification standards or application settings +that are more appropriate for industrial scenes, namely few-shot anomaly +detection, noisy anomaly detection, and 3D anomaly detection. +4.1 Few-shot Anomaly Detection +Few-shot visual anomaly detection (FSAD) [128, 129] is still in its infancy. +There are two settings in FSAD. The first setting is meta-learning [130]. In +other words, this setting requires a large amount of images as meta-training +dataset. Wu et al. [128] propose a novel architecture, called MetaFormer, that +employs meta-learned parameters to achieve high model adaptation capabil- +ity and instance-aware attention to localize abnormal regions. RegAD [130] +trains a model for detecting category-agnostic anomalies. In the test phase, the +anomalies are identified by comparing the registered features of the test image +and its corresponding normal images. The second setting relies on the vanilla +few-shot image learning. PatchCore [62], SPADE [57], PaDim [67] conduct the +ablation study on 16 normal training samples. None of them, however, are spe- +cialized in few-shot anomaly detection. Hence, it is necessary to develop new +algorithms that concentrate on native few-shot anomaly detection tasks. + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +21 +4.2 Noisy Anomaly Detection +Noisy learning is a classical problem for anomaly detection. Tan et al. [131] +employ a novel trust region memory update scheme to keep noise feature +point away from the memory bank. Yoon et al. [132] use a data refinement +approach to improve the robustness of one-class classification model. Qiu et +al. [133] propose a strategy for training an anomaly detector in the presence of +unlabeled anomalies, which is compatible with a broad class of models. They +create labelled anomalies synthetically and jointly optimize the loss function +with normal data and synthesis abnormal data. Chen et al. [134] introduce +an interpolated Gaussian descriptor that learns a one-class Gaussian anomaly +classifier trained with adversarially interpolated training samples. However, +the majority of the aforementioned approaches have not been verified on real +industrial image datasets. In other words, the effectiveness of the existing +anomaly detection methods may not be suitable for industrial manufacturing. +4.3 3D Anomaly Detection +Since the release of MVTec 3D-AD [7] dataset, several papers have focused +on anomaly detection in 3D industrial images. Bergmann [135] introduces +a teacher-student model for 3D anomaly detection. The teacher network is +trained to acquire general local geometric descriptors by recreating local +receptive fields. While the student network is taught to match the local 3D +descriptors of the pre-trained teacher network. Horwitz et al. [136] propose +BTF, a method that combines hand-crafted 3D representations (FPFH [137]) +with the representation method of 2D features (PatchCore [62]). Reiss et +al. [138] propose that the representational ability of self-supervised learning +is temporarily inferior to that of handcrafted features for 3D anomaly detec- +tion. Nevertheless, self-supervised characterization still has great potential +if large-scale 3D anomaly detection dataset are available. AST [17] employs +RGB image with depth information to enhance anomaly detection perfor- +mance. However, most of 3D VAD methods are specialized in RGB-D images. +While the 3D dataset in real-world industrial manufacturing consists of point +clouds, which means current 3D VAD methods cannot be directly deployed +in industrial manufacturing. Thus, there are still opportunities for 3D VAD +advancement. +5 Datasets and metrics +Datasets. Data is a crucial driving factor for machine learning, particu- +larly for deep learning. Principally, the difficulty of getting industrial photos +hampers the advancement of visual anomaly detection in industrial vision. +Table 7 demonstrates that the number and the size of VAD dataset are grad- +ually increasing, but most of them are not generated in a real production line. +The promising alternative approach is to fully utilize the industrial simulator + +Springer Nature 2021 LATEX template +22 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +to generate anomalous images, possibly reducing the gap between academic +research and the demands of industrial manufacturing. +Table 7: Comparison of datasets for anomaly detection. +Dataset +Class +Normal +Abnormal +Total +Annotation Level +Real or Synthetic +AITEX [139] +1 +140 +105 +245 +Segmentation mask +Real +BTAD [98] +3 +- +- +2,830 +Segmentation mask +Real +DAGM [140] +10 +- +- +11,500 +Segmentation mask +Synthetic +DEEPPCB [141] +1 +- +- +1,500 +Bounding box +Synthetic +Eycandies [142] +10 +13,250 +2,250 +15,500 +Segmentation mask +Synthetic +Fabric dataset [143] +1 +25 +25 +50 +Segmentation mask +Synthetic +GDXray [144] +1 +0 +19,407 +19,407 +Bounding box +Real +KolektorSDD [145] +1 +347 +52 +399 +Segmentation mask +Real +KolektorSDD2 [114] +1 +2,979 +356 +3,335 +Segmentation mask +Real +MIAD [146] +7 +87,500 +17,500 +105,000 +Segmentation mask +Synthetic +MPDD [147] +6 +1,064 +282 +1,346 +Segmentation mask +Real +MTD [148] +1 +- +- +1,344 +Segmentation mask +Real +MVTec AD [6] +15 +4,096 +1,258 +5,354 +Segmentation mask +Real +MVTec 3D-AD [149] +10 +2,904 +948 +3,852 +Segmentation mask +Real +MVTec LOCO-AD [7] +5 +2,347 +993 +3,340 +Segmentation mask +Real +NanoTwice [150] +1 +5 +40 +45 +Segmentation mask +Real +NEU-CLS [151] +1 +0 +1,800 +1,800 +Bounding box +Real +RSDD [152] +2 +- +- +195 +Segmentation mask +Real +Steel Defect Detection [153] +1 +- +- +18,076 +Image +Real +Steel Tube Dataset [154] +1 +0 +3,408 +3,408 +Bounding box +Real +VisA [61] +12 +9,621 +1,200 +10,821 +Segmentation mask +Real +Table 8: A summary of metrics used for anomaly detection. +Metric/Level +Formula +Remarks/Usage +Precision (P) ↑ +P = TP/(TP + FP) +True Positive (TP), False Positive (FP) +Recall (R) ↑ +R = TP/(TP + FN) +False Negative (FN) +True Positive Rate (TPR) ↑ +TPR = TP/(TP + FN) +False Positive Rate (FPR) ↓ +FPR = FP/FP + TN) +True Negative (TN) +Area Under the Receiver Operating Charac- +teristic curve (AU-ROC) ↑ +� 1 +0 (TPR) d(FPR) +Classification +Area Under Precision-Recall (AU-PR) ↑ +� 1 +0 (P) d(R) +Localization, Segmentation +Per-Region Overlap (PRO) [155] ↑ +PRO = +1 +N +� +i +� +k +Pi∩Ci,k +Ci,k +Total ground truth number (N)/ +Predicted abnormal pixels (P)/ +Defect ground truth regions (C)/ +Segmentation +Saturated Per-Region Overlap (sPRO) [149] ↑ +sPRO(P) = +1 +m +m +� +i=1 +min( Ai∩P +si +, 1) +Total ground truth number (m)/ +Predicted abnormal pixels (P)/ +Defect ground truth regions (A)/ +Corresponding saturation thresholds (s) / +Segmentation +F1 score ↑ +F1 = 2(P · R)/(P + R) +Classification +Intersection over Union (IoU) [156] ↑ +IoU = (H ∩ G)/(H ∪ G) +Prediction (H), Ground truth (G)/ +Localization, Segmentation +Metrics. Table 8 offers a comprehensive review of the metrics in cross- +modality neuroimage synthesis. The first column denotes the name of the +metric and the second column denotes the level. In other words, if the level +is up, the larger the metrics value, the better the performance. If the level +is down, the lower the metrics value, the better the performance. The third +column gives the detail for each metric, especially on how the metric accurately +indicates the performance of visual anomaly detection. From Table 8, it can +be easily observed that most of novel metrics are the variants of natural image + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +23 +segmentation and detection metrics, such as F1 score, AU-ROC or AU-PR. +However, these metrics can not correspond to the performance of VAD because +the tiny size of anomalies requires a greater weighting than the anomaly-free +regions. Hence, the validity of these metrics for VAD remains to be explored. +Table 9: Image AUROC Performance of Different Methods on MVTec AD. +The highest and second places are marked in red and blue. +Taxonomy +Method +Bottle +Cable +Capsule +Carpet +Grid +Hazelnut +Leather +Metal Nut +Pill +Screw +Tile +Toothbrush +Transistor +Wood +Zipper +Avg. +Memory Bank +PatchCore [62] +1.000 +0.997 +0.981 +0.982 +0.983 +1.000 +1.000 +1.000 +0.971 +0.990 +0.989 +0.989 +0.997 +0.999 +0.997 +0.992 +PatchCore Ensemble [62] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.996 +CFA [63] +1.000 +0.998 +0.973 +0.973 +0.992 +1.000 +1.000 +1.000 +0.979 +0.973 +0.994 +1.000 +1.000 +0.997 +0.996 +0.993 +FAPM [64] +1.000 +0.995 +0.986 +0.993 +0.980 +1.000 +1.000 +1.000 +0.960 +0.952 +0.994 +1.000 +1.000 +0.993 +0.995 +0.990 +N-pad [65] +1.000 +0.995 +0.994 +0.993 +0.987 +1.000 +1.000 +1.000 +0.980 +0.974 +1.000 +1.000 +0.996 +0.996 +0.993 +0.994 +N-pad Ensemble [65] +1.000 +0.998 +0.995 +1.000 +0.986 +1.000 +1.000 +1.000 +0.972 +0.989 +1.000 +0.997 +1.000 +0.994 +0.998 +0.995 +MSPB [60] +1.000 +0.988 +0.972 +0.934 +1.000 +0.996 +0.993 +0.978 +0.977 +0.941 +0.962 +1.000 +0.989 +0.997 +0.995 +0.981 +SPD [61] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.997 +SPADE [57] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.855 +[56] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.921 +SOMAD [58] +1.000 +0.988 +0.988 +1.000 +0.939 +1.000 +1.000 +0.997 +0.986 +0.955 +0.987 +0.986 +0.945 +0.992 +0.977 +0.979 +Teacher-Student +RD4AD [15] +1.000 +0.950 +0.963 +0.989 +1.000 +0.999 +1.000 +1.000 +0.966 +0.970 +0.993 +0.995 +0.967 +0.992 +0.985 +0.985 +STFPM [14] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.955 +Uninformed Students [11] +0.918 +0.865 +0.916 +0.695 +0.819 +0.937 +0.819 +0.895 +0.935 +0.928 +0.912 +0.863 +0.701 +0.725 +0.933 +0.857 +MKD [12] +0.994 +0.892 +0.805 +0.793 +0.780 +0.984 +0.951 +0.736 +0.827 +0.833 +0.916 +0.922 +0.856 +0.943 +0.932 +0.877 +STPM [13] +1.000 +0.996 +0.930 +0.987 +1.000 +0.998 +1.000 +1.000 +0.981 +0.968 +0.999 +0.979 +0.983 +0.993 +0.993 +0.987 +AST [17] +1.000 +0.985 +0.997 +0.975 +0.991 +1.000 +1.000 +0.985 +0.991 +0.997 +1.000 +0.966 +0.993 +1.000 +0.991 +0.992 +Distribution Map +[39] +0.998 +0.955 +0.938 +1.000 +0.817 +0.996 +0.997 +0.947 +0.884 +0.854 +0.998 +0.964 +0.963 +0.986 +0.978 +0.953 +[40] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.971 +PEDENet [41] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.928 +PFM [42] +1.000 +0.988 +- +1.000 +0.980 +1.000 +1.000 +1.000 +0.965 +0.918 +0.996 +0.886 +0.978 +0.995 +0.974 +0.975 +FYD [44] +1.000 +0.953 +0.925 +0.988 +0.989 +0.999 +1.000 +0.999 +0.945 +0.901 +0.988 +1.000 +0.992 +0.994 +0.975 +0.977 +FastFlow [50] +1.000 +1.000 +1.000 +1.000 +0.997 +1.000 +1.000 +1.000 +0.994 +0.978 +1.000 +0.944 +0.998 +1.000 +0.995 +0.994 +DifferNet [45] +0.990 +0.959 +0.869 +0.929 +0.840 +0.993 +0.971 +0.961 +0.888 +0.963 +0.994 +0.986 +0.911 +0.998 +0.951 +0.949 +CS-Flow [46] +0.998 +0.991 +0.971 +1.000 +0.990 +0.996 +1.000 +0.991 +0.986 +0.976 +1.000 +0.919 +0.993 +1.000 +0.997 +0.987 +CFLOW-AD [47] +0.989 +0.975 +0.988 +0.990 +0.988 +0.990 +0.996 +0.988 +0.984 +0.991 +0.965 +0.988 +0.952 +0.950 +0.991 +0.982 +CAINNFlow [48] +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +0.988 +0.987 +1.000 +0.972 +1.000 +0.998 +0.996 +0.996 +CS-Flow+AltUB [51] +1.000 +0.978 +0.981 +0.992 +1.000 +1.000 +1.000 +0.995 +0.970 +0.917 +0.999 +0.994 +0.952 +0.990 +0.985 +0.984 +One-Class +Classfication +Patch SVDD [20] +0.986 +0.903 +0.767 +0.929 +0.946 +0.920 +0.909 +0.940 +0.861 +0.813 +0.978 +1.000 +0.915 +0.965 +0.979 +0.921 +SE-SVDD [22] +0.986 +0.977 +0.985 +0.989 +0.972 +0.980 +0.987 +0.983 +0.967 +0.986 +0.923 +0.993 +0.972 +0.951 +0.979 +0.975 +MOCCA [23] +0.950 +0.760 +0.820 +0.860 +0.870 +0.800 +0.980 +0.850 +0.820 +0.840 +0.890 +0.970 +0.880 +1.000 +0.840 +0.875 +PANDA [26] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.865 +[34] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.872 +[28] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.865 +[30] +0.918 +0.883 +0.965 +0.894 +0.881 +0.962 +0.985 +0.926 +0.964 +0.972 +0.919 +0.958 +0.883 +0.892 +0.954 +0.930 +CPC-AD [31] +0.998 +0.880 +0.641 +0.809 +0.983 +0.996 +0.990 +0.845 +0.921 +0.897 +0.957 +0.878 +0.925 +0.803 +0.993 +0.901 +CutPaste [29] +0.982 +0.812 +0.982 +0.939 +1.000 +0.983 +1.000 +0.999 +0.949 +0.887 +0.946 +0.994 +0.961 +0.991 +0.999 +0.961 +MemSeg [19] +1.000 +0.982 +1.000 +0.996 +1.000 +1.000 +1.000 +1.000 +0.990 +0.978 +1.000 +1.000 +0.992 +0.996 +1.000 +0.994 +Reconst.-AE +UTAD [71] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.900 +ALT [73] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.910 +[75] +0.980 +0.890 +0.740 +0.890 +0.970 +0.940 +0.890 +0.730 +0.840 +0.740 +0.990 +1.000 +0.910 +0.950 +0.940 +0.890 +[76] +1.000 +0.983 +0.916 +0.968 +0.956 +0.994 +0.918 +0.977 +0.895 +0.981 +0.964 +1.000 +0.913 +0.983 +0.961 +0.961 +[77] +0.976 +0.844 +0.767 +0.866 +0.957 +0.921 +0.862 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+1.000 +0.997 +0.978 +0.963 +1.000 +0.982 +NSA [87] +0.977 +0.945 +0.952 +0.956 +0.999 +0.947 +0.999 +0.987 +0.992 +0.902 +1.000 +1.000 +0.951 +0.975 +0.998 +0.972 +[83] +0.950 +0.960 +0.980 +0.990 +0.990 +0.980 +0.990 +0.950 +0.980 +0.990 +0.970 +0.980 +0.970 +0.970 +0.990 +0.980 +DREAM+SSPCAB [88] +0.984 +0.969 +0.993 +0.982 +1.000 +1.000 +1.000 +1.000 +0.998 +0.979 +1.000 +1.000 +0.929 +0.995 +1.000 +0.989 +DREAM+SSMCTB [89] +0.994 +0.941 +0.971 +0.968 +1.000 +1.000 +1.000 +1.000 +0.988 +0.990 +1.000 +1.000 +0.960 +1.000 +1.000 +0.987 +NSA+SSPCAB [88] +0.977 +0.956 +0.954 +0.975 +0.999 +0.942 +0.999 +0.990 +0.992 +0.911 +1.000 +1.000 +0.956 +0.977 +0.998 +0.975 +NSA+SSMCTB [89] +0.977 +0.961 +0.955 +0.961 +1.000 +0.971 +1.000 +0.995 +0.995 +0.904 +1.000 +1.000 +0.962 +0.978 +0.999 +0.977 +FAVAE [93] +0.999 +0.950 +0.804 +0.671 +0.970 +0.993 +0.675 +0.852 +0.821 +0.837 +0.805 +0.958 +0.932 +0.948 +0.972 +0.879 +[94] +0.990 +0.720 +0.680 +0.710 +0.910 +0.940 +0.960 +0.830 +0.680 +0.800 +0.950 +0.920 +0.730 +0.960 +0.970 +0.850 +Reconst.-GAN +SCADN [95] +0.957 +0.856 +0.765 +0.504 +0.983 +0.833 +0.659 +0.624 +0.814 +0.831 +0.792 +0.981 +0.863 +0.968 +0.846 +0.818 +Anoseg [96] +0.980 +0.980 +0.840 +0.960 +0.990 +0.980 +0.990 +0.950 +0.870 +0.970 +0.980 +0.990 +0.960 +0.990 +0.990 +0.960 +OCR-GAN [97] +0.996 +0.991 +0.962 +0.994 +0.996 +0.985 +0.971 +0.995 +0.983 +1.000 +0.955 +0.987 +0.983 +0.957 +0.990 +0.983 +Reconst.-Transformer +ADTR [99] +1.000 +0.925 +0.925 +1.000 +0.978 +0.999 +1.000 +0.945 +0.933 +0.942 +1.000 +0.939 +0.980 +0.999 +0.970 +0.969 +AnoViT [100] +0.830 +0.740 +0.730 +0.500 +0.520 +0.880 +0.850 +0.860 +0.720 +1.000 +0.890 +0.740 +0.830 +0.950 +0.730 +0.780 +HaloAE [101] +1.000 +0.846 +0.884 +0.697 +0.951 +0.998 +0.978 +0.884 +0.901 +0.896 +0.957 +0.972 +0.844 +1.000 +0.997 +0.914 +InTra [102] +1.000 +0.703 +0.865 +0.988 +1.000 +0.957 +1.000 +0.969 +0.902 +0.957 +0.982 +1.000 +0.958 +0.975 +0.994 +0.950 +MSTUnet [103] +1.000 +0.914 +0.984 +0.999 +1.000 +1.000 +1.000 +1.000 +0.974 +1.000 +1.000 +1.000 +0.963 +1.000 +1.000 +0.989 +MeTAL [104] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.863 +Reconst.-Diffusion +VDD [106] +1.000 +0.968 +0.961 +0.969 +1.000 +0.999 +0.996 +0.972 +0.953 +0.996 +0.986 +0.998 +0.954 +0.988 +0.998 +0.982 +Few-Shot +Metaformer [128] +0.991 +0.971 +0.875 +0.940 +0.859 +0.994 +- +0.962 +0.901 +0.975 +0.990 +1.000 +0.944 +0.992 +0.986 +0.958 +RegAD [130] +0.998 +0.806 +0.763 +0.985 +0.915 +0.965 +1.000 +0.983 +0.806 +0.634 +0.974 +0.985 +0.934 +0.994 +0.940 +0.912 +[157] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.780 +RFS [129] +1.000 +0.920 +0.894 +0.984 +0.896 +0.999 +1.000 +0.999 +0.945 +0.700 +0.969 +0.992 +0.919 +0.981 +0.987 +0.945 +MAEDAY [158] +0.937 +0.690 +0.649 +0.979 +0.839 +0.941 +0.999 +0.734 +0.817 +0.614 +0.984 +0.925 +0.753 +0.995 +0.943 +0.853 +Noisy +TrustMAE [131] +0.970 +0.851 +0.788 +0.974 +0.991 +0.985 +0.951 +0.761 +0.833 +0.824 +0.973 +0.969 +0.875 +0.998 +0.875 +0.908 +[132] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.937 +[133] +- +- +- +- +- +- +- +- +- +- +- +- +- +- +- +0.959 +IGD [134] +1.000 +0.997 +0.915 +0.913 +0.958 +0.873 +0.946 +0.828 +0.991 +0.978 +0.906 +0.906 +0.997 +0.825 +0.970 +0.934 +SoftPatch [159] +0.937 +0.995 +0.963 +0.991 +0.968 +1.000 +1.000 +0.999 +0.963 +0.960 +0.993 +0.997 +0.990 +0.987 +0.978 +0.986 +Supervised +CGVGA [113] +0.960 +0.970 +0.930 +0.820 +0.810 +0.920 +0.840 +0.880 +0.970 +0.790 +0.860 +0.990 +0.890 +0.890 +0.960 +0.900 +DevNet [115] +0.993 +0.892 +0.865 +0.867 +0.967 +1.000 +0.999 +0.991 +0.866 +0.970 +0.987 +0.860 +0.924 +0.999 +0.990 +0.945 +DRA [117] +1.000 +0.909 +0.935 +0.940 +0.987 +1.000 +1.000 +0.997 +0.904 +0.977 +0.994 +0.826 +0.915 +0.998 +1.000 +0.959 + +Springer Nature 2021 LATEX template +24 +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +6 Total Performance Analysis +Table 9 shows the statistical result of current VAD performance on MVTec +AD. We provide a deep analysis of the performance of current VAD methods +and unlock meaningful insights as below: +• Regarding the identification of image-level anomaly detection tasks, mem- +ory bank-based approaches are the most effective neural network design. +However, they are inadequate at detecting pixel-level anomalies. +• Ensemble learning can dramatically improve the performance of state-of- +the-art anomaly detection methods. +• SSPCAB [88] can be seamlessly integrated into cutting-edge method and +significantly enhance the performance of reconstruction-based methods. +• The gap between few-shot VAD and vanilla VAD is narrowing. In other +words, we may utilize data distillation algorithms to lower the amount of +the dataset used for anomaly detection. +7 Future Directions +We outline several intriguing future directions as following: +• We should build up a mutli-modalities VAD Dataset. In actual assem- +bly lines, RGB images are insufficient to detect anomalies. Hence, we may +employ additional modalities information, such as X-ray and ultrasound, to +enhance anomaly detection performance. +• Given that test samples are sequentially streamed on the product line, most +of VAD methods are incapable of making instantaneous predictions upon +the arrival of a new test sample. In industrial manufacturing, the inference +speed of VAD should be addressed in addition to its accuracy. Adopting +multi-objective evolutionary neural architecture search algorithms to find +the optimal trade-off architecture is thus a promising approach. +• The majority of VAD methods use ImageNet pre-trained models to extract +the features from industrial images, which inevitably results in the feature +drift issue. Consequently, there is a pressing need to construct a pre-trained +model for industrial images. +• Previously, we focused on developing data augmentation method for normal +image. However, we have not made much effort on synthesizing abnor- +mal samples via data augmentation. In industrial manufacturing, it is very +difficult to collect a large number of abnormal samples since most of produc- +tion lines are faultless. Hence, more attention should be paid to abnormal +synthesis methods in the future, like CutPaste [29]. +8 Conclusions +In this paper, we provide a literature review on visual anomaly detection +in industrial manufacturing, focusing on the level of supervision, the design of + +Springer Nature 2021 LATEX template +Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey +25 +neural network architecture, the types and properties of datasets and the eval- +uation metrics. In particular, we characterize the new setting from industrial +manufacturing and review current VAD algorithms in our proposed new set- +ting. In addition, we investigate in depth which network architecture design can +considerably improve anomaly detection performance. 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Advances +in Neural Information Processing Systems + diff --git a/hdFJT4oBgHgl3EQfVyxJ/content/tmp_files/load_file.txt b/hdFJT4oBgHgl3EQfVyxJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e0ad309f47984d2b6b481b74087438dc71daae7 --- /dev/null +++ b/hdFJT4oBgHgl3EQfVyxJ/content/tmp_files/load_file.txt @@ -0,0 +1,2702 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf,len=2701 +page_content='Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey Jiaqi Liu1†, Guoyang Xie1,2†, Jingbao Wang1†, Shangnian Li1, Chengjie Wang3, Feng Zheng1 and Yaochu Jin2,4 1Research Institute of Trustworthy Autonomous Systems, Southern University of Science and Technology, Shenzhen 518055, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 2NICE Group, University of Surrey, Guildford GU2 7YX, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 3Youtu Lab, Tencent, Shenzhen 10000, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 4NICE Group, Bielefeld University, Bielefeld 33619, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Abstract The recent rapid development of deep learning has laid a milestone in visual anomaly detection (VAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In this paper, we provide a comprehen- sive review of deep learning-based visual anomaly detection techniques, from the perspectives of neural network architectures, levels of supervi- sion, loss functions, metrics and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, we extract the new setting from industrial manufacturing and review the current VAD approaches under our proposed our new setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Moreover, we highlight several opening challenges for visual anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The merits and downsides of representative network architectures under varying supervision are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Finally, we summarize the research findings and point out future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' More resources are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='com/M-3LAB/awesome-industrial-anomaly-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Keywords: Visual anomaly detection, Industrial manufacturing, Deep learning 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='11514v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='CV] 27 Jan 2023 Springer Nature 2021 LATEX template 2 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 1 Introduction We review the recent advances of deep learning based visual anomaly detec- tion since the rapid development of deep learning can bring the capabilities of visual anomaly detection into the factory floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In modern manufacturing, VAD is always performed at the end of the manufacturing process and tries to identify product defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The price of a product is significantly affected by the defect’s severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, if the flaw reaches a certain threshold, the prod- uct will be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Historically, the majority of anomaly detection tasks are performed by humans, which suffers from the following many disadvantages: It is impossible to avoid human fatigue, resulting in a false positive phe- nomenon (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=',the ground truth is abnormal, while the human’s judgment is normal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Long and intensive working on anomaly detection may cause health prob- lems, such as visual impairment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Locating anomalies requires a significant number of employees, raising operational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Thus, the goal of VAD algorithms is to reduce human labour and improve pro- ductivity and product quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In the pre deep learning era, the performance of VAD cannot fulfill the demands of industrial manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' But nowadays most of deep learning-based visual anomaly detection is approaching 97% accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' So more careful analysis of VAD applied on industrial manufacturing is called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 1 demonstrates clearly the merits of our survey in terms of dataset, metric, neural network architecture, levels of supervision and new setting for industrial manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Firstly, our study uses twice as many VAD datasets as Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Secondly, we analyze the performance of VAD using the most comprehensive image level and pixel level metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Nevertheless, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [2] and Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [1] only employ image level metrics, neglecting the anomalies localization performance of VAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Thirdly, our study develops a taxonomy based on the design of neural network architecture with varying degrees of supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Finally, to bridge the gap between academic research and real- world industry needs, we review the current VAD algorithms under industrial manufacturing settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 1: Our survey and existing surveys for VAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Content [3] [4] [1] [2] Ours VAD mataset 5 9 7 20 VAD metric 3 1 6 Neural network architecture � � � � � Levels of supervision � � � � � Industrial manufacturing setting � � � � � Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 3 As an emerging field, research on VAD must fully consider industrial man- ufacturing requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The following is a summary of the challenging issues that need to be investigated: VAD dataset should be gathered from actual manufacturing lines, not labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The public cannot access the real-world anomalous dataset due to privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The majority of open-source VAD datasets generate anomalies from anomaly-free products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In other words, the abnormalities from open- source VAD datasets may not occur in actual production lines, which makes deploying VADs in industrial manufacturing very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' It is challenging to enable the creation of a unified VAD model in the absence of multiple domain VAD datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Recently, You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [5] propose a unified VAD model for multiple class objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, they disregard the notion that commodities produced in the same plant should be of the same sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' For example, an automaker manufactures several types of workpieces but does not produce fruit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Current popular VAD datasets, like MVTec AD [6] and MVTec LOCO [7], consist of numerous classes but not multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' To simulate a realistic manufacturing process, we must create a new VAD dataset collected from multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' It is urgent to set up a uniform assessment for the image-level and pixel level of VAD performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The majority of VAD metrics shrink the anoma- lous mask (ground truth) into the size of feature map for evaluation, which inevitably reduces the precision of assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Moreover, we discover that certain VAD methods perform well on image AUROC but poorly on pixel AP, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Therefore, it is essential to develop a uniform metric for assessment VAD performance at both image and pixel level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' We should design a more efficient loss function that can leverage both the guidance of labelled data and the exploration of unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In realistic manufacturing scenario, limited number of anomalous samples are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, most of unsupervised VAD methods outperform semi-supervised VAD methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' By observing the failure of semi-supervised VAD, we would call for more attention to the feature extraction and loss function, which can leverage both the guidance from labels efficiently and the exploration from the unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Regarding the key problem mentioned above, improving feature extraction from abnormal samples and redesigning deviation loss function can fully use labelled anomalies and diverge the feature space of abnormal samples from those of normal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1 Contribution The main contributions of this survey can be summarized as following: We provide an in-depth review of visual anomaly detection by consid- ering the design of neural network architecture with varying degrees of supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey Visual Anomaly Detection (VAD) in Industrial Manufacturing Feature Embedding Based Methods §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1 Reconstruction Based Methods §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2 Unsupervised Anomaly Detection §2 Supervised Anomaly Detection §3 Different Settings in Anomaly Detection §4 Few-shot Anomaly Detection §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1 Noisy Anomaly Detection §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2 3D Anomaly Detection §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='3 Datasets and Metrics §5 Performance Analysis and Experiments §6 Future Directions §7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 1: Framework of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' It provides a comprehensive review of the current VAD algorithms in the industrial manufacturing setting to bridge the gap between academic research and real-world industrial manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' It summarizes the main issues and potential challenges in VAD, which outlines the underlying research directions for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The rest of this paper is organized as Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In Section 2 and Section 3, we review VAD on the basis of the neural network architecture with different levels of supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Next, we review the recent advances of VAD under our proposed setting from industrial manufacturing in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' We describe the popular dataset in Section 5 and take a retrospective view of the metrics function in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Then, we provide an analysis of the performance of current VAD methods on various datasets in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Finally, we provide future research directions for VAD in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 2 Unsupervised Anomaly Detection The majority of current research focuses on unsupervised anomaly detec- tion, based on the assumption that the collection of abnormal samples incurs massive human and financial costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This indicates that only normal samples are included in the training set, whereas both abnormal and normal samples are included in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Anomaly detection in industrial images is a subset of problems with out-of-distribution (OOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Before the rise of deep learning, differential detection and filtering were frequently used to detect anomalies in industrial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Following the release of the MVTec AD [6], methods for anomaly detection in industrial images can be divided into two categories: feature-embedding and reconstructed-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Currently, more AD techniques are based on feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1 Feature Embedding based Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1 Teacher-Student Architecture The performance of these methods is outstanding, but they depend on pre- trained models such as ResNet [8] VGG [9] and EfficientNet [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The selection of the ideal teacher model is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This type of instructional strategy is sum- marized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The structure of the network and the method of distillation are the primary distinctions between various techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 2: A summary of teacher-student methods regarding loss function, pre-trained model, and highlights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Method Loss Function Pre-trained Highlights Uninformed Students [11] L2, Compactness ResNet The paper designs a basic approach to anomaly detection problems using a teacher-student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MKD [12] L2 VGG The paper uses multi-scale features and lighter networks for distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' STPM [13] L2 ResNet The paper uses multi-scale features under different network layers for distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' STFPM [14] L2 ResNet The paper adds another teacher-student pair to get different feature recon- struction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' RD4AD [15] Cosine Similarity ResNet The paper designs the teacher-student model of reverse distillation in a similar way to reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' IKD [16] Context Similarity ResNet The paper adds context similarity loss and adaptive hard sample mining module to prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AST [17] L2, Log-Likelihood EfficientNet The paper uses a asymmetric teacher-student network to make the represen- tation of anomaly more different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The teacher-student network architecture depicted in Figure 2 is the most standard technique for detecting industrial image anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This method typ- ically selects a partial layer of a backbone network pre-trained on a large-scale dataset as a fixed-parameter teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During training, the teacher model imparts to the student model the knowledge of extracting normal sample fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During inference, the characteristics of normal images extracted from the test set by the teacher network and the student network are comparable, whereas the characteristics of abnormal images extracted from the test set are quite distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' By comparing the feature maps generated by the two networks, it is possible to generate anomaly score maps with the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Then, by enlarging the anomaly score map to the same proportion as the input image, we can obtain the anomaly scores of various input image locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' On the justification of this model, it is possible to determine whether the test image is abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Bergmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [11] is the first to use teacher-student architecture for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The model is straightforward and effective, significantly outperforming other benchmark methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' While STPM [13] and MKD [12] both use multi-scale features under different network layers for distillation, they do so in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In this instance, the normal sample features extracted by the student network are more similar to those extracted by the teacher network, whereas the abnormal sample features are more dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, MKD finds that the lighter student network structure performs better than the student network structure identical to that of the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Based on STPM, RSTPM [14, 18] adds a pair of teacher-student Springer Nature 2021 LATEX template 6 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey Teacher Network Student Network Distillation Anomaly Map Normal feature from teacher network Normal feature from student network Abnormal feature from teacher network Abnormal feature from teacher network Distance of image feature from two networks Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 2: Teacher-student model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During reasoning, the new teacher network is placed behind the original teacher-student network and is responsible for recreating the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' When anomalous images are presented, the student network typically recon- structs normal features that can be distinguished from those of the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' RSTPM also includes a mechanism for transferring features from the teacher network to the student network in order to facilitate feature recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' RD4AD [15] and RSTPM share certain similarities in their learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' RSTPM employs two pairs of teacher-student networks for feature reconstruc- tion, whereas RD4AD only employs one pair of teacher-student networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' RD4AD proposes a Multi-scale Feature Fusion (MFF) block and One-Class Bottleneck (OCB) to form an embedding, which is used to eliminate redundant features at multiple scales so that a single pair of teacher-student networks can perform feature reconstruction effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The abnormal image features extracted by the teacher-student network of RD4AD differ significantly dur- ing inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AST [17] concludes that the abnormal image features extracted by the teacher-student model with the same structure are significantly sim- ilar, so they propose an asymmetric teacher-student architecture to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AST also introduces a normalized flow to avoid this problem and prevent estimation bias caused by the inconsistency of the two network struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Previous teacher-student architecture anomaly detection methods suffer from overfitting as a result of inconsistency between neural network capacity and knowledge amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' By incorporating the Context Similarity Loss (CSL) and Adaptive Hard Sample Mining (AHSM) modules, Informative Knowledge Distillation (IKD) [16] hopes to reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CSL can assist the stu- dent network in comprehending the structure of a context-containing data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The AHSM can concentrate on difficult samples containing a lot of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2 One-Class Classification One-class classification techniques rely more heavily on abnormal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' If the generated abnormal samples are of poor quality, the method’s perfor- mance will be severely compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' As demonstrated in Table 3, with the exception of MemSeg [19], the training of other methods relies on SVDD and Cross-Entropy loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' consequently, the performance of the vast majority of methods is marginally inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 3: A summary of one-class classification methods regarding loss function, pre-trained model, and highlights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Method Loss Function Pre-trained Highlights Patch SVDD [20] Cross-Entropy, SVDD The paper divides image into patches and sends them to SVDD for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DDSPSVDD [21] L2, SVDD VGG DSPSVDD takes reconstruction error into model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SE-SVDD [22] SVDD ResNet The paper proposes a Semantic Correlation module (SCB) to represent abnormal semantics information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MOCCA [23] L2, SVDD The paper extends a single boundary to a hard boundary and a soft boundary, it also trains AE as feature exactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [24] Cross-Entropy Xception [25] The paper uses Xception to train a classification network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PANDA [26] SVDD, Log-Likelihood DN2 [27] The paper introduces a method to avoid combating collapse in model adaption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [28] Cross-Entropy, Contrastive ResNet The paper presents a novel distribution-augmented contrastive learning to enhance the representing ability of network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CutPaste [29] Cross-Entropy EfficientNet The paper applies “cut and paste” augmentation into binary anomaly classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [30] Cosine Similarity, Contrastive The paper applies some dynamic local augmentation to generate negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CPC-AD [31] InfoNCE The paper applies Contrastive Predictive Coding (CPC) model to AD and get an anomaly score through pixel-wise loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MemSeg [19] L1, Focal ResNet The paper artificially creates anomalies in the foreground of products and makes detecting artificial anomalies a segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Anomaly detection can also be viewed as a One-Class Classification (OCC) problem, which has inspired some research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' As depicted in Figure 3, the method finds a hypersphere to distinguish normal sample features from abnor- mal sample features during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During inference, the method determines whether the sample is abnormal based on the relative position of the test sample’s features and the hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Since the training set does not con- tain abnormal samples, some methods create abnormal samples artificially to improve the accuracy of the hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SVDD [32] is a classic algorithm in the OCC problem, PatchSVDD [20] DSPSVDD [21] and SE-SVDD [22] improve it for industrial image AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PatchSVDD [20] divides the image into uniform patches and sends them to the model for training, which significantly enhances the model’s ability to detect anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DSPSVDD [21] designs an improved comprehensive optimization objective for the deep SVDD model that simultaneously considers hypersphere volume minimization and network reconstruction error minimization to extract deep data features more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SE-SVDD proposes a Semantic Correla- tion module (SCB) to improve the representation of abnormal semantics and the accuracy of anomaly localization by extracting multi-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MOCCA [23] employs multi-layer features for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MOCCA, unlike SE-SVDD, uses an autoencoder to extract features and locates the boundary position of normal features at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' And Sauter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [24] Springer Nature 2021 LATEX template 8 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey Normal Sample Artificial Abnormal Sample Abnormal Sample Normal Sample Training Testing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 3: One-class classification model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' attempt to use the Xception network for classification and obtained results comparable to SVDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FCDD [33] employs a fully convolutional neural net- work for OCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Since the relative positions of the features of each image layer do not change during the convolution process, FCDD yields more interpretable results than alternative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PANDA [26] examines the migration method of pre-trained features and introduces the early stopping mechanism to the OCC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, Reiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [34] investigate the issue of catastrophic forgetting in PANDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' They propose a new loss function capable of overcoming the failure modes of both center-loss and contrastive-loss methods and replacing Euclidean distance with a confidence-invariant angular center loss for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DisAug CLR [28] proposes a two-stage anomaly detection framework, in which the first stage hinders the uniformity of contrastive representations by means of a novel distribution-enhanced contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' After comparative learning, abnormal and normal sample representations are easier to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' While the second stage builds a one-class classifier using the representations learned in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Yoa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [30] presents a novel dynamic local aug- mentation to generate negative image pairs from a normal training dataset, which is effective for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Contrastive Predictive Coding (CPC) [35] model is utilized by De et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [31] for anomaly detection and segmentation, which uses patch-wise contrastive loss as anomaly score to localize anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CutPaste [29] is a representative example of an OCC method for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' It generates abnormal images by cutting and pasting por- tions of normal images, allowing the network to distinguish abnormal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Additionally, segmentation-based methods are useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This method puts more emphasis on pixel-level anomaly localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' When the flow is known, Ique- bal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [36] demonstrate that the maximum a posterior estimation of image labels can be formulated as a continuous max-flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Then, anomaly segmentation is accomplished by obtaining flows iteratively using a novel Markov random field on the image domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The technique shows its adaptabil- ity using a dataset for metal additive manufacturing anomaly detection [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MemSeg [19] stores the features of normal images in a memory bank in order to Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 9 improve the segmentation network’s ability to distinguish abnormal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In order to prevent the influence of background factors, MemSeg only introduces anomalies in external data sets in the foreground of items, which is another reason for its excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='3 Distribution Map Distribution-map based methods necessitate a suitable mapping objective for training, and the choice of mapping method impacts model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' As shown in Table 4, Normalizing Flows (NF)-based methods predominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' As a generative model, NF has a strong mapping ability, and it has also demonstrated good performance in AD tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 4: A summary of distribution-map based methods regarding loss function, pre-trained model, and highlights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Method Loss Function Pre-trained Highlights [38] PCA ResNet The paper uses PCA and ResNet to extract features and count their distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [39] Cross-Entropy ResNet, EfficientNet The paper establishes a model of normality by fitting a multivariate Gaussian to feature representations of a pre-trained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [40] Mahalanobis Distance EfficientNet The paper generates a multi-variate Gaussian distribution for the normal class and mitigates the catastrophic forgetting in past research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PEDENet [41] Log-Likelihood, Cross-Entropy, Regularization The model can predict the location of the patch and compare it with the actual location to judge the abnormality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PFM [42] L2 ResNet The paper proposes the bidirectional and multi-hierarchical bidirectional pre- trained feature mapping based on the vanilla feature mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PEFM [43] L2 ResNet The paper introduces position encoding into PFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FYD [44] L2 ResNet The paper aligns samples at image and feature levels to detect anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DifferNet [45] Log-Likelihood ResNet The paper is the first one to introduce normalizing flow into anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CS-Flow [46] Log-Likelihood ResNet The paper uses information of multi-scale feature maps and improve Differ- Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CFlow-AD [47] Log-Likelihood ResNet The paper introduces positional encoding into the conditional normalizing flow framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CAINNFlow [48] Log-Likelihood ViT [49] The paper uses VIT to replace ResNet and achieve better result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FastFlow [50] Log-Likelihood ResNet The paper introduces an alternate stacking of large and small convolution kernels in the NF module to model global and local distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AltUB [51] Log-Likelihood ResNet The paper designs a module for normalizing flow based methods and imporve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Distribution-map based methods are very similar to OCC-based methods, with the exception that OCC-based methods concentrate on finding feature boundaries, whereas mapping-based methods attempt to map features into desired distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' A common framework for those methods is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This expected distribution is typically a MultiVariate Gaussian (MVG) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This type of method first employs a strong pre-trained network to extract the features of normal images, and then maps the extracted features to the Gaussian distribution using a mapping module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This distribu- tion will be deviated from by the features of abnormal images that appear during the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The abnormal probability can be calculated based on the level of deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Tailanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [38] propose a contrario framework that applies statistical analysis to feature maps produced by patch PCA and ResNet, which per- forms well on leather samples, to detect anomalies in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' By fitting a multivariate Gaussian to the feature representations of a pre-trained network, Springer Nature 2021 LATEX template 10 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Normal Samples Pre-trained Network Origin Features Mapping Module Well-distributed Features Normal Feature Abnormal Feature Abnormal Sample Origin Distribution Mapped Distribution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 4: Distribution-map based method architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Rippel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [39] establish a model of normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Nonetheless, the issue of catastrophic forgetting remains unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Based on the relationship between generative and discriminative modeling, Rippel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [40] generate a multi- variable Gaussian distribution for the normal class and prove the efficacy of this concept on Deep SVDD and FCDD, which mitigates the catastrophic for- getting observed in previous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PEDENet [41] framework consists of a Patch Embedding (PE) network, a Density Estimation (DE) network, and a Location Prediction (LP) network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' At first, the PE module is used to reduce the size of the features that the pre-trained network has extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Then, using the DE module, which was inspired by the Gaussian mixture model, and the LP module, the model can predict the relative position of the patch embedding and, based on the difference between the predicted result and the actual result during inference, decide if the image is abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Pre-trained Feature Map- ping (PFM) [42] proposes bidirectional and multi-hierarchical bidirectional pre-trained feature mapping to enhance the performance of vanilla feature mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [43] add position encoding to the PFM framework and propose a novel Position Encoding enhanced Feature Map- ping (PEFM) [43] to further enhance PFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FYD [44] introduce registration to industrial image AD for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FYD suggests a coarse-to-fine align- ment method that starts with aligning the foreground of objects at the image level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Next, in the refinement alignment stage, non-contrastive learning is used to increase the similarity of features between all corresponding positions in a batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Normalizing Flows (NF) [52] is a technique for constructing complex dis- tributions by transforming a probability density via a series of invertible mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' NF methods extract features from normal images from a pre-trained model, such as ResNet [53] or Swin Transformer [54], and transform the fea- ture distribution as a Gaussian distribution during the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In the test phase, after passing through NF, the features of abnormal images will Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 11 deviate from the Gaussian distribution of the training phase, which is the most important principle for classifying anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DifferNet [45] is the first research to use NF to address the industrial image AD issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' By incorporating cross-convolution blocks within the normalizing flow to assign probabilities, CS-Flow [46] makes use of the context within and between multi-scale fea- ture maps to improve DifferNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CFlow-AD [47] adds positional encoding to the framework for conditional normalizing flow to achieve superior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, CFlow-AD [47] analyzes in depth why the multivariate Gaussian assumption is a reasonable prior in earlier models and why the more gen- eral NF framework aims to converge to similar results with less computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FastFlow [50] introduces an alternate stacking of large and small convolution kernels in the NF module to model global and local distribution efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CAINNFlow [48] enhances the performance of the model by introducing the attention mechanism CBAM [55] to the NF module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In techniques such as FastFlow and CFlow-AD, the feature distribution center is not 0 and their per- formance is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [56] propose a simple solution AltUB [51] that uses alternating training to update the base distribution of normalizing flow for anomaly detection in order to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AltUB verifies the effect of CFlow-AD and FastFlow using AltUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='4 Memory Bank As illustrated in Table 5, memory-based methods regularly do not require the loss function for training, and models are constructed quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Their per- formance is ensured by a robust pre-training network and additional memory space, and this type of method is currently the most effective in VAD tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 5: A summary of memory bank based methods regarding loss function, pre-trained model, and highlights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Method Loss Function Pre-trained Highlights SPADE [57] ResNet The paper uses multi-resolution feature to detect anomalies based on KNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [56] ResNet The paper reduces the computational cost for the inverse of multi-dimensional covariance tensor so that bigger resolution image can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SOMAD [58] ResNet The paper maintains normal characteristics by using topological memory based on multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' GCPF [59] ResNet The paper processes normal features into multiple independent multivariate Gaussian clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MSPB [60] Kmeans, Cosine Similarity, SVDD VGG The paper enhances network representation capabilities by learning patch position relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SPD [61] Focal, InfoNCE, SPD, Cosine Similarity Design a contrastive learning method to retrain ResNet to enhance the ability of defect representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PatchCore [62] ResNet The paper introduces a core-set sampling method to build a memory bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CFA [63] SVDD ResNet The paper improves PatchCore so that image features are distributed on a hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FAPM [64] ResNet The paper puts different position features of the image into different memory banks to speed up retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' N-pad [65] Mahalanobis Distance, Log-Likelihood ResNet The paper allows for possible edge misalignment by estimating a nominal distribution for each pixel using the pixel’s neighborhood features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The primary distinction between memory bank-based methods and OCC- based methods, is that memory-based methods, such as SVDD, require additional memory space to store image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' As shown in Figure 5, these methods require minimal network training and only require sampling or Springer Nature 2021 LATEX template 12 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Normal Samples Pre-trained Network Abnormal Samples Sampling or Mapping Basic Memory Bank Memory Bank and Testing Samples Normal Samples Training set normal feature Testing set abnormal feature Testing set normal feature Distance from neighbor Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 5: Memory bank based method architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' mapping the collected normal image features for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During inference, features of the test image are compared to features in the memory bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The abnormal probability of the test image is equal to the spatial distance from the normal features in the memory bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' K Nearest Neighbors (KNN) [66] is a widely used algorithm for unsuper- vised anomaly detection, but it operates only at the sample level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Semantic Pyramid Anomaly Detection (SPADE) [57] is inspired by KNN and utilizes correspondences based on a multi-resolution feature pyramid to obtain pixel- level anomaly segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PaDim [67] employs multivariate Gaussian distributions to construct a probabilistic representation of the normal class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Consequently, the memory bank size is determined solely by the image resolu- tion and not by the size of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PaDiM requires the batch-inverse of the multidimensional covariance tensor, which makes it challenging to scale up to larger CNNs due to the increased feature size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' To reduce the computa- tional cost of the inverse by a factor of three, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [56] generalize random feature selection into semi-orthogonal embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Meanwhile, Self-organizing Map for Anomaly Detection (SOMAD) [58] and GCPF [59] enhance the storage of normal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SOMAD preserves normal characteristics by employing topological memory based on multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' While GCPF transforms standard characteristics into multiple independent multivariate Gaussian clustering PatchCore [62] is a significant advancement in industrial image AD that significantly raises the performance for MVTec AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PatchCore contains two special points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' First, the memory bank of PatchCore is coreset-subsampled to ensure a low inference cost while maximizing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PatchCore then determines whether the test sample is abnormal based on the distance between the test sample’s nearest neighbor feature in its memory bank and other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This process of reweighting makes PatchCore more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Since PatchCore was proposed, numerous improved methods have been developed atttSpringer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 13 on its foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Coupled-hypersphere-based Feature Adaptation (CFA) is proposed by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [63] to obtain target-oriented features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The center and surface of the hypersphere in the memory bank are obtained through trans- fer learning, and the positional relationship between the test feature and the coupled-hypersphere can be used to determine whether it is abnormal or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FAPM [64] is comprised of numerous patch-wise and layer-wise memory banks located in various places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FAPM calculates the features in different memory banks independently during inference, which significantly accelerates inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' N-pad [68] allows for the possibility of marginal misalignment by esti- mating a per-pixel nominal distribution using neighboring and target pixel features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, anomaly scores are deduced using both Mahalanobis and Euclidean distances between target pixels and the estimated distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Similarly, Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [65] model the cumulative histogram using location infor- mation as conditional probabilities, and neighborhood information was used to establish the normal feature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Furthermore, this work introduces the first refinement approach in the anomaly detection and localization prob- lem, using synthetic anomalous images to improve the anomaly map based on the input image, as well as using neighborhood and location information to estimate the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' By learning the embedding position information and comparing the extracted features with the normal embedding during inference, Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [60] propose a method to improve the network’s ability to represent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' It is also based on the concept of self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [61] use contrastive learning to train the backbone network and propose a new data augmentation method called SPD to push the network to differenti- ate between two images with slight differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, they demonstrate the representation capability of the backbone network using PatchCore [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2 Reconstruction based methods Reconstruction-based methods primarily self-train encoders and decoders to reconstruct images for anomaly detection, which makes them less reliant on the pre-trained model and increases their ability to detect anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, its image classification capability is poor due to its inability to extract high- level semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' As shown in Table 6, the loss functions of various methods are comparable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' however, their performance varies due to different reconstruction model paradigms and abnormal sample construction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The structure of the reconstruction-based technique is depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During the training process, normal or abnormal images are sent to the recon- struction network, and the reconstruction loss function is used to guide the training of the reconstruction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Finally, the reconstruction network can restore the reconstruction image in a manner similar to the original normal image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In the inference stage, the comparison model compares the original image to the reconstructed image to generate a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In contrast to the variety of methods for feature embedding, the majority of reconstruction- based methods only differ in the construction of the reconstruction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey Table 6: A summary of reconstruction based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Method Loss Function Pre-trained Highlights (1) Autoencoder Model [69] L2, SSIM The paper firstly takes SSIM as a loss to reconstruct image and detect anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [70] L2, SSIM The paper proposes two AEs and reduces style change during image recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' UTAD [71] L1, Adversarial VGG The paper uses two-stage reconstruction to generate high-fidelity images to avoid reconstruction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DFR [72] L2 VGG The paper proposes to reconstruct and compare at the feature level to detect anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' ALT [73] L1, Perceptual, Adversarial VGG The paper proposes an adaptive attention-level transition strategy and uses perceptual loss to improve reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' P-Net [74] L1, Adversarial The paper designs a new architecture for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [75] L1, L2 The paper adds skip-connection in reconstruction network and adds noise during training to improve reconstruction sharpness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [76] L2 VGG The paper proposes a dense feature fusion module to assist reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [77] L2, Adversarial The paper uses memory to help reconstructing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' EdgRec [78] L2, SSIM The paper reconstructs from the gray value edge and preserves the high- frequency information with skip-connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PAE [79] L2, Cross-Entropy The paper gradually increases the resolution of the input image during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SMAI [80] L2, SSIM The paper masks and inpaintings image by superpixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' RIAD [81] L2, MSGMS, SSIM The paper proposes to inpaint and reconstruct images by patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' I3AD [82] L1, Adversarial The paper gradually masks the high anomaly probability areas and recon- structs them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [83] L2 The paper proposes to reconstruct the anomalous area differently from the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [84] L2, SSIM, GMS Similar to I3AD, but the paper adds skip connections to reconstruction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DREAM [85] L2, SSIM, Focal The paper designs a method to generate abnormal images and uses U-Net to distinguish anomalies after reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DSR [86] L2, Focal The paper generates abnormal samples in feature level and perform better than DRAEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' NSA [87] L2, Cross-Entropy The paper generates abnormal samples by pasting parts of other normal samples, which is the SOTA method without extra data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SSPCAB [88] L2 The paper designs a “plug and play” self-supervised block to improve the reconstruction ability of many methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SSMCTB [89] L2 This paper replaces the SE-layer in SSPCAB with transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [90] Cross-Entropy The paper guides reconstruction using gradient descent with VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [91] Attention Disentanglement The paper proposes to use disentanglement VAE to detect anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DGM [92] L2, Log-Likelihood The paper proposes to use non-regularized objective functions for training VAE under heterogeneous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FAVAE [93] Log-Likelihood VGG The paper uses VAE to model the distribution of features extracted by its pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [94] L2, Cross-Entropy The paper uses VQ-VAE to construct a discrete latent space and reconstructs images based on the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' (2) GAN Model SCADN [95] L2, Adversarial The paper masks part of image and reconstruct image with GAN during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AnoSeg [96] L1, L2, Adversarial The paper generates abnormal samples through a GAN and detects anomalies with the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' OCR-GAN [97] L1, L2, Adversarial The paper uses the Frequency Decoupling module to decouple and reconstruct images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' (3) Transformer Model VT-ADL [98] L2, SSIM, Log-Likelihood The paper proposes a transformer-based framework to reconstruct images and detects anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' ADTR [99] L2, Cross-Entropy EfficientNet The paper makes it simple to identify anomalies when reconstruction fails by reconstructing features from pre-trained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AnoViT [100] L2 ViT The paper uses a pre-trained ViT to extract features and reconstruct images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' HaloAE [101] L2, Cross-Entropy, SSIM VGG The paper introduces an auto-encoder architecture based on a transformer with HaloNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' InTra [102] L2, GMS, SSIM The paper leverages more global information to repair images with trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MSTUnet [103] L2, SSIM, Focal The paper uses swin transformer for inpainting masked images and detects anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' MeTAL [104] L1, SSIM The paper uses information from neighbor patches to inpainting images, better accounting for local structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' UniAD [5] L2 EfficientNet The paper trains all categories of products in one model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' (4) Diffusion Model AnoDDPM [105] L2, Log-Likelihood The paper is the first to apply diffusion model for industrial image anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [106] L2, Log-Likelihood The paper significantly speeds up the inference process of anomaly detection using diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 15 Reconstruction Network Reconstruction Network Comparison Model Reconstruction Loss Training Testing Predict Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 6: Reconstruction based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Reconstruction-based methods outperform feature-embedding methods at the pixel level due to their ability to identify anomalies through pixel-level com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, the majority of reconstruction-based methods are trained from scratch without employing robust pre-trained models, which results in inferior performance compared to image-level feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1 Autoencoder Autoencoder (AE) is the most prevalent reconstruction network for AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Numerous other reconstruction networks also consist of encoder and decoder components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Bergmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [69] investigate the influence of Structure Simi- larity Index Measure (SSIM) and L2 loss on AE reconstruction and anomaly segmentation, providing numerous suggestions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' How to resolve the difference between the reconstructed image and the orig- inal image is the most foundational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' There are regularly differences in style between the reconstructed image and the original image, resulting in over-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [70] present an Outlier-Exposed Style Dis- tillation Network (OE-SDN) to preserve the style translation and suppress the content translation of the AE in order to avoid over-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' As the anomaly prediction, Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' replaced the difference between the original image and the reconstruction image of AE with the difference between the reconstruction image of OE-SDN and the reconstruction image of AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Unsu- pervised Two-stage Anomaly Detection (UTAD) [71] brings an IE-Net and Expert-Net to extract and utilize impressions for anomaly-free and high-fidelity reconstructions, thereby offering the framework interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Reconstruction-based methods are nearly effective as feature embedding methods when utilizing features at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Similar to teacher-student 1Springer Nature 2021 LATEX template 16 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey architecture, Deep Feature Reconstruction (DFR) [72] method detects anoma- lous through reconstruction at the level of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DFR obtains multiple spa- tial context-aware representations from a network that has been pre-trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Then, DFR reconstructs features using a deep yet efficient convolutional AE and detects anomalous regions by comparing the original features to the reconstruction features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [73] propose a novel Multi-Level Image Reconstruction (MLIR) framework that forms the reconstruction process as an image denoising task at different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Thus, MLIR accounts for the detection of both global structure anomalies and detail anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Modifying the structure of AE can also improve its capacity for recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [74] introduce P-Net to compare the difference in structure between the original and reconstruction images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Collin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [75] include skip-connections between encoder and decoder to improve the recon- struction’s sharpness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, they propose corrupting them with a synthetic noise model to prevent the network from convergently mapping identities, and they introduce the innovative Stain noise model for this pur- pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [76] also operate at the feature level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' they employ a dense feature fusion module to obtain a dense feature representation of double input in order to help reconstruction in the dual-Siamese framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [77] also use skip-connections to enhance the quality of reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition to achieving expected results, they add a memory module to skip- connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [78] reconstruct the original RGB image from its gray value edges, with the skip-connections in the model preserving the image’s high-frequency information to better guide the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Progressive Autoencoder (PAE) [79] improves autoencoder reconstruction performance through progressive learning and modified CutPaste augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During training, PAE achieves progressive learning by gradually increasing the input image’s resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Masking and repainting is an effective method for self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The Superpixel Masking And Inpainting (SMAI) technique was developed by Li et al [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SMAI divides the image into multiple blocks of superpixels and trains the inpainting module to reconstruct a superpixel within a mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SMAI performs masking and inpainting superpixel-by-superpixel on the test image during inference, and then compares the reconstruction image to the test image to distinguish abnormal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Iterative Image Inpainting Anomaly Detection (I3AD) is a method proposed by Nakanishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [82] that reconstructs partial regions based on the anomaly map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' I3AD improves reconstruction quality by only reconstructing inpainting masks over images, and only masking regions with a high probability of abnormality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SSM [84] is conceptually similar to I3AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SSM adds skip-connections to the reconstruction network and predicts the mask region as the training target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' RIAD [81] randomly masks a portion of the training set image at the patch level and reconstructs it using a U- Net encoder-decoder network [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During inference, RIAD combines multiple random masks and reconstruction patches to generate a reconstructed image, which is then compared to the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Multi-Scale Gradient Magnitude Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 17 Similarity (MSGMS) outperforms SSIM as an anomaly score, according to RIAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DRAEM [85] is representative of reconstruction-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DRAEM synthesizes abnormal images and reconstructs them as normal by introduc- ing external datasets, which greatly improves the reconstruction network’s generalization capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, DRAEM feeds the original image and the reconstructed image into the segmentation network to predict abnor- mal regions, significantly enhancing the model’s ability to segment anomalous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Nevertheless, DRAEM is susceptible to failure when synthesizing near-in-distribution anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DSR [86] proposes an architecture based on quantized feature space representation and dual decoders to circumvent the requirement for image-level anomaly generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' By sampling the learned quantized feature space at the feature level, the near-in-distribution anomalies are generated in a controlled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' NSA [87] does not use external data for data data augmentation and adopts more data augmentation methods, allowing it to outperform all previous methods that learned without utilizing additional datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In contrast to other methods that attempt to reconstruct abnormal images into normal images, Bauer [83] proposes reconstructing the abnormal areas of the image so that they deviate from the original image’s appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This approach produces comparable results to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In contrast to classical reconstruction-based methods, Ristea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [88] pro- pose integrating reconstruction-based functionality into a Self-Supervised Pre- dictive Architectural Building Block (SSPCAB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SSPCAB can be incorporated into models such as DRAEM and CutPaste to enhance those models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Self- Supervised Masked Convolutional Transformer Block (SSMCTB) [89] trans- forms the SE-layer [108] in SSPCAB into a channel-wise transformer block and achieves superior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' VAE is a variant of AE, with the difference that the intermediate variables of VAE are data from a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Naturally, VAE has superior inter- pretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Dehaene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [90] iteratively guide reconstruction using gradient descent with energy defined by the reconstruction loss, thereby overcoming the tendency of VAE to produce blurry reconstructions and preserving the nor- mal high-frequency structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The variational autoencoder is trained with an attention disentanglement loss by Liu et al [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Anomaly inputs in this VAE will result in Gaussian-deviating latent variables during gradient backpropaga- tion and attention generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This deviation can be used to locate anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' According to Matsubara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [92], datasets are commonly heterogeneous rather than regularized, and non-regularized objective functions are more suit- able for training VAE models on heterogeneous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FAVAE [93] employs VAE to model the distribution of features extracted by the pre-trained model, implicitly simulating richer anomalies and enhancing the model’s generaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [94] use VQ-VAE to create a discrete latent space, resample the discrete latent code deviate from the normal distribution, and reconstruct the image using the resampled latent code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' VQ-VAE reconstructs images that are closer to the training set’s normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 18 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2 Generative Adversarial Networks The stability of the reconstruction model based on Generative Adversarial Networks (GANs) is not as good as that of AE, but the discriminant network has a better effect on some scenes described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During training, Semantic Context based Anomaly Detection Network (SCADN) [95] masks a portion of the image and reconstructs it with GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SCADN detects anomalies for inference by comparing the input image to the reconstruction image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition to masking images, AnoSeg [96] utilizes hard augmentation, adversarial learning, and channel concatenation to gener- ate abnormal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AnoSeg then trains GAN to generate normal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AnoSeg differs from the AE reconstruction model in that its objective func- tion incorporates both reconstruction loss and adversarial loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' OCR-GAN [97] utilizes the Frequency Decoupling (FD) module to decouple the image into information combinations of different frequencies, and then reconstructs and combines the information of these different frequencies to yield reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' During inference, the model can identify a statistically significant difference between the frequency distributions of normal and abnormal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='3 Transformer Transformer has a higher capacity to represent global information, which gives it the potential to surpass AE and become a new reconstruction network foundation for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [98] propose a transformer- based framework to reconstruct images at the patch level and employ a gaussian mixture density network to localize anomalous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [99] propose ADTR for reconstructing pre-trained features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' According to them, the use of transformers prevents well-reconstructed anomalies, making it easy to identify anomalies when reconstruction fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [100] intro- duce a vision transformer-based encoder-decoder model (AnoViT) and assert that AnoViT is superior to the CNN-based l2-CAE in the issue of anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' HaloAE [101] implements transformer into HaloNet [109] and facil- itates image reconstruction by reconstructing features to achieve competitive results on the MVTec AD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' A common self-supervised learning method for reconstruction-based anomaly detection is the reconstruction of masked images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, traditional CNNs find it difficult to extract global context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In order to accomplish this, Pirnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [102] propose Inpaint- ing Transformer (InTra), which integrates information from larger regions of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' InTra is representative of trained-from-scratch methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Masked Swin Transformer Unet (MSTUnet) [103] is comparable to InTra, but MSTUnet employs additional enhancements [110] when simulating anomalies, thereby achieving superior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' De et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [104] used the neighbor patch to reconstruct the masked patch and also achieved a powerful reconstruction ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='4 Diffusion Model Diffusion model [111] is a recently popular generative model that can also be utilized for reconstruction-based anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AnoDDPM [105] is, to the best of our knowledge, the first to apply the diffusion model to industrial image anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In comparison to GAN-based methods, AnoDDPM with simplex noise can also capture large anomaly regions without the need for large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' When applying the diffusion model to anomaly detection, Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [106] primarily make two improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' As a replacement metric for reconstruction loss, a time-dependent gradient value of normal data dis- tribution is used to measure the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, they develop a novel T-scales method to reduce the required number of iterations and accelerate the inference process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 3 Supervised Anomaly Detection Despite the fact that abnormal data is diverse and difficult to collect, it is still possible to collect abnormal samples in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Therefore, some research focuses on how to train models for anomaly detection using a small number of abnormal samples and a large number of normal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [112] propose a semi-supervised framework for detecting anoma- lies in the presence of significant data imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' They assume that changes in loss values during training can be used to identify abnormal data as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' To achieve this, they train a reinforcement learning-based aneural batch sampler to amplify the difference in loss curves between anomalous and non-anomalous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FCDD [33] is an unsupervised method that synthesizes abnormal sam- ples for training the OCC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' FCDD [33] is an unsupervised method that synthesizes abnormal samples for training the OCC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' This concept is transferable to other OCC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Venkataramanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [113] propose a Convolutional Adversarial Variational Autoencoder (CAVGA) with Guided Attention that can be applied equally to cases with and without abnormal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In an unsupervised setting, CAVGA is guided to focus on all normal regions of an image by an attention expansion loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' CAVGA uses a comple- mentary guided attention loss in the weakly supervised setting to minimize the attention map corresponding to abnormal regions of the image while focusing on normal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Bovzivc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [114] examine the influence of image-level supervision information, mixed supervision information, and pixel-level super- vision information on surface defect detection tasks within the same deep learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Bovzivc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' find that a small number of pixel-level anno- tations can help the model achieve performance comparable to full supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DevNet [115] uses a small number of abnormal samples to realize fine-grained end-to-end differentiable learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [116] propose a Logit Inducing Loss (LIS) for training with imbalanced data distribution and an Abnormality Capturing Module (ACM) for characterizing anomalous features in order to effectively utilize a small amount of anomalous information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' DRA [117] pro- poses a framework for learning disentangled representations of seen, pseudo, Springer Nature 2021 LATEX template 20 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey and latent residual anomalies in order to detect both visible and invisible anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Besides, a number of studies fail to account for the unbalanced distribution of normal and abnormal samples and rely primarily on abnormal samples for supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Sindagi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [118] investigate the domain transfer prob- lem of datasets for anomaly detection in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Dual Weighted PCA (DWPCA) is an algorithm proposed by Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [119] for image registration and surface defect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' An interleaved Deep Artifacts-aware Attention Mechanism (iDAAM) is proposed by Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [120] propose to classify multi-object and multi-class defects in abnormal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [121] view anomaly detection as a subset of target detection and designed a Reference-based Defect Detection Network (RDDN) to detect anomalies using template reference and context reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [122] investigate defect detection in a tactile image, which has obvious benefits for fabric structure defect detection in RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, there are methods that refer to the concept of semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' To detect defects in infrared thermal volumetric data, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [123] propose a hybrid multi-dimensional space and temporal segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [124] use Mask Region-based CNN architecture to detect and segment defects in X-ray images simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' There are also numerous modified models on anomaly detection based on the object detection and semantic segmentation model of natural images under full supervision [125–127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Here we will not discuss them one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 4 Industrial Manufacturing Setting This section introduces the classification standards or application settings that are more appropriate for industrial scenes, namely few-shot anomaly detection, noisy anomaly detection, and 3D anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='1 Few-shot Anomaly Detection Few-shot visual anomaly detection (FSAD) [128, 129] is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' There are two settings in FSAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The first setting is meta-learning [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In other words, this setting requires a large amount of images as meta-training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [128] propose a novel architecture, called MetaFormer, that employs meta-learned parameters to achieve high model adaptation capabil- ity and instance-aware attention to localize abnormal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' RegAD [130] trains a model for detecting category-agnostic anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In the test phase, the anomalies are identified by comparing the registered features of the test image and its corresponding normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The second setting relies on the vanilla few-shot image learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' PatchCore [62], SPADE [57], PaDim [67] conduct the ablation study on 16 normal training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' None of them, however, are spe- cialized in few-shot anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Hence, it is necessary to develop new algorithms that concentrate on native few-shot anomaly detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='2 Noisy Anomaly Detection Noisy learning is a classical problem for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [131] employ a novel trust region memory update scheme to keep noise feature point away from the memory bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [132] use a data refinement approach to improve the robustness of one-class classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [133] propose a strategy for training an anomaly detector in the presence of unlabeled anomalies, which is compatible with a broad class of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' They create labelled anomalies synthetically and jointly optimize the loss function with normal data and synthesis abnormal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [134] introduce an interpolated Gaussian descriptor that learns a one-class Gaussian anomaly classifier trained with adversarially interpolated training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, the majority of the aforementioned approaches have not been verified on real industrial image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In other words, the effectiveness of the existing anomaly detection methods may not be suitable for industrial manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='3 3D Anomaly Detection Since the release of MVTec 3D-AD [7] dataset, several papers have focused on anomaly detection in 3D industrial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Bergmann [135] introduces a teacher-student model for 3D anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The teacher network is trained to acquire general local geometric descriptors by recreating local receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' While the student network is taught to match the local 3D descriptors of the pre-trained teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Horwitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [136] propose BTF, a method that combines hand-crafted 3D representations (FPFH [137]) with the representation method of 2D features (PatchCore [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Reiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' [138] propose that the representational ability of self-supervised learning is temporarily inferior to that of handcrafted features for 3D anomaly detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Nevertheless, self-supervised characterization still has great potential if large-scale 3D anomaly detection dataset are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' AST [17] employs RGB image with depth information to enhance anomaly detection perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, most of 3D VAD methods are specialized in RGB-D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' While the 3D dataset in real-world industrial manufacturing consists of point clouds, which means current 3D VAD methods cannot be directly deployed in industrial manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Thus, there are still opportunities for 3D VAD advancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 5 Datasets and metrics Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Data is a crucial driving factor for machine learning, particu- larly for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Principally, the difficulty of getting industrial photos hampers the advancement of visual anomaly detection in industrial vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 7 demonstrates that the number and the size of VAD dataset are grad- ually increasing, but most of them are not generated in a real production line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The promising alternative approach is to fully utilize the industrial simulator Springer Nature 2021 LATEX template 22 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey to generate anomalous images, possibly reducing the gap between academic research and the demands of industrial manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 7: Comparison of datasets for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Dataset Class Normal Abnormal Total Annotation Level Real or Synthetic AITEX [139] 1 140 105 245 Segmentation mask Real BTAD [98] 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='830 Segmentation mask Real DAGM [140] 10 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='500 Segmentation mask Synthetic DEEPPCB [141] 1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='500 Bounding box Synthetic Eycandies [142] 10 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='250 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='250 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='500 Segmentation mask Synthetic Fabric dataset [143] 1 25 25 50 Segmentation mask Synthetic GDXray [144] 1 0 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='407 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='407 Bounding box Real KolektorSDD [145] 1 347 52 399 Segmentation mask Real KolektorSDD2 [114] 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='979 356 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='335 Segmentation mask Real MIAD [146] 7 87,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='500 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='500 105,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='000 Segmentation mask Synthetic MPDD [147] 6 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='064 282 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='346 Segmentation mask Real MTD [148] 1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='344 Segmentation mask Real MVTec AD [6] 15 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='096 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='258 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='354 Segmentation mask Real MVTec 3D-AD [149] 10 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='904 948 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='852 Segmentation mask Real MVTec LOCO-AD [7] 5 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='347 993 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='340 Segmentation mask Real NanoTwice [150] 1 5 40 45 Segmentation mask Real NEU-CLS [151] 1 0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='800 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='800 Bounding box Real RSDD [152] 2 195 Segmentation mask Real Steel Defect Detection [153] 1 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='076 Image Real Steel Tube Dataset [154] 1 0 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='408 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='408 Bounding box Real VisA [61] 12 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='621 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='200 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='821 Segmentation mask Real Table 8: A summary of metrics used for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Metric/Level Formula Remarks/Usage Precision (P) ↑ P = TP/(TP + FP) True Positive (TP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' False Positive (FP) Recall (R) ↑ R = TP/(TP + FN) False Negative (FN) True Positive Rate (TPR) ↑ TPR = TP/(TP + FN) False Positive Rate (FPR) ↓ FPR = FP/FP + TN) True Negative (TN) Area Under the Receiver Operating Charac- teristic curve (AU-ROC) ↑ � 1 0 (TPR) d(FPR) Classification Area Under Precision-Recall (AU-PR) ↑ � 1 0 (P) d(R) Localization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Segmentation Per-Region Overlap (PRO) [155] ↑ PRO = 1 N � i � k Pi∩Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='k Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='k Total ground truth number (N)/ Predicted abnormal pixels (P)/ Defect ground truth regions (C)/ Segmentation Saturated Per-Region Overlap (sPRO) [149] ↑ sPRO(P) = 1 m m � i=1 min( Ai∩P si ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 1) Total ground truth number (m)/ Predicted abnormal pixels (P)/ Defect ground truth regions (A)/ Corresponding saturation thresholds (s) / Segmentation F1 score ↑ F1 = 2(P · R)/(P + R) Classification Intersection over Union (IoU) [156] ↑ IoU = (H ∩ G)/(H ∪ G) Prediction (H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Ground truth (G)/ Localization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Segmentation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 8 offers a comprehensive review of the metrics in cross- modality neuroimage synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The first column denotes the name of the metric and the second column denotes the level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In other words, if the level is up, the larger the metrics value, the better the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' If the level is down, the lower the metrics value, the better the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The third column gives the detail for each metric, especially on how the metric accurately indicates the performance of visual anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' From Table 8, it can be easily observed that most of novel metrics are the variants of natural image Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 23 segmentation and detection metrics, such as F1 score, AU-ROC or AU-PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, these metrics can not correspond to the performance of VAD because the tiny size of anomalies requires a greater weighting than the anomaly-free regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Hence, the validity of these metrics for VAD remains to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Table 9: Image AUROC Performance of Different Methods on MVTec AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The highest and second places are marked in red and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='998 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content='959 Springer Nature 2021 LATEX template 24 Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 6 Total Performance Analysis Table 9 shows the statistical result of current VAD performance on MVTec AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' We provide a deep analysis of the performance of current VAD methods and unlock meaningful insights as below: Regarding the identification of image-level anomaly detection tasks, mem- ory bank-based approaches are the most effective neural network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, they are inadequate at detecting pixel-level anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Ensemble learning can dramatically improve the performance of state-of- the-art anomaly detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' SSPCAB [88] can be seamlessly integrated into cutting-edge method and significantly enhance the performance of reconstruction-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The gap between few-shot VAD and vanilla VAD is narrowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In other words, we may utilize data distillation algorithms to lower the amount of the dataset used for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 7 Future Directions We outline several intriguing future directions as following: We should build up a mutli-modalities VAD Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In actual assem- bly lines, RGB images are insufficient to detect anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Hence, we may employ additional modalities information, such as X-ray and ultrasound, to enhance anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Given that test samples are sequentially streamed on the product line, most of VAD methods are incapable of making instantaneous predictions upon the arrival of a new test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In industrial manufacturing, the inference speed of VAD should be addressed in addition to its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Adopting multi-objective evolutionary neural architecture search algorithms to find the optimal trade-off architecture is thus a promising approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' The majority of VAD methods use ImageNet pre-trained models to extract the features from industrial images, which inevitably results in the feature drift issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Consequently, there is a pressing need to construct a pre-trained model for industrial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Previously, we focused on developing data augmentation method for normal image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' However, we have not made much effort on synthesizing abnor- mal samples via data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In industrial manufacturing, it is very difficult to collect a large number of abnormal samples since most of produc- tion lines are faultless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' Hence, more attention should be paid to abnormal synthesis methods in the future, like CutPaste [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' 8 Conclusions In this paper, we provide a literature review on visual anomaly detection in industrial manufacturing, focusing on the level of supervision, the design of Springer Nature 2021 LATEX template Deep Visual Anomaly Detection in Industrial Manufacturing: A Survey 25 neural network architecture, the types and properties of datasets and the eval- uation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In particular, we characterize the new setting from industrial manufacturing and review current VAD algorithms in our proposed new set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In addition, we investigate in depth which network architecture design can considerably improve anomaly detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdFJT4oBgHgl3EQfVyxJ/content/2301.11514v1.pdf'} +page_content=' In the end, we highlight several exciting future research directions for visual anomaly detection.' 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a/k9AyT4oBgHgl3EQfk_gh/content/tmp_files/2301.00442v1.pdf.txt b/k9AyT4oBgHgl3EQfk_gh/content/tmp_files/2301.00442v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..baf7669baa9325291c31215676b9d12e0e7a6e8c --- /dev/null +++ b/k9AyT4oBgHgl3EQfk_gh/content/tmp_files/2301.00442v1.pdf.txt @@ -0,0 +1,1456 @@ +arXiv:2301.00442v1 [cs.CR] 1 Jan 2023 +An Overview of Limitations and Approaches in Identity +Management +Daniela Pöhn +Wolfgang Hommel +daniela.poehn@unibw.de +wolfgang.hommel@unibw.de +Universität der Bundeswehr München, Research Institute CODE +Munich, Germany +ABSTRACT +Identity and access management (I&AM) is the umbrella term for +managing users and their permissions. It is required for users to ac- +cess different services. These services can either be provided from +their home organization, like a company or university, or from +external service providers, e. g., cooperation partners. I&AM pro- +vides the management of identifiers with the attributes, creden- +tials, roles, and permissions the user has. Today, the requirements +have evolved from simply accessing individual web services in the +internet or at a company to the majority of all IT services from +different service providers with various accounts. Several identity +management models have been created with different approaches +within. +In order to adjust to heterogeneous environments, use cases, +and the evolution of identity management, this paper extends known +requirements for identity management. Existing models and ap- +proaches for identity management are mapped to the derived re- +quirements. Based on the mapping, advantages, disadvantages, and +gaps are identified. Current approaches suffer, as an example, from +trustworthiness and liability issues. Interoperability issues are even +more inherent as the approaches partly develop apart, forming an +heterogeneous environment. The results from this analysis empha- +size the need for one holistic identity management framework. +CCS CONCEPTS +• Security and privacy → Security services; Access control; +Authentication; Authorization. +KEYWORDS +Security, Identity, Identity Management, Federated Identity Man- +agement, Blockchain +ACM Reference Format: +Daniela Pöhn and Wolfgang Hommel. 2020. An Overview of Limitations +and Approaches in Identity Management . In The 15th International Confer- +ence on Availability, Reliability and Security (ARES 2020), August 25–28, 2020, +Virtual Event, Ireland. ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/3407023.3407026 +ARES 2020, August 25–28, 2020, Virtual Event, Ireland +© 2020 Copyright held by the owner/author(s). Publication rights licensed to ACM. +This is the author’s version of the work. It is posted here for your personal use. Not +for redistribution. The definitive Version of Record was published in The 15th Inter- +national Conference on Availability, Reliability and Security (ARES 2020), August 25–28, +2020, Virtual Event, Ireland, https://doi.org/10.1145/3407023.3407026. +1 +INTRODUCTION +Medium-sized and large organizations, such as universities and +companies, typically provide several information and communica- +tions technology services to their members. Usually, a username +as identifier is assigned to each member. All services, e. g., email, +file, and web collaboration, can then be used by the combination of +username and some sort of credential. In the backend, username, +salted and hashed password, and further information, called at- +tributes, are stored centrally. Organizations either use lightweight +directory access protocol (LDAP) or other database management +systems for this. Even though a local Identity & Access Manage- +ment (I&AM) seems trivial, it gets challenging, if interfaces to other +entities appear, e.g., with collaborations or joint ventures. +When members of an organization want to access external ser- +vices, like collaboration platforms, either their user accounts are +doubled or Federated Identity Management (FIM) is used. FIM is +an arrangement between multiple entities, in order to let users +use the same identification data as in their home organization to +obtain access to the services of the provided entities within the +trust boundaries. FIM is mostly based on the protocol Security As- +sertion Markup Language (SAML) [59], which is common in the +R&E field as well as in national federations, or on OpenID Con- +nect (OIDC) [64]. OIDC is the authentication layer on top of the +authorization protocol OAuth [29], while SAML incorporates both +aspects. OAuth and OIDC are widely used in today’s web services, +e. g., Google Application Interfaces (APIs). Each user has at least +some kind of home organization, called Identity Provider (IdP) in +SAML or Relying Party (RP) in OIDC. The user wants to access a +service from a Service Provider (SP), respectively OpenID Provider +(OP). The set of IdPs and SPs that collaborate for a specific reason is +commonly referred to as federation. Federations are especially cus- +tomary in SAML [59], however an OIDC specification is currently +developed [30]. While federations in industry are relatively small, +many national research and education networks (NRENs) operate +large authentication and authorization infrastructures (AAIs) with +hundreds of organizations. Technical trust is established by pre- +sharing metadata, i. e., information about the entity in extensible +markup language (XML) files. Another form of trust is set into con- +tracts, e. g. between federation operator and participating entity. +Due to limitations of the rather static SAML federations, inter- +federations, like eduGAIN [26], were established and dynamic con- +cepts, e. g. interfederation metadata exchange [84] and TrustBro- +ker [57], were developed. As industry tends to use OIDC, organi- +zations set up OIDC in parallel or even changed completely. At the + +ARES 2020, August 25–28, 2020, Virtual Event, Ireland +Pöhn and Hommel +same time eID federations with SAML were established [34]. While +SAML federations have trust established in out-of-bounds mecha- +nisms, other mechanisms need to be set up with OIDC. This led to +the growth of different assurance frameworks and registries [39], +while, at the same time, a Domain Name System (DNS) like struc- +ture for discovery of the SP and verification of trust scheme mem- +bership, i. e. LIGHTest [62], was developed. +I&AM as well as FIM focus on organizations. Since some users +are at least concerned about their identity and privacy, user-centric +solutions were developed. A privacy policy of a service provider +summarizes the usage of collected personal user data, which can +now be gained by EU general data protection regulation (GDPR). +At that time, most sites described user information in little under- +standable way. User-centric identity management (UCIM) targeted +these shortcomings by a clear presentation and intuitive evalua- +tion of privacy policies. One first application was Microsoft Pass- +port, an IdP in a Microsoft environment, which could be used as +universal IdP in the internet. As Microsoft was integrated in ev- +ery communication, trust and privacy were drawbacks [54]. As a +next step, Windows Cardspace was developed, which could make +use of other IdPs with SAML. The user had several infocards, ei- +ther created by himself or managed by an IdP. The application +possessed several security bugs though [5]. User Managed Access +(UMA) [41], an OAuth-based standard, even goes one step further. +It enables the user to control the authorization of data sharing and +other protected resources. Sovrin [75] sees Self-Sovereign Identi- +ties as the next step of evolution. The user must be the ultimate +owner of a self-sovereign identity. Therefore, a self-sovereign iden- +tity is fully autonomous in management as well as administration, +while the user is the single source of truth. This paradigm is re- +alized by decentralized domains; inherent this means the use of +distributed ledger technologies (DLTs), i. e., blockchain [21]. +Hence, orthogonal approaches are currently developed and ap- +plied. SAML is rather static with solid trust boundaries, in contrast +to OIDC. DNS-based structures facilitate stronger trust, based on +certificates and eIDs. While UMA can be applied to OIDC, self- +sovereign identities (SSIs) can be used for different use cases. The +bouquet of identity management approaches lead to different prob- +lems. Administrators need to take care of different solutions in or- +ganizations. Due to this, getting an overview and applying similar +policies gets cumbersome. Also, if companies join each other or +establish projects, it might be the case, that their identity manage- +ment solutions are incompatible. This would require yet another +system to manage. In addition, not all requirements from the or- +ganizations might been fulfilled by the systems, leading to further +tools. Besides administrators, also end-users and other stakehold- +ers have lots of solutions to decide from. The end user might end up +using several different systems to manage his identities. Therefore, +it is important to have an overview, showing advantages and disad- +vantages, as well as gaps. This overview can help the stakeholders +to choose the best fitting approach and identify needed tools. +In order provide this overview, we extend known requirements +for identity management in Section 2. The related work is divided +into the established models [85] centralized identity management, +FIM, and UCIM in Sections 3 to 5. The existing approaches are anal- +ysed based on the requirements and gaps are identified. The paper +is concluded by an outlook and summary of the analysis. +2 +REQUIREMENTS OF IDENTITY +MANAGEMENT +In order to understand the usage of the different identity manage- +ment approaches and discover possible gaps, various requirements +are identified. According to Torres et al. [76], the main require- +ments are Usability, Interoperability, Functionality, Trustworthi- +ness, Security, Mobility, Privacy, Law Enforcement, and Affordabil- +ity. Boujezza et al. [16] refer to these requirements as bases for IoT. +Ferdous and Poet leaf out Mobility [22], while Gao et al. [25] con- +centrate on the core functionalities. The authors of [27] see User +control and consent, Minimal disclosure for a constrained use, Jus- +tifiable parties, Directed Identities, Design for a pluralism of oper- +ators and technology, Human integration, and Consistent experi- +ence across contexts as requirements for SSI. +Regarding SSIs, the management of users has to be another re- +quirement. Besides management of identities, other functionali- +ties might be needed for today’s identity management models, ap- +proaches, and protocols to be interoperable. Therefore, the require- +ment Functionality is divided into Management and Functionality. +The location of the user data is renamed to Portability. With the +GDPR in Europe, the user can ask to get his account data exported +and then imported to another platform. Therefore, the user data +needs to be portable from one system to another. Additionally, the +user should be able to make use of different devices, see require- +ment Design for a pluralism of operators and technology. Afford- +ability is split into Integration into the existing system and Scala- +bility. As more and more devices and thereby identities are created, +scalability becomes increasingly important. +In consequence, the following requirements are used in further +sections to evaluate the different approaches. These requirements +extend the requirements gathered from Torres et al. [76]. They +are already condensed and comprise different sub-requirements, +as shown in Table 1. +3 +CENTRALIZED IDENTITY MANAGEMENT +As identity management advances in different directions, selected +areas of related work are covered in the following. In order to struc- +ture the work, we use different identity management models: +• centralized / network-centric, +• federated / application-centric, +• decentralized / user-centric. +These models have further characteristics, used in different use +cases with different protocols. Even though centralized identity +management was the first evolutionary step after single instances, +it is still used. Mostly companies, which control all their own iden- +tities, use single sign-on between services. If they outsourced IdM, +it is most likely FIM. Also, companies with different locations may +have one federation. Within a centralized identity management, +the company is in control of law-enforcement [REQ10] and there +is no need for further interoperability [REQ3]. If cooperations are +established, it looks differently. +3.1 +Centralized Identity Management with DNS +The DNS is the hierarchical and decentralized naming system for +computers, services, and other resources connected to a network. + +An Overview of Limitations and Approaches in Identity Management +ARES 2020, August 25–28, 2020, Virtual Event, Ireland +Table 1: Requirements for Identity Management +No. +Name +Description +REQ1 +Management +of Identities, Automation +REQ2 +Usability +User Interface, Reduced Complex- +ity, Consistent Experience +REQ3 +Interoperability +Interoperability between Proto- +cols, Models, Silos +REQ4 +Scalability +Mobility of Environment +REQ5 +Functionality +Translation +Services, +Proxies, +Bridges, Group Management +REQ6 +Trustworthiness +Trust Management, Segregation +of Power, Policies +REQ7 +Security +Basic and Multi-Lateral +REQ8 +Portability +of Accounts and Systems +REQ9 +Privacy +Anonymity, +Pseudonymity, +Transparency, +Controlability, +Consent, Data Minimization +REQ10 +Liability +Law-enforcement, +Digital +Evi- +dence, Data Retention +REQ11 +Integration +Affordability, +Integration +into +Processes and Existing Infrastruc- +ture +Most commonly used, it translates memorable domains to numeric +internet protocol (IP) addresses needed for locating and identifying +computer services and devices with the underlying network proto- +cols. In an identity federation, the home organization is discovered +by specific protocols and partly also services, i. e., a trusted third +party having a list of possible entities in form of metadata. Further- +more, in order to make use of different protocols at one entity, at +least bridges need to be established. A way to improve the situation +could be the usage of DNS. The EU project LIGHTest [63] [62] is +based on DNS and DNS Secure Extensions (DNSSEC) as a method +of discovery and trust. Therefore, a trust scheme publication au- +thority (TSPA) [81] needs to be set up. The approach can make use +of role-based access control (RBAC) as a form of delegation, while +it harmonizes the delegation data format [80]. As drawbacks, eI- +DAS or some sort of certificates are needed, which also means fixed +Level of Assurance (LoA), therefore [REQ6] is only partly fulfilled, +and a centralized structure, similar to SAML. Even though it can +be used with internet of things (IoT) [38], [REQ3] is not fulfilled. +3.2 +Centralized Identity Management with +DNS for IoT +Wang and Wen [83] present a privacy-enhanced DNS-scheme for +IoT. DNS was not designed with all the security complications of to- +day’s worldin mind. Therefore, the authors introduce new features. +Nevertheless, the approach does not take caching into account, +which contradicts [REQ7] and [REQ9]. Additionally, fresh domain +names are needed after several steps [REQ1]. The approach of Lee +et al. [48] describes the auto-configuration of IoT devices with the +device’s category and model in IPv6 environment, as the neighbor +discovery is used. It is, in contrast, inefficient due to manual con- +figuration of the DNS names for IoT devices [REQ1]. Though the +discovery via DNS is widely known as well as possible attacks, no +Table 2: Overview of Approaches for Centralized Identity +Management +Ref +Name +Requirements not fulfilled +Centralized IdM +[REQ3] +[63] [62] +LIGHTest +[REQ6] [REQ3] +[81] +TSPA +[REQ6] [REQ3] +[80] +Delegation +[REQ6] [REQ3] +[38] +LIGHTest for IoT +[REQ6] [REQ3] +[83] +Privacy-enhanced +DNS-scheme +[REQ1] [REQ7] [REQ9] +[48] +Autoconfiguration +of IoT devices +[REQ1] +[69] +ACE +[REQ3] +[87] +Natural context +[REQ11] +[51] +PhD thesis +[REQ1] [REQ11] +[33] +Cloud-based IoT +[REQ11] +authorization is provided. The size of messages can get a problem +as well as the scalability [REQ4], if too many features are added +to DNS. Therefore, changes are difficult [REQ11]. According to +the presented approaches, identity management with DNS can be +used in use cases, where no anonymity is needed, but where eIDAS, +fixed trust values, and a centralized structure for certificates is in +place. +3.3 +Centralized Identity Management for IoT +Not only humans need to be identified, but also things, like comput- +ers and IoT devices. IoT devices often have limitations, like band- +width or central processing unit (CPU). Therefore, different proto- +cols are used, as shown in [23]. Belran and Skarmeta [9] give an +overview of the IETF protocol Authentication and Authorization +for Constrained Environments (ACE) [69], which is extended by +several Internet-Drafts (I-Ds) [REQ3]. The authors of [87] identify +and authenticate IoT objects in a natural context using DNS with +time property, name, and location resolution. Efficiency degrada- +tion is avoided, but the life cycle is not taken into account [REQ11]. +P. N. Mahalle’s thesis [51] is about identity management for IoT de- +vices. Even though the context of the identifier is specified, users +and their relationship to objects are not mentioned [REQ1] [REQ11]. +[33] focuses on cloud-based IoT. While it takes the life cycle of IoT +things into account, it does not consider the owner [REQ11]. +3.4 +Conclusion from Centralized Identity +Management +The results are summarized in Table 2. Many approaches do not +fulfill the requirement Interoperability [REQ3]. This is comprehen- +sible, as they are solely centralized. Additionally, centralized IdM +approaches seem to lack Trustworthiness [REQ6], Management +[REQ1], and Integration [REQ11] in several cases. Especially Trust- +worthiness and Integration come inherent with the model. Porta- +bility [REQ8] is difficult to estimate, as the approaches are mostly +described for computer use cases. + +ARES 2020, August 25–28, 2020, Virtual Event, Ireland +Pöhn and Hommel +4 +FEDERATED IDENTITY MANAGEMENT +Federated identity management is an arrangement between mul- +tiple entities within certain trust boundaries. It lets users use the +same identification as in their home organization to obtain access +to the services of the provided entities. In many constellations, a +trusted third party is needed. Especially SAML federations tend +to have scalability problems [REQ4]. Between different FIM ap- +proaches, interoperability is a problem [REQ3]. Liability [REQ10] +depends on all entities, while the trusted third party could provide +additional functionality, though it is not the case with the follow- +ing approaches [REQ5]. Bargh et al. [8] show that the growing +number of IdPs and SPs is a well-known problem of SAML in aca- +demic world. Different approaches, therefore, try to improve the +scalability of trust and metadata exchange as well as trust negotia- +tion in federated identity management. IdMRep by Arias Cabarcos +et al. [6] introduces dynamic trust values. The presented approach +does not work with new members [REQ6]. Additionally, the accu- +mulated amount of data at the trust engine can become to a sin- +gle point of failure in large-scale infrastructures [REQ4]. The ap- +proach Dynamic Identity Federations by Md. Sadek Ferdous and +Ron Poet [20] focuses on fully automated establishment of feder- +ations and trust. Though the technical establishment is fully auto- +mated, the user has to enter the EntityID of the IdP and needs to +generate a code, which is not user-friendly [REQ2]. Furthermore, +the division into trusted,semi-trusted, and untrustedis rather coarse- +grained [REQ6],and a further database per entity is needed [REQ11]. +Another approach is called TrustBroker by Pöhn et al. [58], which +is based on SAML, though the generic concepts can be applied to +FIM [REQ3]. There are also other federation approaches. OLYM- +PUS by Moreno et al. [53] is privacy preserving, while it allows dis- +tributed IdPs. Nevertheless, it misses [REQ3]. Kang and Khashno- +bish [40] approach for a peer-to-peer (P2P) federation is decentral- +ized, but has scalability issues in large-scale infrastructures with +many entities and federation servers [REQ4]. Dhungana et al. [19] +provide an IDM framework for cloud networking based on UMA +with OIDC. It thereby allows federated identity management with- +out the standard of OIDC federations [30] [REQ11]. UnlimitID by +Isaakidis et al. [36] utilizes attribute-based anonymous credentials, +which are based on algebraic Message Authentication Codes (MACs). +The approach is usable with OAuth and let pseudonyms expire. +Though for this short time users might be linkable [REQ9]. Fur- +thermore, in some use cases pseudonyms might not be allowed +[REQ11]. The usage of MACs seems like a step back. These ap- +proaches show that certain aspects are regarded, which can be +solved by federated identity management, but not all. Additionally, +especially FIM with SAML is difficult to port to mobile applications +and IoT devices [REQ8]. +4.1 +Federated Identity Management with DNS +ID4me [71] combines DNS with OIDC. It utilizes the protocol Au- +tomated Certificate Management Environment (ACME), which au- +tomatically verifies the ownership of a domain and also simplifies +the issuance of certificates. The DNS challenge is used as discov- +ery method, secured with DNSSEC. The communication is secured +by DNS-based Authentication of Named Entities (DANE), which is +only rarely used in practice. ID4me has the roles identity agent, +which administrates the attributes and is identified by DNS, and +identity authority, which is used for authentication. As a conse- +quence, the user has to trust the identity agent [REQ6]. The idea +to use DNS is not new. Hulsebosch et al. [35] present the combina- +tion for effective and trusted user controlled light-path establish- +ment in grid environment. Thereby, DNS tree is used to establish +technical trust between IdP and Optical Network Service Provider +(ONSP) dynamically within a virtual organization (VO), a kind of +federation [REQ3]. +4.2 +Federated Identity Management with eID +The EU Regulation 910/2014, also known as eIDAS, is effective +since September 2018. The authors of [37] show the federated iden- +tity architecture with eIDs. The approach of Berbecaru et al. [13] +describes the connect between eIDAS and STORK. STORK [31] +[60] is the eID federation based on eID. It features the European +electronic identity interoperability platform, which relies on SAML +with one node per country. Thereby, it is heterogeneous and a +large-scale infrastructure. Several approaches describe the integra- +tion of academic attributes in the eIDAS infrastructure without re- +garding the inter-federation eduGAIN, which is the de-facto stan- +dard federation in the academic world [14] [15] [REQ11]. Alonso +et al. [4] proposed an identity model to connect FIWARE services +authenticated with OAuth 2.0 to eIDAS nodes and thereby meet +the regulations. It nevertheless is the solution for one specific use +case [REQ3]. +4.3 +Federated Identity Management as a Service +Other approaches focus on federation as a service, which utilizes +cloudinfrastructure. The framework AIDF by Zouari and Hamdi [88] +makes use of a trust broker, like in [58], in cloud. IdP and SP reg- +ister at a central party. The SP can tell, which IDs and attributes +he wants and he can at the same time subscribe to claim transfor- +mation and attribute mapping service. Attribute mapping is nec- +essary, if different semantic or/and syntax of attributes is used. +A simple example is date format. The central service automati- +cally establishes a link between IdP and SP after consent. Some +sort of trust assurance is established by National Institute of Stan- +dards and Technology (NIST) LoA, which is currently published +in the third revision of the special publication 800-63 from 2017. +As the LoA is revised, the framework needs updates as well. Fur- +thermore, not only the NIST LoA could be applied, but also other +standards [REQ6]. Additionally, there exists no trust assurance for +IdPs, solely for SPs. The same drawback with NIST LoA has the ap- +proach of Zefferer et al. [86], which links identities to eIDs. Even +though the federations have a strong legal foundation as eIDs are +used, it is highly customized for this one use case [REQ3]. +4.4 +Conclusion from Federated Identity +Management +A summary of FIM approaches is found in Table 3. FIM itself lacks +Interoperability [REQ3], Functionality [REQ5], and Liability [REQ10]. +Liability often comes with policies and other agreements. Interop- +erability can be archived with additional tools, but the protocols +as they are act like silos. Looking at the protocol SAML, Scalability +[REQ4] and Portability [REQ8] are also missing. The approaches + +An Overview of Limitations and Approaches in Identity Management +ARES 2020, August 25–28, 2020, Virtual Event, Ireland +Table 3: Overviewof Approaches for FederatedIdentity Man- +agement +Ref +Name +Requirements not fulfilled +FIM +[REQ3] [REQ5] [REQ10] +[6] +IdMRep +[REQ4] [REQ6] +[20] +Dynamic +Identity +Federations +[REQ2] [REQ6] [REQ11] +[58] +TrustBroker +[REQ3] +[53] +OLYMPUS +[REQ3] +[40] +P2P federation +[REQ4] +[19] +IDM framework +[REQ11] +[36] +UnlimitID +[REQ9] [REQ11] +[71] +ID4me +[REQ6] +[35] +VO +[REQ3] +[14] [15] +Academic eIDAS +[REQ11] +[4] +FIWARE eIDAS +[REQ3] +[88] +AIDF +[REQ6] +[86] +eID federations +[REQ3] [REQ6] +described in this paper often have drawbacks in Interoperability +[REQ3] and Trustworthiness [REQ6]. Also, Integration [REQ11] is +often stated, while it is partly difficult to estimate Liability [REQ10] +as it depends on the implementation and established processes. As +the users access services from other organizations, trust needs to +be established between home organization and service provider, +user and service provider, and user and home organization. If ad- +ditional entities are involved, they need to be involved in the trust +establishment as well. +5 +USER-CENTRIC IDENTITY MANAGEMENT +User-Centric Identity Management puts the user in control of their +own identities. It has evolved from simple clients on a PC to UMA +and SSI, which are described in the following. Although it gives +more power to the user and thereby adds value to [REQ9], the in- +teroperability with other approaches is low [REQ3], as [REQ5] is +missing. [REQ10] depends on the client and the provider. +5.1 +User-Centric Identity Management for IoT +Do van Thuan et al. [78] present a user centric identity manage- +ment for IoT with four identifiers, i. e., idpID, domIDPart, devID- +Part, and userIDPart. They describe different relationships as well, +e. g., serving people in general, serving one person permanently, +serving multiple persons successively, and serving multiple per- +sons simultaneously. Though the relationship between device and +people is characterized in all variants and the approach has sev- +eral identifier, the life cycle itself is not regarded [REQ11]. It is fur- +thermore unknown how the approach should interact with other +approaches and protocols [REQ3]. +5.2 +User-Centric Identity Management with +UMA +The works published by Kumar et al. [47] and Slomovic [70] make +it clear that pseudonymity and anonymity can be important in cer- +tain use cases. Identities from different social networks can be ag- +gregated and lead to one person. UMA manages to decouple iden- +tity resolution from the maintenance of identity information [44] +[50] [52] and even one application for IoT by Cruz-Piris et al. [17] is +published. Although UMA is based on OAuth, the principle can be +applied to SAML as well [68]. Nevertheless, the principle of UMA +might not be possible to apply to all use cases [REQ3]. +5.3 +User-Centric Identity Management with SSI +The principles of SSI management are described by Toth and Anderson- +Priddy [77]. Ferdous et al. [21] broaden the laws of identities from +existence, autonomy, ownership, access, single source, protection, +availability, and persistence to user control and consent, minimal +disclosure for a constrained use, justifiable parties, directed iden- +tity, pluralism of operators and technologies, human integration, +and consistent experience across contexts for self-sovereign iden- +tities. This step is seen as an evolution of identity management by +Sovrin [75]. +SSIs are typically implemented by blockchain, which have de- +centralized identifiers (DIDs), proposed by the W3C, as a new type +of identifier for verifiable, decentralized digital identities. Claim +Chain by Stokkink and Pouwelse [73] is a pure self-sovereign build- +ing on claim model, which is practically used, but also rather re- +strictive. No data is sent without consent, but also at the same time +only attested attributes are possible [REQ3]. DecentID by Friebe et +al. [24] and Steichen et al. [72] is based on Ethereum smart con- +tracts and tries to preserve the privacy of the attributes. The ap- +proach utilizes external storage like distributed hash tables (DHTs) +for attributes for scalability reasons. The nodes do not mine ef- +ficiently with large files [REQ4]. Additionally, a linkage between +identities to centralized registration contracts is needed [REQ9]. +Sora Identity [74] uses the mobile phone of the user as one of the +main actors. Though this approach preserves the privacy, the life +cycle of a device and the usage of different devices can get problem- +atic [REQ8]. Another blockchain-based approach combines it with +PKI [82] [7] [REQ3]. The approaches support the revocation of cer- +tificates, but have scalability problems [REQ4]. The Horcux Proto- +col [55][56] is a decentralized biometric credential storage using +SSI with Biometric Open Protocol Standard (BOPS). This is done al- +though biometrics tend to have false-positives and false-negatives +and their implementations are often broken [REQ7]. In [65], Schanzen- +bach and Banse describe an approach, which is adapted from UMA +applying NameID [46]. As they use a secure name system - GNU +Name System (GNS) - to store data, they do not require a global, +publicledger and do not compromise the privacy of users. Ahadipour +and Schanzenbach [1] present a survey on authorization in dis- +tributed systems. In [66], the authors describe the decentralized +attribute-based delegation (ABD). reclaimID by Schanzenbach et +al. [67] is based on the work published beforehand. Attributes are +encrypted by attribute-based encryption (ABE), which means the +attributes are self-signed with the private key associated with the +identity of the user. The approach allows the use of one or more + +ARES 2020, August 25–28, 2020, Virtual Event, Ireland +Pöhn and Hommel +identities with different attributes. Nevertheless, privacy might be +one drawback as well as trust into the attributes [REQ6] [REQ9]. +5.4 +User-Centric Identity Management with SSI +for Cloud Environments +The identity wallet platform CREDENTIAL [79] [45] [42] [32] aims +at the development of a secure and privacy-preserving data shar- +ing platform that puts the user under full control. [32] shows how +proxy-re-encryption and redactable signature technology can be +integrated into their workflows, which is basically a man in the +middle. The platform has three main actors: wallet platform, user, +and the data receiver, which can be either user or SP, as described +in [42]. A metadata preserving high-level architecture is proposed, +which might be practically challenging due to log files leaking in- +formation [REQ7]. Kostopoulos et al. [45] describe the architec- +ture of CREDENTIAL. Interfaces to existing protocols are missing +[REQ3] [REQ11]. Finally, Veseli et al. [79] focus on privacy by de- +sign. The LINDDUN (Linkability, Identifiability, Non-repudiation, +Detectability, information Disclosure, content Unawareness, and +policy and consent Non-compliance) method [18], a privacy threat +analysis framework, was used to establish privacy by design. Thereby, +the data is encrypted, though the approach is cloud-based. The user +has to trust the wallet provider [REQ3] [REQ6] [REQ11]. +5.5 +User-Centric Identity Management with SSI +as Federation +NEXTLEAP by Halpin et al. [28] is an approach for FIM with blind +signatures based on MACs to improve OIDC. The identity itself is +a privacy-enhanced based blockchain identity with a decentralized +PKI. Therefore, access control is performed in a decentralized man- +ner. Though the purpose is secure messaging, the scalability can be +a drawback in large-scale infrastructures [REQ4]. [49] is designed +with having a cross-organizational authentication system in mind. +One-way hashed enforce the concept of one-time pad passwords, +which can provide unlinkability. Unfortunately, the performance +is not clearly described [REQ4]. +5.6 +User-Centric Identity Management with SSI +as Cloud Federation +The goal of the EU project SUNFISH was the establishment of a +cloud federation, q. v. Alansari et al. [2] [3]. The cloud federation +is a collaboration of organizations sharing data hosted on their pri- +vate cloud infrastructures for business opportunity. While the ap- +proach introduces distributed access in a privacy-preserving way, +the user must be identified and authenticated beforehand. eIDAS +can be used as a source of trust, though the trust within a cloud fed- +eration is not clearly described [REQ6]. Although the anonymity +for users by the usage of tokens instead of attributessounds promis- +ing, tokens might not in every setting. Additionally, the approach +is solely based on Intel Software Guard Extensions (SGX) [REQ11]. +The approach of dynamic cloudfederation by Bendiab evolves from +a dynamic cloudfederation to blockchain with fuzzy cognitive maps +dynamic trust model [43] [11] [10] [12]. In [43], the authors de- +scribe the basic approach with the entities user, cloudservice provider +(CSP), IdP, federation provider (FP), and certificate authority. The +token is sent from one FP to another FP for validation and trans- +formation. The CSP verifies the token’s legitimacy, before the user +can access the cloud service. This could also be done by a simple +proxy between two protocols, like SAML and OIDC in eduGAIN +[REQ4]. Trust is described in [11]. Each user gets his identity from +a trusted IdP. If the CSP does not trust this IdP, the trust can be +computed for each other in real time. Based on the result, a con- +nection is established. The authors describe a complex trust evalu- +ation, which is derived from different characteristics, like security, +and privacy, though these factors are difficult to measure and ac- +cess [REQ6]. The trust is then put into blockchain [10]. The CSP +can manage the trust relationships in a dynamic and distributed +manner without the need for centralized authorities, like IdP. Nev- +ertheless, a Trust Management Platform (TMP) is needed and the +CSPs are responsible for authenticating their registered users. [12] +is the next step with fuzzy cognitive maps for modeling and eval- +uating trust relationships. It uses a trust computation model, sim- +ilar to the characteristics introduced beforehand. It can calculate +the trust of unknown entities, if further information are available. +Unfortunately, LoA methods like Vectors of Trust (VoT) [61] are +not taken into account [REQ6] [REQ11]. The approach eases the +creation of secure Infrastructure as a Service (IaaS) cloud federa- +tion. Though the approach makes use of anonymity approaches, +it still stores personal information in an external database [REQ9]. +Additionally, the approach concentrates on one use case [REQ3]. +5.7 +Conclusion from User-Centric Identity +Management +A summary of user-centric identity management approaches is +shown in Table 4. UCIM has drawbacks in Interoperability [REQ3], +Functionality [REQ5], and Liability [REQ10], though there are dif- +ferences between older approaches, UMA, and SSI. Within SSI, it +partly depends on use cases, centralized / decentralized manage- +ment, among others. The presented approaches for UMA miss In- +teroperability [REQ3]. The principle of UMA should work for many +use cases, although this is not further explored to the knowledge +of the authors. UCIM for IoT has issues with Integration [REQ11] +and presumably with Interoperability [REQ3]. Looking at the pre- +sented SSI approaches, Scalability [REQ4] and Privacy [REQ9] are +often stated. Many approaches are tested within smaller environ- +ments, leading to problems with scalability. Privacy seems con- +tradictory as the approaches focus on users. Not all approaches +concentrate on privacy and others seem to have issues as well. +Focusing on cloud and federation use cases, Scalability [REQ4], +Trustworthiness [REQ6], and Integration [REQ11] are often miss- +ing. Trustworthiness is difficult, as the user has more control over +his data while the service provider needs to trust the user’s at- +tributes. +6 +CONCLUSION AND FUTURE WORK +Nowadays, security is one key requirement in a network environ- +ment. Identities are everywhere and with the growing identity theft, +secure identity management gets more relevant than before. In this +paper, we extend known requirements. Based on new approaches +as well as the heterogeneous environments and possible use cases, +new requirements were added. At the same time, others were newly + +An Overview of Limitations and Approaches in Identity Management +ARES 2020, August 25–28, 2020, Virtual Event, Ireland +Table 4: Overview of Approaches for User-Centric Identity +Management +Ref +Name +Requirements not fulfilled +UCIM +[REQ3] [REQ5] [REQ10] +[78] +UCIM for IoT +[REQ3] [REQ11] +[44] +UMA +[REQ3] +[17] +UMA for IoT +[REQ3] +[73] +Claim Chain +[REQ3] +[24] +DecentID +[REQ4] [REQ9] +[72] +SSI for IPFS +[REQ4] [REQ9] +[74] +Sora Identity +[REQ8] +[82] [7] +Blockchain-based +PKI +management +framework +[REQ3] [REQ4] +[55] [56] +Horcux Protocol +[REQ7] +[67] +reclaimID +[REQ6] [REQ9] +[79] +CREDENTIAL +[REQ3] +[REQ6] +[REQ7] +[REQ11] +[28] +NEXTLEAP +[REQ4] +[49] +Blockchain-based +AAA system +[REQ4] +[2] [3] +SUNFISH +[REQ6] [REQ11] +[43] +Dynamic cloud fed- +eration +[REQ3] +[REQ4] +[REQ6] +[REQ9] [REQ11] +defined. Management [REQ1] of identities is a new requirement, +while Functionality [REQ5] is still needed for interoperability be- +tween all these approaches. The GDPR leads to the change from +Mobility to Portability [REQ8], as users can request to port their +account and more and more different devices are used. This also +leads into the division Scalability [REQ4] and Integration [REQ11], +which was originally Affordability. +As a next step, we give a broad overview of different identity +management approaches structured in identity management mod- +els and use cases. This ranges from centralized identity manage- +ment for specific use cases, over federated identity management, to +user-centric approaches, like user-managed access and self-sovereign +identities. We compare these approaches with the derived require- +ments, which gives us an overview of advantages as well as draw- +backs. +Centralized approaches lack Interoperability [REQ3], Trustwor- +thiness [REQ6], Management [REQ1], and Integration [REQ11]. Porta- +bility [REQ8] cannot be estimated. The shortcomings are inher- +ent of the model. Federated Identity Management has general is- +sues with Interoperability [REQ3], Functionality [REQ5], and Li- +ability [REQ10]. SAML misses Scalability [REQ4] and Portability +[REQ8] as well. Trustworthiness [REQ6] and Integration [REQ11] +are additional shortcomings of the described FIM approaches. User- +Centric Identity Management has drawbacks in Interoperability +[REQ3], Functionality [REQ5], and Liability [REQ10]. Also Trust- +worthiness [REQ6] is an issue. Depending on the approach, Scal- +ability [REQ4] and Integration [REQ11] are unavailable as well. +No approach has managed to meet all requirements. Two require- +ments were out of scope of the described approaches: Liability +[REQ10] and Functionality [REQ5]. Liability is mainly endorsed +by the home organization, while additional approaches try to ful- +fill Functionality [REQ5]. Interoperability [REQ3] and Trustwor- +thiness [REQ6] might be met within the approach or model but +not crosswise. Thereby, Integration [REQ11] is also difficult. Sim- +ply combining different approaches does not satisfy the stated re- +quirements. Running several approaches in parallel binds more +manpower and time. Additionally, it can lead to different policies +and further inconsistent database. Therefore, at least an overview +and connectors between the approaches are needed to interact be- +tween the approaches and provide an overview. In order to meet +all requirements, further work is needed. +As a result, stakeholders would profit from a framework includ- +ing different models and, thereby, approaches. 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Identifying and Authenticating IoT Objects in a Natural Context. +Computer 48, 8 (Aug 2015), 81–83. https://doi.org/10.1109/MC.2015.213 +[88] Jaweher +Zouari +and +Mohamed +Hamdi. +2016. +AIDF: +An +Identity +as +a +Service +Framework +for +the +Cloud. +In +2016 +International +Sym- +posium +on +Networks, +Computers +and +Communications +(ISNCC). +1–5. +https://doi.org/10.1109/ISNCC.2016.7746120 + diff --git a/k9AyT4oBgHgl3EQfk_gh/content/tmp_files/load_file.txt b/k9AyT4oBgHgl3EQfk_gh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe599b033af58aae16ecdc66203d943b245504ea --- /dev/null +++ b/k9AyT4oBgHgl3EQfk_gh/content/tmp_files/load_file.txt @@ -0,0 +1,1387 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf,len=1386 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='00442v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='CR] 1 Jan 2023 An Overview of Limitations and Approaches in Identity Management Daniela Pöhn Wolfgang Hommel daniela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='poehn@unibw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='de wolfgang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='hommel@unibw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='de Universität der Bundeswehr München, Research Institute CODE Munich, Germany ABSTRACT Identity and access management (I&AM) is the umbrella term for managing users and their permissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It is required for users to ac- cess different services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' These services can either be provided from their home organization, like a company or university, or from external service providers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', cooperation partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' I&AM pro- vides the management of identifiers with the attributes, creden- tials, roles, and permissions the user has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Today, the requirements have evolved from simply accessing individual web services in the internet or at a company to the majority of all IT services from different service providers with various accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Several identity management models have been created with different approaches within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In order to adjust to heterogeneous environments, use cases, and the evolution of identity management, this paper extends known requirements for identity management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Existing models and ap- proaches for identity management are mapped to the derived re- quirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Based on the mapping, advantages, disadvantages, and gaps are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Current approaches suffer, as an example, from trustworthiness and liability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Interoperability issues are even more inherent as the approaches partly develop apart, forming an heterogeneous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The results from this analysis empha- size the need for one holistic identity management framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' CCS CONCEPTS Security and privacy → Security services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Access control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Authentication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Authorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' KEYWORDS Security, Identity, Identity Management, Federated Identity Man- agement, Blockchain ACM Reference Format: Daniela Pöhn and Wolfgang Hommel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' An Overview of Limitations and Approaches in Identity Management .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In The 15th International Confer- ence on Availability, Reliability and Security (ARES 2020), August 25–28, 2020, Virtual Event, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' ACM, New York, NY, USA, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1145/3407023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='3407026 ARES 2020, August 25–28, 2020, Virtual Event, Ireland © 2020 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Publication rights licensed to ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This is the author’s version of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It is posted here for your personal use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Not for redistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The definitive Version of Record was published in The 15th Inter- national Conference on Availability, Reliability and Security (ARES 2020), August 25–28, 2020, Virtual Event, Ireland, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1145/3407023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='3407026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 1 INTRODUCTION Medium-sized and large organizations, such as universities and companies, typically provide several information and communica- tions technology services to their members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Usually, a username as identifier is assigned to each member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' All services, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', email, file, and web collaboration, can then be used by the combination of username and some sort of credential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In the backend, username, salted and hashed password, and further information, called at- tributes, are stored centrally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Organizations either use lightweight directory access protocol (LDAP) or other database management systems for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Even though a local Identity & Access Manage- ment (I&AM) seems trivial, it gets challenging, if interfaces to other entities appear, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', with collaborations or joint ventures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' When members of an organization want to access external ser- vices, like collaboration platforms, either their user accounts are doubled or Federated Identity Management (FIM) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' FIM is an arrangement between multiple entities, in order to let users use the same identification data as in their home organization to obtain access to the services of the provided entities within the trust boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' FIM is mostly based on the protocol Security As- sertion Markup Language (SAML) [59], which is common in the R&E field as well as in national federations, or on OpenID Con- nect (OIDC) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' OIDC is the authentication layer on top of the authorization protocol OAuth [29], while SAML incorporates both aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' OAuth and OIDC are widely used in today’s web services, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', Google Application Interfaces (APIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Each user has at least some kind of home organization, called Identity Provider (IdP) in SAML or Relying Party (RP) in OIDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The user wants to access a service from a Service Provider (SP), respectively OpenID Provider (OP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The set of IdPs and SPs that collaborate for a specific reason is commonly referred to as federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Federations are especially cus- tomary in SAML [59], however an OIDC specification is currently developed [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' While federations in industry are relatively small, many national research and education networks (NRENs) operate large authentication and authorization infrastructures (AAIs) with hundreds of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Technical trust is established by pre- sharing metadata, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', information about the entity in extensible markup language (XML) files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Another form of trust is set into con- tracts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' between federation operator and participating entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Due to limitations of the rather static SAML federations, inter- federations, like eduGAIN [26], were established and dynamic con- cepts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' interfederation metadata exchange [84] and TrustBro- ker [57], were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As industry tends to use OIDC, organi- zations set up OIDC in parallel or even changed completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' At the ARES 2020, August 25–28, 2020, Virtual Event, Ireland Pöhn and Hommel same time eID federations with SAML were established [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' While SAML federations have trust established in out-of-bounds mecha- nisms, other mechanisms need to be set up with OIDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This led to the growth of different assurance frameworks and registries [39], while, at the same time, a Domain Name System (DNS) like struc- ture for discovery of the SP and verification of trust scheme mem- bership, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' LIGHTest [62], was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' I&AM as well as FIM focus on organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Since some users are at least concerned about their identity and privacy, user-centric solutions were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' A privacy policy of a service provider summarizes the usage of collected personal user data, which can now be gained by EU general data protection regulation (GDPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' At that time, most sites described user information in little under- standable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' User-centric identity management (UCIM) targeted these shortcomings by a clear presentation and intuitive evalua- tion of privacy policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' One first application was Microsoft Pass- port, an IdP in a Microsoft environment, which could be used as universal IdP in the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As Microsoft was integrated in ev- ery communication, trust and privacy were drawbacks [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As a next step, Windows Cardspace was developed, which could make use of other IdPs with SAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The user had several infocards, ei- ther created by himself or managed by an IdP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The application possessed several security bugs though [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' User Managed Access (UMA) [41], an OAuth-based standard, even goes one step further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It enables the user to control the authorization of data sharing and other protected resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Sovrin [75] sees Self-Sovereign Identi- ties as the next step of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The user must be the ultimate owner of a self-sovereign identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, a self-sovereign iden- tity is fully autonomous in management as well as administration, while the user is the single source of truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This paradigm is re- alized by decentralized domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' inherent this means the use of distributed ledger technologies (DLTs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', blockchain [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Hence, orthogonal approaches are currently developed and ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' SAML is rather static with solid trust boundaries, in contrast to OIDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' DNS-based structures facilitate stronger trust, based on certificates and eIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' While UMA can be applied to OIDC, self- sovereign identities (SSIs) can be used for different use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The bouquet of identity management approaches lead to different prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Administrators need to take care of different solutions in or- ganizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Due to this, getting an overview and applying similar policies gets cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Also, if companies join each other or establish projects, it might be the case, that their identity manage- ment solutions are incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This would require yet another system to manage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In addition, not all requirements from the or- ganizations might been fulfilled by the systems, leading to further tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Besides administrators, also end-users and other stakehold- ers have lots of solutions to decide from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The end user might end up using several different systems to manage his identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, it is important to have an overview, showing advantages and disad- vantages, as well as gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This overview can help the stakeholders to choose the best fitting approach and identify needed tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In order provide this overview, we extend known requirements for identity management in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The related work is divided into the established models [85] centralized identity management, FIM, and UCIM in Sections 3 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The existing approaches are anal- ysed based on the requirements and gaps are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The paper is concluded by an outlook and summary of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2 REQUIREMENTS OF IDENTITY MANAGEMENT In order to understand the usage of the different identity manage- ment approaches and discover possible gaps, various requirements are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' According to Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [76], the main require- ments are Usability, Interoperability, Functionality, Trustworthi- ness, Security, Mobility, Privacy, Law Enforcement, and Affordabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Boujezza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [16] refer to these requirements as bases for IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Ferdous and Poet leaf out Mobility [22], while Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [25] con- centrate on the core functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The authors of [27] see User control and consent, Minimal disclosure for a constrained use, Jus- tifiable parties, Directed Identities, Design for a pluralism of oper- ators and technology, Human integration, and Consistent experi- ence across contexts as requirements for SSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Regarding SSIs, the management of users has to be another re- quirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Besides management of identities, other functionali- ties might be needed for today’s identity management models, ap- proaches, and protocols to be interoperable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, the require- ment Functionality is divided into Management and Functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The location of the user data is renamed to Portability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' With the GDPR in Europe, the user can ask to get his account data exported and then imported to another platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, the user data needs to be portable from one system to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, the user should be able to make use of different devices, see require- ment Design for a pluralism of operators and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Afford- ability is split into Integration into the existing system and Scala- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As more and more devices and thereby identities are created, scalability becomes increasingly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In consequence, the following requirements are used in further sections to evaluate the different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' These requirements extend the requirements gathered from Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' They are already condensed and comprise different sub-requirements, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 3 CENTRALIZED IDENTITY MANAGEMENT As identity management advances in different directions, selected areas of related work are covered in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In order to struc- ture the work, we use different identity management models: centralized / network-centric, federated / application-centric, decentralized / user-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' These models have further characteristics, used in different use cases with different protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Even though centralized identity management was the first evolutionary step after single instances, it is still used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Mostly companies, which control all their own iden- tities, use single sign-on between services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' If they outsourced IdM, it is most likely FIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Also, companies with different locations may have one federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Within a centralized identity management, the company is in control of law-enforcement [REQ10] and there is no need for further interoperability [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' If cooperations are established, it looks differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1 Centralized Identity Management with DNS The DNS is the hierarchical and decentralized naming system for computers, services, and other resources connected to a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' An Overview of Limitations and Approaches in Identity Management ARES 2020, August 25–28, 2020, Virtual Event, Ireland Table 1: Requirements for Identity Management No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Name Description REQ1 Management of Identities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Automation REQ2 Usability User Interface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Reduced Complex- ity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Consistent Experience REQ3 Interoperability Interoperability between Proto- cols,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Silos REQ4 Scalability Mobility of Environment REQ5 Functionality Translation Services,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Proxies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Bridges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Group Management REQ6 Trustworthiness Trust Management,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Segregation of Power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Policies REQ7 Security Basic and Multi-Lateral REQ8 Portability of Accounts and Systems REQ9 Privacy Anonymity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Pseudonymity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Transparency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Controlability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Consent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Data Minimization REQ10 Liability Law-enforcement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Digital Evi- dence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Data Retention REQ11 Integration Affordability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Integration into Processes and Existing Infrastruc- ture Most commonly used,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' it translates memorable domains to numeric internet protocol (IP) addresses needed for locating and identifying computer services and devices with the underlying network proto- cols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In an identity federation, the home organization is discovered by specific protocols and partly also services, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', a trusted third party having a list of possible entities in form of metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Further- more, in order to make use of different protocols at one entity, at least bridges need to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' A way to improve the situation could be the usage of DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The EU project LIGHTest [63] [62] is based on DNS and DNS Secure Extensions (DNSSEC) as a method of discovery and trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, a trust scheme publication au- thority (TSPA) [81] needs to be set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approach can make use of role-based access control (RBAC) as a form of delegation, while it harmonizes the delegation data format [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As drawbacks, eI- DAS or some sort of certificates are needed, which also means fixed Level of Assurance (LoA), therefore [REQ6] is only partly fulfilled, and a centralized structure, similar to SAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Even though it can be used with internet of things (IoT) [38], [REQ3] is not fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='2 Centralized Identity Management with DNS for IoT Wang and Wen [83] present a privacy-enhanced DNS-scheme for IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' DNS was not designed with all the security complications of to- day’s worldin mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, the authors introduce new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Nevertheless, the approach does not take caching into account, which contradicts [REQ7] and [REQ9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, fresh domain names are needed after several steps [REQ1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approach of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [48] describes the auto-configuration of IoT devices with the device’s category and model in IPv6 environment, as the neighbor discovery is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It is, in contrast, inefficient due to manual con- figuration of the DNS names for IoT devices [REQ1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Though the discovery via DNS is widely known as well as possible attacks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' no ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Table 2: Overview of Approaches for Centralized Identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Ref ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Requirements not fulfilled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Centralized IdM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[63] [62] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='LIGHTest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] [REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[81] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='TSPA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] [REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[80] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Delegation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] [REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[38] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='LIGHTest for IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] [REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[83] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Privacy-enhanced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='DNS-scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ1] [REQ7] [REQ9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[48] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Autoconfiguration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='of IoT devices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[69] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='ACE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[87] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Natural context ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[51] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='PhD thesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ1] [REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[33] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Cloud-based IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='authorization is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The size of messages can get a problem as well as the scalability [REQ4], if too many features are added to DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, changes are difficult [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' According to the presented approaches, identity management with DNS can be used in use cases, where no anonymity is needed, but where eIDAS, fixed trust values, and a centralized structure for certificates is in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='3 Centralized Identity Management for IoT Not only humans need to be identified, but also things, like comput- ers and IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' IoT devices often have limitations, like band- width or central processing unit (CPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, different proto- cols are used, as shown in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Belran and Skarmeta [9] give an overview of the IETF protocol Authentication and Authorization for Constrained Environments (ACE) [69], which is extended by several Internet-Drafts (I-Ds) [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The authors of [87] identify and authenticate IoT objects in a natural context using DNS with time property, name, and location resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Efficiency degrada- tion is avoided, but the life cycle is not taken into account [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Mahalle’s thesis [51] is about identity management for IoT de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Even though the context of the identifier is specified, users and their relationship to objects are not mentioned [REQ1] [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [33] focuses on cloud-based IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' While it takes the life cycle of IoT things into account, it does not consider the owner [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='4 Conclusion from Centralized Identity Management The results are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Many approaches do not fulfill the requirement Interoperability [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This is comprehen- sible, as they are solely centralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, centralized IdM approaches seem to lack Trustworthiness [REQ6], Management [REQ1], and Integration [REQ11] in several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Especially Trust- worthiness and Integration come inherent with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Porta- bility [REQ8] is difficult to estimate, as the approaches are mostly described for computer use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' ARES 2020, August 25–28, 2020, Virtual Event, Ireland Pöhn and Hommel 4 FEDERATED IDENTITY MANAGEMENT Federated identity management is an arrangement between mul- tiple entities within certain trust boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It lets users use the same identification as in their home organization to obtain access to the services of the provided entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In many constellations, a trusted third party is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Especially SAML federations tend to have scalability problems [REQ4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Between different FIM ap- proaches, interoperability is a problem [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Liability [REQ10] depends on all entities, while the trusted third party could provide additional functionality, though it is not the case with the follow- ing approaches [REQ5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Bargh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [8] show that the growing number of IdPs and SPs is a well-known problem of SAML in aca- demic world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Different approaches, therefore, try to improve the scalability of trust and metadata exchange as well as trust negotia- tion in federated identity management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' IdMRep by Arias Cabarcos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [6] introduces dynamic trust values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The presented approach does not work with new members [REQ6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, the accu- mulated amount of data at the trust engine can become to a sin- gle point of failure in large-scale infrastructures [REQ4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The ap- proach Dynamic Identity Federations by Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Sadek Ferdous and Ron Poet [20] focuses on fully automated establishment of feder- ations and trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Though the technical establishment is fully auto- mated, the user has to enter the EntityID of the IdP and needs to generate a code, which is not user-friendly [REQ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Furthermore, the division into trusted,semi-trusted, and untrustedis rather coarse- grained [REQ6],and a further database per entity is needed [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Another approach is called TrustBroker by Pöhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [58], which is based on SAML, though the generic concepts can be applied to FIM [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' There are also other federation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' OLYM- PUS by Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [53] is privacy preserving, while it allows dis- tributed IdPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Nevertheless, it misses [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Kang and Khashno- bish [40] approach for a peer-to-peer (P2P) federation is decentral- ized, but has scalability issues in large-scale infrastructures with many entities and federation servers [REQ4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Dhungana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [19] provide an IDM framework for cloud networking based on UMA with OIDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It thereby allows federated identity management with- out the standard of OIDC federations [30] [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' UnlimitID by Isaakidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [36] utilizes attribute-based anonymous credentials, which are based on algebraic Message Authentication Codes (MACs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approach is usable with OAuth and let pseudonyms expire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Though for this short time users might be linkable [REQ9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Fur- thermore, in some use cases pseudonyms might not be allowed [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The usage of MACs seems like a step back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' These ap- proaches show that certain aspects are regarded, which can be solved by federated identity management, but not all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, especially FIM with SAML is difficult to port to mobile applications and IoT devices [REQ8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1 Federated Identity Management with DNS ID4me [71] combines DNS with OIDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It utilizes the protocol Au- tomated Certificate Management Environment (ACME), which au- tomatically verifies the ownership of a domain and also simplifies the issuance of certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The DNS challenge is used as discov- ery method, secured with DNSSEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The communication is secured by DNS-based Authentication of Named Entities (DANE), which is only rarely used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' ID4me has the roles identity agent, which administrates the attributes and is identified by DNS, and identity authority, which is used for authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As a conse- quence, the user has to trust the identity agent [REQ6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The idea to use DNS is not new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Hulsebosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [35] present the combina- tion for effective and trusted user controlled light-path establish- ment in grid environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Thereby, DNS tree is used to establish technical trust between IdP and Optical Network Service Provider (ONSP) dynamically within a virtual organization (VO), a kind of federation [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='2 Federated Identity Management with eID The EU Regulation 910/2014, also known as eIDAS, is effective since September 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The authors of [37] show the federated iden- tity architecture with eIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approach of Berbecaru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [13] describes the connect between eIDAS and STORK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' STORK [31] [60] is the eID federation based on eID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It features the European electronic identity interoperability platform, which relies on SAML with one node per country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Thereby, it is heterogeneous and a large-scale infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Several approaches describe the integra- tion of academic attributes in the eIDAS infrastructure without re- garding the inter-federation eduGAIN, which is the de-facto stan- dard federation in the academic world [14] [15] [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [4] proposed an identity model to connect FIWARE services authenticated with OAuth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='0 to eIDAS nodes and thereby meet the regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It nevertheless is the solution for one specific use case [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='3 Federated Identity Management as a Service Other approaches focus on federation as a service, which utilizes cloudinfrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The framework AIDF by Zouari and Hamdi [88] makes use of a trust broker, like in [58], in cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' IdP and SP reg- ister at a central party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The SP can tell, which IDs and attributes he wants and he can at the same time subscribe to claim transfor- mation and attribute mapping service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Attribute mapping is nec- essary, if different semantic or/and syntax of attributes is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' A simple example is date format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The central service automati- cally establishes a link between IdP and SP after consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Some sort of trust assurance is established by National Institute of Stan- dards and Technology (NIST) LoA, which is currently published in the third revision of the special publication 800-63 from 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As the LoA is revised, the framework needs updates as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Fur- thermore, not only the NIST LoA could be applied, but also other standards [REQ6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, there exists no trust assurance for IdPs, solely for SPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The same drawback with NIST LoA has the ap- proach of Zefferer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [86], which links identities to eIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Even though the federations have a strong legal foundation as eIDs are used, it is highly customized for this one use case [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='4 Conclusion from Federated Identity Management A summary of FIM approaches is found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' FIM itself lacks Interoperability [REQ3], Functionality [REQ5], and Liability [REQ10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Liability often comes with policies and other agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Interop- erability can be archived with additional tools, but the protocols as they are act like silos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Looking at the protocol SAML, Scalability [REQ4] and Portability [REQ8] are also missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approaches An Overview of Limitations and Approaches in Identity Management ARES 2020,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' August 25–28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2020,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Virtual Event,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Ireland ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Table 3: Overviewof Approaches for FederatedIdentity Man- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='agement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Ref ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Requirements not fulfilled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='FIM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] [REQ5] [REQ10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='IdMRep ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ4] [REQ6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Dynamic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Federations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ2] [REQ6] [REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[58] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='TrustBroker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[53] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='OLYMPUS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[40] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='P2P federation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[19] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='IDM framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[36] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='UnlimitID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ9] [REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[71] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='ID4me ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[35] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='VO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[14] [15] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Academic eIDAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='FIWARE eIDAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='AIDF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[86] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='eID federations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] [REQ6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='described in this paper often have drawbacks in Interoperability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] and Trustworthiness [REQ6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Also, Integration [REQ11] is often stated, while it is partly difficult to estimate Liability [REQ10] as it depends on the implementation and established processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As the users access services from other organizations, trust needs to be established between home organization and service provider, user and service provider, and user and home organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' If ad- ditional entities are involved, they need to be involved in the trust establishment as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 5 USER-CENTRIC IDENTITY MANAGEMENT User-Centric Identity Management puts the user in control of their own identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It has evolved from simple clients on a PC to UMA and SSI, which are described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Although it gives more power to the user and thereby adds value to [REQ9], the in- teroperability with other approaches is low [REQ3], as [REQ5] is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [REQ10] depends on the client and the provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1 User-Centric Identity Management for IoT Do van Thuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [78] present a user centric identity manage- ment for IoT with four identifiers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', idpID, domIDPart, devID- Part, and userIDPart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' They describe different relationships as well, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=', serving people in general, serving one person permanently, serving multiple persons successively, and serving multiple per- sons simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Though the relationship between device and people is characterized in all variants and the approach has sev- eral identifier, the life cycle itself is not regarded [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It is fur- thermore unknown how the approach should interact with other approaches and protocols [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='2 User-Centric Identity Management with UMA The works published by Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [47] and Slomovic [70] make it clear that pseudonymity and anonymity can be important in cer- tain use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Identities from different social networks can be ag- gregated and lead to one person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' UMA manages to decouple iden- tity resolution from the maintenance of identity information [44] [50] [52] and even one application for IoT by Cruz-Piris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [17] is published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Although UMA is based on OAuth, the principle can be applied to SAML as well [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Nevertheless, the principle of UMA might not be possible to apply to all use cases [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='3 User-Centric Identity Management with SSI The principles of SSI management are described by Toth and Anderson- Priddy [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Ferdous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [21] broaden the laws of identities from existence, autonomy, ownership, access, single source, protection, availability, and persistence to user control and consent, minimal disclosure for a constrained use, justifiable parties, directed iden- tity, pluralism of operators and technologies, human integration, and consistent experience across contexts for self-sovereign iden- tities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This step is seen as an evolution of identity management by Sovrin [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' SSIs are typically implemented by blockchain, which have de- centralized identifiers (DIDs), proposed by the W3C, as a new type of identifier for verifiable, decentralized digital identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Claim Chain by Stokkink and Pouwelse [73] is a pure self-sovereign build- ing on claim model, which is practically used, but also rather re- strictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' No data is sent without consent, but also at the same time only attested attributes are possible [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' DecentID by Friebe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [24] and Steichen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [72] is based on Ethereum smart con- tracts and tries to preserve the privacy of the attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The ap- proach utilizes external storage like distributed hash tables (DHTs) for attributes for scalability reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The nodes do not mine ef- ficiently with large files [REQ4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, a linkage between identities to centralized registration contracts is needed [REQ9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Sora Identity [74] uses the mobile phone of the user as one of the main actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Though this approach preserves the privacy, the life cycle of a device and the usage of different devices can get problem- atic [REQ8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Another blockchain-based approach combines it with PKI [82] [7] [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approaches support the revocation of cer- tificates, but have scalability problems [REQ4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The Horcux Proto- col [55][56] is a decentralized biometric credential storage using SSI with Biometric Open Protocol Standard (BOPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This is done al- though biometrics tend to have false-positives and false-negatives and their implementations are often broken [REQ7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In [65], Schanzen- bach and Banse describe an approach, which is adapted from UMA applying NameID [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As they use a secure name system - GNU Name System (GNS) - to store data, they do not require a global, publicledger and do not compromise the privacy of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Ahadipour and Schanzenbach [1] present a survey on authorization in dis- tributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In [66], the authors describe the decentralized attribute-based delegation (ABD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' reclaimID by Schanzenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [67] is based on the work published beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Attributes are encrypted by attribute-based encryption (ABE), which means the attributes are self-signed with the private key associated with the identity of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approach allows the use of one or more ARES 2020, August 25–28, 2020, Virtual Event, Ireland Pöhn and Hommel identities with different attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Nevertheless, privacy might be one drawback as well as trust into the attributes [REQ6] [REQ9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='4 User-Centric Identity Management with SSI for Cloud Environments The identity wallet platform CREDENTIAL [79] [45] [42] [32] aims at the development of a secure and privacy-preserving data shar- ing platform that puts the user under full control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [32] shows how proxy-re-encryption and redactable signature technology can be integrated into their workflows, which is basically a man in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The platform has three main actors: wallet platform, user, and the data receiver, which can be either user or SP, as described in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' A metadata preserving high-level architecture is proposed, which might be practically challenging due to log files leaking in- formation [REQ7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Kostopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [45] describe the architec- ture of CREDENTIAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Interfaces to existing protocols are missing [REQ3] [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Finally, Veseli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [79] focus on privacy by de- sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The LINDDUN (Linkability, Identifiability, Non-repudiation, Detectability, information Disclosure, content Unawareness, and policy and consent Non-compliance) method [18], a privacy threat analysis framework, was used to establish privacy by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Thereby, the data is encrypted, though the approach is cloud-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The user has to trust the wallet provider [REQ3] [REQ6] [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='5 User-Centric Identity Management with SSI as Federation NEXTLEAP by Halpin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [28] is an approach for FIM with blind signatures based on MACs to improve OIDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The identity itself is a privacy-enhanced based blockchain identity with a decentralized PKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, access control is performed in a decentralized man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Though the purpose is secure messaging, the scalability can be a drawback in large-scale infrastructures [REQ4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [49] is designed with having a cross-organizational authentication system in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' One-way hashed enforce the concept of one-time pad passwords, which can provide unlinkability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Unfortunately, the performance is not clearly described [REQ4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='6 User-Centric Identity Management with SSI as Cloud Federation The goal of the EU project SUNFISH was the establishment of a cloud federation, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Alansari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [2] [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The cloud federation is a collaboration of organizations sharing data hosted on their pri- vate cloud infrastructures for business opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' While the ap- proach introduces distributed access in a privacy-preserving way, the user must be identified and authenticated beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' eIDAS can be used as a source of trust, though the trust within a cloud fed- eration is not clearly described [REQ6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Although the anonymity for users by the usage of tokens instead of attributessounds promis- ing, tokens might not in every setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, the approach is solely based on Intel Software Guard Extensions (SGX) [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approach of dynamic cloudfederation by Bendiab evolves from a dynamic cloudfederation to blockchain with fuzzy cognitive maps dynamic trust model [43] [11] [10] [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In [43], the authors de- scribe the basic approach with the entities user, cloudservice provider (CSP), IdP, federation provider (FP), and certificate authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The token is sent from one FP to another FP for validation and trans- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The CSP verifies the token’s legitimacy, before the user can access the cloud service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This could also be done by a simple proxy between two protocols, like SAML and OIDC in eduGAIN [REQ4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Trust is described in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Each user gets his identity from a trusted IdP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' If the CSP does not trust this IdP, the trust can be computed for each other in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Based on the result, a con- nection is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The authors describe a complex trust evalu- ation, which is derived from different characteristics, like security, and privacy, though these factors are difficult to measure and ac- cess [REQ6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The trust is then put into blockchain [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The CSP can manage the trust relationships in a dynamic and distributed manner without the need for centralized authorities, like IdP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Nev- ertheless, a Trust Management Platform (TMP) is needed and the CSPs are responsible for authenticating their registered users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [12] is the next step with fuzzy cognitive maps for modeling and eval- uating trust relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It uses a trust computation model, sim- ilar to the characteristics introduced beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' It can calculate the trust of unknown entities, if further information are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Unfortunately, LoA methods like Vectors of Trust (VoT) [61] are not taken into account [REQ6] [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The approach eases the creation of secure Infrastructure as a Service (IaaS) cloud federa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Though the approach makes use of anonymity approaches, it still stores personal information in an external database [REQ9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, the approach concentrates on one use case [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='7 Conclusion from User-Centric Identity Management A summary of user-centric identity management approaches is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' UCIM has drawbacks in Interoperability [REQ3], Functionality [REQ5], and Liability [REQ10], though there are dif- ferences between older approaches, UMA, and SSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Within SSI, it partly depends on use cases, centralized / decentralized manage- ment, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The presented approaches for UMA miss In- teroperability [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The principle of UMA should work for many use cases, although this is not further explored to the knowledge of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' UCIM for IoT has issues with Integration [REQ11] and presumably with Interoperability [REQ3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Looking at the pre- sented SSI approaches, Scalability [REQ4] and Privacy [REQ9] are often stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Many approaches are tested within smaller environ- ments, leading to problems with scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Privacy seems con- tradictory as the approaches focus on users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Not all approaches concentrate on privacy and others seem to have issues as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Focusing on cloud and federation use cases, Scalability [REQ4], Trustworthiness [REQ6], and Integration [REQ11] are often miss- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Trustworthiness is difficult, as the user has more control over his data while the service provider needs to trust the user’s at- tributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 6 CONCLUSION AND FUTURE WORK Nowadays, security is one key requirement in a network environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Identities are everywhere and with the growing identity theft, secure identity management gets more relevant than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In this paper, we extend known requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Based on new approaches as well as the heterogeneous environments and possible use cases, new requirements were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' At the same time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' others were newly An Overview of Limitations and Approaches in Identity Management ARES 2020,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' August 25–28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2020,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Virtual Event,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Ireland ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Table 4: Overview of Approaches for User-Centric Identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Ref ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Requirements not fulfilled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='UCIM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] [REQ5] [REQ10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[78] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='UCIM for IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] [REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[44] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='UMA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[17] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='UMA for IoT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[73] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Claim Chain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[24] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='DecentID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ4] [REQ9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[72] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='SSI for IPFS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ4] [REQ9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[74] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Sora Identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[82] [7] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Blockchain-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='PKI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] [REQ4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[55] [56] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Horcux Protocol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ7] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[67] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='reclaimID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] [REQ9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[79] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='CREDENTIAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ7] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[28] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='NEXTLEAP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[49] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Blockchain-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='AAA system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[2] [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='SUNFISH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] [REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[43] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='Dynamic cloud fed- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='eration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='[REQ9] [REQ11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Management [REQ1] of identities is a new requirement, while Functionality [REQ5] is still needed for interoperability be- tween all these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The GDPR leads to the change from Mobility to Portability [REQ8], as users can request to port their account and more and more different devices are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This also leads into the division Scalability [REQ4] and Integration [REQ11], which was originally Affordability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As a next step, we give a broad overview of different identity management approaches structured in identity management mod- els and use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' This ranges from centralized identity manage- ment for specific use cases, over federated identity management, to user-centric approaches, like user-managed access and self-sovereign identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' We compare these approaches with the derived require- ments, which gives us an overview of advantages as well as draw- backs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Centralized approaches lack Interoperability [REQ3], Trustwor- thiness [REQ6], Management [REQ1], and Integration [REQ11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Porta- bility [REQ8] cannot be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The shortcomings are inher- ent of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Federated Identity Management has general is- sues with Interoperability [REQ3], Functionality [REQ5], and Li- ability [REQ10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' SAML misses Scalability [REQ4] and Portability [REQ8] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Trustworthiness [REQ6] and Integration [REQ11] are additional shortcomings of the described FIM approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' User- Centric Identity Management has drawbacks in Interoperability [REQ3], Functionality [REQ5], and Liability [REQ10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Also Trust- worthiness [REQ6] is an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Depending on the approach, Scal- ability [REQ4] and Integration [REQ11] are unavailable as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' No approach has managed to meet all requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Two require- ments were out of scope of the described approaches: Liability [REQ10] and Functionality [REQ5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Liability is mainly endorsed by the home organization, while additional approaches try to ful- fill Functionality [REQ5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Interoperability [REQ3] and Trustwor- thiness [REQ6] might be met within the approach or model but not crosswise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Thereby, Integration [REQ11] is also difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Sim- ply combining different approaches does not satisfy the stated re- quirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Running several approaches in parallel binds more manpower and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, it can lead to different policies and further inconsistent database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Therefore, at least an overview and connectors between the approaches are needed to interact be- tween the approaches and provide an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In order to meet all requirements, further work is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' As a result, stakeholders would profit from a framework includ- ing different models and, thereby, approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The framework needs to be flexible enough for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In order to meet the miss- ing requirements, it needs to comprise of additional tools and rec- ommended processes, including connectors between different ap- proaches and an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' For future work, we plan to develop missing components in the design of a holistic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' The framework will be adapted for different use cases in a further step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Additionally, a categorization for identity management models will be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' REFERENCES [1] Ava Ahadipour and Martin Schanzenbach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' A Survey on Au- thorization in Distributed Systems: Information Storage, Data Retrieval and Trust Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In 16th 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Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' RFC 6749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' RFC Editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='rfc-editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='org/rfc/rfc6749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='txt http://www.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1109/ICITCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='7021724 [79] Fatbardh Veseli, Jetzabel Serna Olvera, Tobias Pulls, and Kai Rannenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Engineering Privacy by Design: Lessons from the Design and Implementation of an Identity Wallet Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In Proceedings of the 34th ACM/SIGAPP Symposium on Applied Computing (Limassol, Cyprus) (SAC ’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' ACM, New York, NY, USA, 1475–1483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1145/3297280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='3297429 [80] Georg Wagner, Olamide Omolola, and Stefan More.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Harmonizing Delega- tion Data Formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In Open Identity Summit 2017, Lothar Fritsch, Heiko Roßnagel, and Detlef Hühnlein (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Gesellschaft für Informatik, Bonn, 25–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [81] 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' [82] Alexander Yakubov, Wazen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Shbair, Anders Wallbom, David Sanda, and Radu State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' A blockchain-based PKI management framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In NOMS 2018 - 2018 IEEE/IFIP Network Operations and Management Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1109/NOMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='8406325 [83] Yanjiong Wang and Qiaoyan Wen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2011.' metadata={'source': 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+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' A survey of Identity Management technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In 2010 IEEE International Conference on Information Theory and Information Secu- rity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 287–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1109/ICITIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='5689468 [86] Thomas Zefferer, Dominik Ziegler, and Andreas Reiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' Best of Two Worlds: Secure Cloud Federations meet eIDAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' In 2017 12th International Con- ference for Internet Technology and Secured Transactions (ICITST).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='1109/ISNCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} +page_content='7746120' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AyT4oBgHgl3EQfk_gh/content/2301.00442v1.pdf'} diff --git a/kdFRT4oBgHgl3EQfYDcO/vector_store/index.faiss b/kdFRT4oBgHgl3EQfYDcO/vector_store/index.faiss new file mode 100644 index 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Technological University, Singapore +{hongyu.zhou,xin.zhou,ZQShen}@ntu.edu.sg +Abstract. User interaction data in recommender systems is a form of +dyadic relation that reflects the preferences of users with items. Learn- +ing the representations of these two discrete sets of objects, users and +items, is critical for recommendation. Recent multimodal recommenda- +tion models leveraging multimodal features (e.g., images and text de- +scriptions) have been demonstrated to be effective in improving rec- +ommendation accuracy. However, state-of-the-art models enhance the +dyadic relations between users and items by considering either user-user +or item-item relations, leaving the high-order relations of the other side +(i.e., users or items) unexplored. Furthermore, we experimentally reveal +that the current multimodality fusion methods in the state-of-the-art +models may degrade their recommendation performance. That is, with- +out tainting the model architectures, these models can achieve even bet- +ter recommendation accuracy with uni-modal information. On top of +the finding, we propose a model that enhances the dyadic relations by +learning Dual RepresentAtions of both users and items via construct- +ing homogeneous Graphs for multimOdal recommeNdation. We name +our model as DRAGON. Specifically, DRAGON constructs the user-user graph +based on the commonly interacted items and the item-item graph from +item multimodal features. It then utilizes graph learning on both the +user-item heterogeneous graph and the homogeneous graphs (user-user +and item-item) to obtain the dual representations of users and items. +To capture information from each modality, DRAGON employs a simple +yet effective fusion method, attentive concatenation, to derive the rep- +resentations of users and items. Extensive experiments on three pub- +lic datasets and seven baselines show that DRAGON can outperform the +strongest baseline by 22.03% on average. Various ablation studies are +conducted on DRAGON to validate its effectiveness in modality fusing and +learning dual representations for recommendation. +Keywords: Multimodal Recommendation · Dyadic Relation · Dual Rep- +resentation Learning · Graph Neural Network · Multimodal Fusion. +1 +Introduction +As society evolves, recommender systems have become indispensable tools to +assist users to find products and services of their choice. Previous work [1,16,7] +arXiv:2301.12097v1 [cs.IR] 28 Jan 2023 + +2 +Hongyu Zhou, Xin Zhou, and Zhiqi Shen +explores historical user-item interactions that can be considered as a form of +dyadic relation to capturing user preferences. However, these methods show in- +ferior performance due to the sparse nature of interactions between users and +items in real-world datasets. +To alleviate the data sparsity problem, recent multimodal recommender sys- +tems utilizing multimodal information (e.g., item descriptive texts, product im- +ages) to improve recommendation performance have gained considerable atten- +tion. A line of work [11,6] integrates multimodal features as side information +to enhance latent item representations under the classic collaborative filtering +framework. Inspired by the success of graph neural networks (GNNs) on rec- +ommendation [19,7], recent works focus on modeling user-item interactions as a +bipartite graph and integrating multimodal information with graph structure. +For example, MMGCN [23] builds a user-item bipartite graph for every modal- +ity to obtain modal-specific representation to understand user preference better. +GRCN [22] presents a graph refine layer that could identify noisy edges and +corrupt false-positive edges to clarify the structure of the user-item interaction +graph. DualGNN [18] and LATTICE [24] introduce either user-user or item-item +relations into the user-item interactions and achieve state-of-the-art recommen- +dation performance. Although these models show effective recommendation ac- +curacy, we argue the high-order relations in both sides of the dyadic relations +can be explored simultaneously to fully address the data sparsity issue. Inspired +by the dual representation learning mechanism [25], we enhance the representa- +tion learning of users and items by incorporating their dual representations to +capture both the inter- and intra-relations between users and items. +Furthermore, we experimentally reveal that these methods fail to fuse the +modality features effectively. Specifically, we conduct an ablation study of mul- +timodal features on two competitive multimodal models, DualGNN [18] and +LATTICE [24]. The results shown in Table 1 demonstrate that the performance +of these models fed with a single modality, especially textual features, outper- +forms that with both modalities. This finding poses a meaningful question: How +can we effectively fuse the multimodal information for recommendation? +To address the above question, we study the performance of different modal- +ity fusion methods, including Attentively Sum, Max-pooling, Mean-pooling, +and Concatenation. Our experiments show that the late-fusion approach Con- +catenation which directly concatenates the textual and visual features as the +multimodal representation achieves the best performance. +Toward this end, we propose a framework that learns Dual RepresentAtions +of both users and items via constructing homogeneous Graphs for multimOdal +recommeNdation (DRAGON). To be specific, DRAGON constructs a heterogeneous +user-item bipartite graph for each modality to learn the modality-specific rep- +resentations. It then utilizes the direct Concatenation fusion method to better +exploit the learned modality-specific information. In order to learn the dual +representations, we construct two homogeneous graphs based on the user co- +occurrence and the item semantic features to capture the user preference from +the neighbor users and the latent item content semantic from the neighbor items. + +DRAGON +3 +Table 1. Performance of DualGNN [18] and LATTICE [24] utilizing features in differ- +ent modalities. R and N denote evaluation metrics Recall and NDCG. T and V denote +textual and visual information. +Dataset Metric +DualGNN +LATTICE +V&T +T +V +V&T +T +V +Baby +R@10 0.0448 0.0612 0.0511 0.0547 0.0546 0.0492 +R@20 0.0716 0.0943 0.0830 0.0850 0.0874 0.0781 +N@10 0.0240 0.0331 0.0278 0.0292 0.0287 0.0265 +N@20 0.0309 0.0417 0.0360 0.0370 0.0371 0.0339 +Sports +R@10 0.0568 0.0697 0.0615 0.0620 0.0625 0.0572 +R@20 0.0859 0.1060 0.0926 0.0953 0.0971 0.0887 +N@10 0.0310 0.0379 0.0335 0.0335 0.0336 0.0312 +N@20 0.0385 0.0473 0.0415 0.0421 0.0425 0.0393 +Clothing +R@10 0.0454 0.0524 0.0420 0.0492 0.0521 0.0408 +R@20 0.0683 0.0798 0.0636 0.0733 0.0749 0.0614 +N@10 0.0241 0.0281 0.0229 0.0268 0.0290 0.0221 +N@20 0.0299 0.0351 0.0283 0.0330 0.0348 0.0273 +Finally, DRAGON utilizes the learned dual representations of users and items to +make recommendations. Extensive experiments are conducted on three public +datasets to show the effectiveness of our proposed method. +2 +Related Work +2.1 +Multimodal Recommendation +Collaborative filtering (CF) based models are widely used [8,19] for recommender +systems. The CF-based methods leverage the historical interactions between +users and items to predict users’ preferences. However, they usually suffer from +the data sparsity problem since user-item interactions are generally limited for +real-world datasets. +To alleviate the data sparsity problem, massive multimodal content informa- +tion has been utilized to improve recommendation performance. For example, +VBPR [6] leverages the visual features from the pre-trained Convolutional Neu- +ral Network (CNN) to enhance the matrix factorization by incorporating the +visual features with ID embeddings. VECF [4] models the user’s attention on +different regions of images and reviews. Recently, GNNs have drawn more at- +tention when applied to multimodal-based recommender systems. MMGCN [23] +improves the quality of learned user and item representations by constructing +a modality-specific user-item bipartite graph and adapting the message-passing +mechanism of GNNs. Following MMGCN, GRCN [22] presents a graph refine +layer that can locate noisy edges and corrupt false-positive edges to refine the +structure of the user-item interaction graph. DualGNN[25] adds an attention +mechanism to capture the user’s preference on different modalities. Meanwhile, + +4 +Hongyu Zhou, Xin Zhou, and Zhiqi Shen +it constructs a user-user graph to learn the user preference from neighbor users. +LATTICE [24] builds an item-item graph for each modality, then combines them +to create a latent modality-fused item graph. With graph convolution operations, +items can share information from highly linked affinities in the graph to enhance +their representations. SLMRec [17] incorporates self-supervised learning tasks +into the GNNs to uncover the latent patterns from multi-modalities to learn +powerful representations. +2.2 +Multimodal Fusion +Finding a fused multi-modal representation that is complimentary and compre- +hensible can greatly improve performance. Technically speaking, multi-modal +fusion integrates information from different modalities to create a multimodal +representation that can be applied to different tasks, such as link prediction [12] +and node classification [26], etc. It can be categorized into early fusion, late +fusion and hybrid fusion [2]. Early fusion incorporates the extracted features +at the beginning. Contrarily, the late fusion integrates after each modality has +completed the decision (e.g., classification or regression). Hybrid fusion combines +the two methods. For example, ACNet [9] adopts an early fusion method based +on the attention mechanism. For the late fusion method, NMCL [21] utilizes +the cooperative networks for each modality to do feature augmentation with the +attention mechanism, then a late fusion is applied to the prediction from vari- +ous modalities. CELFT [20] designs a hybrid fusion method that combines both +early and late fusion together to overcome the shortcomings of the single fusion +methods. +Multimodal recommendation models [23,24,25] normally apply mean-pooling, +attentive sum or max-pooling. Our experimental results suggest that those mod- +els use a single modality representation leads to better performance than utilizing +the multimodal representation learned from these fusion methods. It indicates +that pooling or attention-based methods may result in information loss when +performing multimodal fusion. To mitigate this problem, we adopt attentive +concatenation fusion but without reducing the embedding dimension, which has +been demonstrated to be more effective in joining information from different +modalities for recommendation. +3 +Methodology +This section introduces the details of our proposed model DRAGON. Fig. 1 shows +the overall architecture of DRAGON. There are four main components: (1) graph +learning on a modality-specific heterogeneous graph to learn a uni-modal rep- +resentation; (2) Multimodal representation learning module that captures user +preference on different modalities and complementary information from each +modality; (3) graph learning on homogeneous graphs to capture the co-occurrence +relation between users and the semantic relation between items; (4) a predictor +module that ranks the candidate items based on the scores calculated from final +user and item representations. + +DRAGON +5 +Heterogeneous graph +Homogeneous graph +Multimodal representation +𝒖𝑣 +𝒖𝑡 +𝒊𝑣 +𝒊𝑡 +𝒖𝑓 +𝒊𝑓 +𝑦𝑢𝑖 +𝒛𝑢 +𝒛𝑖 +Visual +Textual +Modality-specific user +item bipartite graph +neighbors of u +Layer 1 +Layer 3 +Layer 2 +Normalized sum +neighbors of i +Layer 1 +Layer 2 +Normalized sum +𝒊𝟏 +𝒊𝟐 +𝒊𝟑 +𝒖𝟐 +𝒖𝟏 +𝒖𝑡 +𝒊𝑡 +𝒖𝑓 +𝒊𝑓 +Layer 1 +Layer 2 +Layer 1 +Layer 2 +𝒉𝑢 +𝒉𝑖 +Item semantic graph +User co-occurrence graph +u +i +Fig. 1. The diagram depicts the main components of DRAGON. We first learn the +modality-specific representations, then the multimodal representation learning mod- +ule fused the single-modality representations and utilizes the homogeneous graph to +capture the internal relations +3.1 +Preliminary +Given a set of N users u ∈ U, a set of M items i ∈ I. We model the dyadic +relations of user interactions as a user-item bipartite graph G = {U, I, E}, where +we regard the historical interactions as the set of edges in the graph denoted by +E. Besides the user-item interactions, each item is associated with multimodal +content information m ∈ {v, t}, where v and t represent the visual and textual +features respectively. We denote the modality feature for an item i as xm +i ∈ Rdm, +where dm denotes the feature dimension of modality m. In this paper, we only +consider the visual and textual modalities denoted by v and t. However, the +proposed framework can be easily extended to scenarios involving more than +two modalities. +3.2 +Dual Representation Learning +Learning the representations of users and items is critical for the recommendation +system. All representation learning-based techniques assume the existence of a +common representation with consistent knowledge of different views of items [25]. +Different views of an item contain specific discriminant information in addition +to consistent knowledge about this item. We construct the heterogeneous and +homogeneous graphs together to learn the dual representations of both user and +item, which could capture both the internal association and the relationship +between users and items. + +6 +Hongyu Zhou, Xin Zhou, and Zhiqi Shen +Heterogeneous Graph To learn modality-specific user and item representa- +tions, we construct a user-item graph for each modality, which is denoted as +Gm. Following MMGCN [23], we maintain the same graph structure G for dif- +ferent Gm, but only keep the node features associated with a specific modality +m. We adopt LightGCN [7] to encode Gm. As shown in [7], LightGCN simpli- +fies the graph convolutional operations by excluding the feature transformation +and nonlinear activation modules to improve recommendation performance and +meanwhile ease the model optimization process. Specifically, the user and item +representations at the (l + 1)-th graph convolution layer of Gm are derived as +follows: +u(l+1) +m += +� +i∈Nu +1 +� +|Nu| +� +|Ni| +i(l) +m , +i(l+1) +m += +� +u∈Ni +1 +� +|Nu| +� +|Ni| +u(l) +m , +(1) +Where Nu and Ni are the set of first hop neighbors of u and i in Gm. u(0) +m +is randomly initialized and i(0) +m is initialized with xm +i . The symmetric normal- +ization +1 +√ +|Nu|√ +|Ni| is used to normalize the modality features learned from each +layer which avoids the increase of scale when operates the graph convolutional +operations. +After L layers of data propagation, we combine the representations from +every GCN layer using element-wise summation to derive the modality-specific +representations for users and items. Formally, +um = +L +� +l=0 +u(l) +m , +im = +L +� +l=0 +i(l) +m . +(2) +In such cases, the historical interactions and the modality information have +been encoded into the final single-modal representations of users and items. +And those operations have been applied to each modality by propagating on the +modality-specific user-item bipartite graphs to learn the representations for each +modality. +Homogeneous Graph In addition to employing the heterogeneous graph, +which encodes the dyadic relation between users and items, we argue that the +recommendation performance can be further enhanced by modeling the inter- +nal relations between users or items. For the two homogeneous graphs, we pre- +established and freeze them to maintain the initial co-occurrence relation and +semantic meaning. +User Co-occurrence Graph. Based on the assumption that users who have +interacted with similar items usually have similar preferences. We argue that +the user’s preference pattern is hidden inside the co-occurrence items and we +construct a homogeneous user co-occurrence graph to learn the internal relations +between users. However, the numbers of co-occurrence items between users are +in a large range. In a general situation, the user will have a high number of +commonly interacted items with a small group of users but with few items with + +DRAGON +7 +other users. We only consider those who have more commonly interacted items +with the user to capture similar preferences. To explicitly model the item co- +occurrence patterns of users, we construct a homogeneous user co-occurrence +graph ˜G = {U, Pu}, where Pu = {eu,u′|u, u′ ∈ U} denotes the edges between +user nodes in ˜G and eu,u′ record the number of items that commonly interacted +with u and u’. +For every user u ∈ U, we retain its top-k users with the highest number +of commonly interacted items. Specifically, we keep the edge weight eu,u′ if u′ +belongs to the top-k users. Otherwise, the edge weight is 0. +eu,u′ = +� +eu,u′ +if eu,u′ ∈ top-k(eu), +0 +otherwise. +(3) +After establishing the user co-occurrence graph, we incorporate the atten- +tion mechanism when performing the graph propagation. The weight used for +aggregating neighboring nodes for a user is computed using the softmax function +to maximize the effect of neighboring users with a higher number of commonly +interacted items. The representation of u learned from ˜G at layer l+1 is denoted +as h(l+1) +u +, which is derived as follows: +h(l+1) +u += +� +u′∈Nu +exp(eu,u′) +� +ˆu∈Nu exp(eu,ˆu)h(l) +u′ , +(4) +Where eu,u′ indicates the number of common interacted items between u and +u′ and Nu denotes the neighbors of user u in ˜G. In this case, the representation +of each user can be enhanced based on neighbors in the co-occurrence graph. +Item Semantic Graph. Multimodal features offer rich and valuable content +information about items, but previous studies [22,18] neglect the significant un- +derlying semantic relations of item features. Inspired by [24], we argue that +item features are objective and we could establish the modality-specific homoge- +neous item graphs based on the raw features to learn the internal relations be- +tween items. Specifically, we construct the modality-aware item semantic graph +ˆGm = {I, Pi +m} for each modality m, where Pi +m = {em +i,i′|i, i′ ∈ I} denotes the +edges between item nodes in ˆGm. For an edge ei,i′, its weight is calculated by +the cosine similarity between original modality features of xm +i +and xm +i′ : +em +i,i′ = (xm +i )⊤xm +i′ +∥xm +i ∥∥xm +i′ ∥. +(5) +The derived ˆGm is a fully connected graph where edge weights are calculated +based on the similarity scores of modality features of connected nodes. Next, we +make the graph sparse by retaining the top-k similar items of every item. As the +ˆGm is a weighted graph, we convert it into an unweighted graph that captures +the fundamental relation structure of the most related items [3]. Formally, +em +i,i′ = +� +1 +if em +i,i′ ∈ top-k(em +i ), +0 +otherwise. +(6) + +8 +Hongyu Zhou, Xin Zhou, and Zhiqi Shen +Since we get one item semantic graph for each modality, we combine them +by performing weighted summation based on the importance score αm that +indicates the contribution of each modality and the summation is 1. Formally, +ˆG = {I, Pi}, where Pi = {ei,i′|i, i′ ∈ I} and ei,i′ = � +m∈M αem +i,i′. +After establishing the item semantic graph, we apply the graph convolution +operation on it to capture the item-item relationship: +h(l+1) +i += +� +i′∈Ni +ei,i′h(l) +i′ , +(7) +Where Ni denotes the neighbors of item i in Gi. Both h(0) +u +and h(0) +i +are +initialized with their fused representations uf and if, which are introduced in +the following section. +3.3 +Multimodal Fusion +An important factor influencing multimodal recommendation accuracy is multi- +modal fusion. As we mentioned in Section 1, previous multimodal recommenda- +tion models utilizing single-modal information perform better than multimodal +information. We speculate their fusion methods may fail to capture modality- +specific characteristics and even corrupt the learned single-modality represen- +tation. We intend to learn the multi-modality that can capture complementary +information which can not be contained in the single modality. Specifically, to +fuse the single modal features derived from modality-specific user-item graphs, +we apply the Attentive Concatenation for user multimodal embeddings that +capture the user preference on different modalities and direct concatenation for +item multimodal embeddings. The attention weight α for users is initialized to +0.5. Formally, +uf = αuv ∥ (1 − α)ut, +if = iv ∥ it, +(8) +Where ∥ denotes the concatenation operation. By performing attentive con- +catenation, we assume the single modality representations carry the richest in- +formation for each modality and this operation can capture the intact comple- +mentary information from each modality. +3.4 +Integration with Dual Representations +We integrate representations of users and items learned from heterogeneous (i.e., +Modality-specific User-Item Graphs) and homogeneous (i.e., User Co-occurrence +Graph & Item Semantic Graph) graphs to form their dual representations such +that interactions between users and items and their internal relations can be well +captured. We perform element-wise summation on the outputs learned from the +three graphs to generate the dual representations for u and i: +zu = uf + hLu +u , +zi = if + hLi +i , +(9) +Where zu and zi denote the final representations of user u and item i. Lu +and Li denote the number of GCN layers for the user co-occurrence graph and +item semantic graph respectively. + +DRAGON +9 +3.5 +Optimization +To optimize the parameters of DRAGON for the recommendation task, we leverage +the Bayesian Personalized Ranking (BPR) loss [16], which aims to score higher +for the positive item than the negative one. We construct a triplet set R including +the triplet (u, i, j) for each user u with the positive item i and a randomly +sampled negative item j that has no interactions with u. The loss function Lrec +is defined as follows, +R = {(u, i, j)|(u, i) ∈ E, (u, j) /∈ E}, +Lrec = +� +(u,i,j)∈R +−lnσ(yui − yuj) + λ∥Θ∥2, +(10) +Where yui = z⊤ +u zi calculates the inner product of zu and zi, λ is the L2 +regularization weight, and Θ denotes model parameters. +4 +Experiments +4.1 +Experimental Settings +Datasets We conduct experiments on three categories Baby, Sports and Cloth- +ing from the Amazon dataset [13], which contains product descriptions and im- +ages as textual and visual features. We retain the 5-core users and items such that +each user or item is associated with at least 5 interactions, which is widely used +in previous studies [6,7,24]. We use the pre-trained sentence-transformers [15] +to extract text features with a dimension equal to 384 and follow [24] to use +the published 4096-dimensional visual features. The statistics of datasets are +summarised in Table 2. In this case, data sparsity is calculated by dividing the +number of interactions by the product of the number of items and users. +Table 2. Statistics of the datasets. +Dataset # Users # Items # Interactions Sparsity +Baby +19,445 +7,050 +160,792 99.88% +Sports +35,598 +18,357 +296,337 99.95% +Clothing +39,387 +23,033 +278,677 99.97% +Baselines To evaluate the performance of our proposed model DRAGON, we +compare it with the following baseline methods which can be divided into two +groups. +1) General recommendation models: +– BPR [16] optimizes the user and item representations utilizing the matrix +factorization method. + +10 +Hongyu Zhou, Xin Zhou, and Zhiqi Shen +– LightGCN [7] simplifies the Graph Convolution Network by discarding the +feature transformation and nonlinear activation modules. +2) Multimodal recommendation models: +– VBPR [6] integrates visual features into item representations. For fair com- +parsion, we combine text and vision features to learn item representations. +– DualGNN [18] proposes to use representations learned from modality- +specific graphs and fuses the representations of neighbors in the user cor- +relation graph. +– GRCN [22] locates and removes the false-positive edges in the graph. It +then learns representations of items and users by conducting information +propagation and aggregation in the refined graph. +– LATTICE [24] introduces an item-item graph on each modality and ob- +tains the latent item semantic graph by aggregating information from all +modalities. +– SLMRec [17] uses self-supervised learning techniques that supplements the +supervised tasks to uncover the hidden signals from the data itself with +contrastive loss. +Evaluation Metrics We follow the settings as previous models +[23,18,24] +to randomly split the historical interactions with the ratio of 8:1:1 as train, +valid and test sets. Moreover, we adopt the widely used metrics Recall@K and +NDCG@K (denoted by R@K and N@K) to evaluate the top-K recommendation +performance. We empirically set K = 10 and 20. For each metric, we compute the +performance of each user in the testing data and report the average performance +over all users. +Implementation Details We implement our proposed model by PyTorch [14] +and embed the users and items with a dimensional size of 64 for all models. +We use the Xavier method [5] to initialize the embedding parameters, utilize +Adam [10] as the optimizer, and fix the mini-batch size to 2048. All models +are evaluated on a Tesla V100 32GB GPU card. The optimal hyper-parameters +are determined via grid searches on the validation set: we do a grid search on +the learning rate in {1e-1, 1e-2, 1e-3, 1e-4, 1e-5,1e-6} and the regularization +weight in {1e-1, 1e-2, 1e-3, 1e-4, 1e-5}. We fix the number of GCN layers in the +heterogeneous graph and homogeneous graph with L = 2 and Lu = Li = 1, +respectively. The important score αm for image weight in the item semantic +graph is empirically fixed as 0.1. The k of top-k in the user co-occurrence graph +is set to 10. We set the maximum number of epochs to 1000 and adopt the early +stopping strategy. That is, the model terminates when R@20 on the validation +set does not increase for 20 successive epochs. +4.2 +Performance Comparison +As shown in Table 3, we compare the recommendation performance of the state- +of-the-art methods with our proposed model. We have the following observations: + +DRAGON +11 +Table 3. Performance of baselines in terms of Recall and NDCG. Best results are in +boldface and the second best is underlined. “%Imp” denotes the relative improvement +of DRAGON over the best baseline. +Dataset +Metric +General Model +Multimodal Model +BPR LightGCN VBPR DualGNN GRCN SLMRec LATTICE DRAGON +%Imp +Baby +R@10 0.0357 +0.0479 +0.0423 +0.0448 +0.0539 +0.0529 +0.0547 +0.0662 21.02% +R@20 0.0575 +0.0754 +0.0663 +0.0716 +0.0833 +0.0775 +0.0850 +0.1021 20.12% +N@10 0.0192 +0.0257 +0.0223 +0.0240 +0.0288 +0.0290 +0.0292 +0.0345 18.15% +N@20 0.0249 +0.0328 +0.0284 +0.0309 +0.0363 +0.0353 +0.0370 +0.0435 17.57% +Sports +R@10 0.0432 +0.0569 +0.0558 +0.0568 +0.0598 +0.0663 +0.0620 +0.0749 12.97% +R@20 0.0653 +0.0864 +0.0856 +0.0859 +0.0915 +0.0990 +0.0953 +0.1124 13.54% +N@10 0.0241 +0.0311 +0.0307 +0.0310 +0.0332 +0.0365 +0.0335 +0.0403 10.41% +N@20 0.0298 +0.0387 +0.0384 +0.0385 +0.0414 +0.0450 +0.0421 +0.0500 11.11% +Clothing +R@10 0.0206 +0.0361 +0.0281 +0.0454 +0.0424 +0.0442 +0.0492 +0.0650 32.11% +R@20 0.0303 +0.0544 +0.0415 +0.0683 +0.0662 +0.0659 +0.0733 +0.0957 30.56% +N@10 0.0114 +0.0197 +0.0158 +0.0241 +0.0223 +0.0241 +0.0268 +0.0357 33.21% +N@20 0.0138 +0.0243 +0.0192 +0.0299 +0.0283 +0.0296 +0.0330 +0.0435 31.82% +– All GCN-based methods perform better than MF-based recommendation +models (i.e., BPR and VBPR), which demonstrates the effectiveness of mod- +eling the historical interactions by a graph with graph convolutional opera- +tions. +– In all evaluation metrics including Recall and NDCG, DRAGON outperforms +all the baseline models across all the datasets. For example, DRAGON makes an +improvement over the strongest baseline in the term of R@10 on the datasets +Baby, Sports and Clothing by 21.02%, 12.97% and 32.11% respectively. The +improvement attributes to the dual representations and the multimodal fu- +sion method. Learning dual representations from the heterogeneous and ho- +mogeneous graphs captures both the historical interactions and the internal +relations among each set of dyadic objects (i.e., users or items). A homo- +geneous graph helps to learn relevant characteristics from the neighbors. +The fusion method helps to learn the complementary information from each +single modal to enhance the multimodal representation. +– The multimodal recommendation models outperform the general recom- +mendation models. GRCN, LATTICE and SLMRec are multimodal mod- +els which outperform all general methods. The performance of VBPR that +builds upon the BPR framework by introducing modality information out- +performs BPR on all datasets. However, there exist multimodal models that +have a strong reliance on the representativeness of multimodal characteristics +of items, which results in inconsistent performance across various datasets. +For example, DualGNN builds based on LightGCN and outperforms it on +the Clothing dataset, but is less effective on Baby and Sports. It is possible +that multimodal features are more important to reveal item characteristics +in the clothing dataset, but are less informative in the other two datasets, +in which DualGNN is inferior to LightGCN. +In addition, we evaluate the scalability of DRAGON on a larger dataset Electronic +of Amazon dataset with around 1.7M interactions, 200K users and 63K items. +LATTICE [24] consumes more memory than the other baselines which could +not be handled by the 32GB GPU card. SLMRec [17] needs to find the optimal + +12 +Hongyu Zhou, Xin Zhou, and Zhiqi Shen +parameter by grid search on more than 200 parameter sets which takes a long +training time. DRAGON is superior to these baselines on the large graph. Although +SLMRec is the strongest baseline on Electronic, DRAGON is capable of gaining an +improvement of 4.76% in terms of R@20 compared with SLMRec. +4.3 +Ablation Study +In this section, we conduct exhaustive experiments to examine the behaviors of +our proposed model under various settings. +Effect of different components of DRAGON We design the following variants of +DRAGON based on the homogeneous graphs used and compared with the strongest +baselines (LATTICE and SLMRec) to investigate the contribution of different +components of DRAGON: +– DRAGONUI excludes the homogeneous graphs and only utilizes the heteroge- +neous graph; +– DRAGONUU incrementally adds the user co-occurrence graph to DRAGONUI. +This variant can capture the relations between users, which means only users +have dual representations. +Baby +Sports +Clothing +Dataset +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +0.11 +R@20 +SLMRec +LATTICE +DRAGONUI +DRAGONUU +DRAGON +(a) Comparison of different +components +Baby +Sports +Clothing +Dataset +0.02 +0.04 +0.06 +0.08 +0.10 +R@20 +DRAGONsum +DRAGONmean +DRAGONmax +DRAGON +(b) Comparison of differ- +ent fusion methods +Baby +Sports +Clothing +Dataset +0.02 +0.04 +0.06 +0.08 +0.10 +R@20 +DRAGONV +DRAGONT +DRAGON +(c) Comparison of differ- +ent modality +Fig. 2. Ablation study of DRAGON. +The full version DRAGON adds item-item homogeneous graph to DRAGONUU. +Fig. 2(a) shows our comparison results in terms of Recall@20. We have the +following observations that show all components of DRAGON contribute to the +performance: (1) LATTICE and SLMRec are the strongest baselines for multi- +modal recommendation, DRAGONUI excludes the dual representation learning and +only utilizes our multimodal fusion can reach almost the same or even higher +accuracy with the strong baselines. (2) The performance accuracy comparison +result DRAGONUI < DRAGONUU < DRAGON shows that adding user co-occurrence +graph gets improvement compared with only utilizing the u-i graph. Based on + +DRAGON +13 +DRAGONUU, adding the item semantic graph could reach higher improvement. +We observe that utilizing the user co-occurrence graph does not get much im- +provement for Baby. It suggests that the graph connectivity also influences the +construction of the user co-occurrence graph. With more interactions between +users and items, more co-occurrence patterns of user relations can be captured. +Effect of different modality fusion method We identify the modality fu- +sion issue for previous works and utilize the direct concatenation with attention +in our proposed model. We compare its performance with the fusion methods +mentioned in [18][24]. We replace the fusion of DRAGON with weighted sum (de- +noted as DRAGONsum), mean (denoted as DRAGONmean) and weighted max (de- +noted as DRAGONmax) to demonstrate the superiority of concatenation fusion +used in DRAGON. Fig. 2(b) shows our comparison results in terms of Recall@20. +Clearly, our fusion method outperforms other methods as it can capture the +complementary information of each modality and attend information from all +modalities for recommendation. +Effect of single modality vs. multi-modalities In Introduction, we reveal +the performance of previous multimodal models might degrade under multimodal +settings. Hence, we compare the performance of DRAGON under uni-modal and +multimodal settings. DRAGONV , DRAGONT and DRAGON denote the models that +utilize visual, textual, and both modalities respectively. Fig. 2(c) shows our +comparison results in terms of Recall@20. We observe that: (1). Models with +different single modality information have different performances. The textual +modality performs better than the visual modality. (2). DRAGON with multimodal +outperforms that utilizing single modality, it demonstrates that fusing different +modalities of information can improve the performance of DRAGON. Thus, the +fusion method in our proposed model is indispensable. +4.4 +Hyper-parameter Sensitivity Study +To guide hyper-parameters selections of DRAGON, we perform the sensitivity study +in terms of Recall@20 and NDCG@20. We vary the learning rate of DRAGON in +{1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1}, and vary the regularization weight λ in {1e-5, +1e-4, 1e-3, 1e-2, 1e-1}. Fig. 3(a) and Fig.3(b) show the performance of DRAGON +under different combinations of the learning rate and regularization weight on +Baby and Clothing datasets. From the figures, we have the following observa- +tions: (1). The suggested learning rate is 1e-4 which obtains the best result across +all regularization weights. Except for the too-large and too-small learning rates, +our model gets stable performance with the middle number of learning rates +in {1e-3, 1e-4, 1e-5}. It further demonstrates the efficiency and stability of our +model. (2). Compared with the learning rate, the performance is less sensitive +to the regularization weight. However, searching the parameter sets of regular- +ization weight in DRAGON also improves its performance. + +14 +Hongyu Zhou, Xin Zhou, and Zhiqi Shen +1e-5 +1e-4 +1e-3 +1e-2 +1e-1 +Regularization Weight +1e-6 +1e-5 +1e-4 +1e-3 +1e-2 +1e-1 +Learning Rate +0.0881 +0.0865 +0.0883 +0.0872 +0.0876 +0.0997 +0.0995 +0.0996 +0.0997 +0.0955 +0.1012 +0.0996 +0.1021 +0.1013 +0.1006 +0.0943 +0.0946 +0.0943 +0.0945 +0.0947 +0.0863 +0.0869 +0.0865 +0.0869 +0.0865 +0.0734 +0.0734 +0.0732 +0.0730 +0.0706 +R@20 +1e-5 +1e-4 +1e-3 +1e-2 +1e-1 +Regularization Weight +1e-6 +1e-5 +1e-4 +1e-3 +1e-2 +1e-1 +Learning Rate +0.0381 +0.0373 +0.0381 +0.0377 +0.0379 +0.0432 +0.0432 +0.0432 +0.0433 +0.0416 +0.0435 +0.0436 +0.0435 +0.0434 +0.0439 +0.0416 +0.0413 +0.0417 +0.0416 +0.0419 +0.0381 +0.0383 +0.0382 +0.0382 +0.0381 +0.0328 +0.0328 +0.0328 +0.0327 +0.0304 +N@20 +(a) Baby +1e-5 +1e-4 +1e-3 +1e-2 +1e-1 +Regularization Weight +1e-6 +1e-5 +1e-4 +1e-3 +1e-2 +1e-1 +Learning Rate +0.0743 +0.0742 +0.0743 +0.0813 +0.0840 +0.0944 +0.0942 +0.0928 +0.0943 +0.0944 +0.0935 +0.0946 +0.0952 +0.0949 +0.0957 +0.0896 +0.0896 +0.0902 +0.0906 +0.0902 +0.0787 +0.0785 +0.0777 +0.0779 +0.0774 +0.0598 +0.0601 +0.0601 +0.0586 +0.0530 +R@20 +1e-5 +1e-4 +1e-3 +1e-2 +1e-1 +Regularization Weight +1e-6 +1e-5 +1e-4 +1e-3 +1e-2 +1e-1 +Learning Rate +0.0329 +0.0328 +0.0329 +0.0364 +0.0379 +0.0429 +0.0429 +0.0423 +0.0429 +0.0429 +0.0427 +0.0431 +0.0434 +0.0432 +0.0435 +0.0411 +0.0411 +0.0410 +0.0412 +0.0409 +0.0352 +0.0356 +0.0347 +0.0351 +0.0348 +0.0264 +0.0266 +0.0267 +0.0262 +0.0237 +N@20 +(b) Clothing +Fig. 3. Sensitivity analyses on the DRAGON hyperparameters. +5 +Conclusion +In this paper, we aim to solve the modality fusion issue and learn better represen- +tations for dyadic-related users and items. Therefore, we develop a novel model, +named DRAGON, to learn the dual representations of users and items by construct- +ing homogeneous and heterogeneous graphs. In particular, we first construct the +modality-specific user-item bipartite graph to learn the modality features. After +getting the representations of each modality, we utilize the late concatenation +fusion method to learn the multimodal features. Then, we construct the user co- +occurrence graph to capture the co-occurrence relations between users and the +item semantic graph to capture the semantic relations between items. Therefore, +we learn both inter- and intra-relations of the dyadic-related users and items. +Finally, we conduct extensive experiments on three datasets to demonstrate the +effectiveness of our proposed model. Our experimental finding on multimodal fu- +sion could shed light on the design of future multimodal recommender systems. + +DRAGON +15 +References +1. Adomavicius, G., Tuzhilin, A.: Toward the next generation of recommender sys- +tems: A survey of the state-of-the-art and possible extensions. IEEE transactions +on knowledge and data engineering 17(6), 734–749 (2005) +2. Baltruˇsaitis, T., Ahuja, C., Morency, L.P.: Multimodal machine learning: A survey +and taxonomy. IEEE transactions on pattern analysis and machine intelligence +41(2), 423–443 (2018) +3. Chen, J., Fang, H.r., Saad, Y.: Fast approximate knn graph construction for high +dimensional data via recursive lanczos bisection. Journal of Machine Learning Re- +search 10(9) (2009) +4. 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IEEE Access 6, 63748–63756 (2018) + diff --git a/ntFLT4oBgHgl3EQffy-_/content/tmp_files/load_file.txt b/ntFLT4oBgHgl3EQffy-_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d113f4bd7702156c5e90a02e50b9a8cf6f12f7ea --- /dev/null +++ b/ntFLT4oBgHgl3EQffy-_/content/tmp_files/load_file.txt @@ -0,0 +1,833 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf,len=832 +page_content='Enhancing Dyadic Relations with Homogeneous Graphs for Multimodal Recommendation Hongyu Zhou1, Xin Zhou2, and Zhiqi Shen3 Alibaba-NTU Singapore Joint Research Institute, Department of Computer Science and Engineering, Nanyang Technological University, Singapore {hongyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='zhou,xin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='zhou,ZQShen}@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='sg Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' User interaction data in recommender systems is a form of dyadic relation that reflects the preferences of users with items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Learn- ing the representations of these two discrete sets of objects, users and items, is critical for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Recent multimodal recommenda- tion models leveraging multimodal features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', images and text de- scriptions) have been demonstrated to be effective in improving rec- ommendation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' However, state-of-the-art models enhance the dyadic relations between users and items by considering either user-user or item-item relations, leaving the high-order relations of the other side (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', users or items) unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Furthermore, we experimentally reveal that the current multimodality fusion methods in the state-of-the-art models may degrade their recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' That is, with- out tainting the model architectures, these models can achieve even bet- ter recommendation accuracy with uni-modal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' On top of the finding, we propose a model that enhances the dyadic relations by learning Dual RepresentAtions of both users and items via construct- ing homogeneous Graphs for multimOdal recommeNdation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We name our model as DRAGON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Specifically, DRAGON constructs the user-user graph based on the commonly interacted items and the item-item graph from item multimodal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' It then utilizes graph learning on both the user-item heterogeneous graph and the homogeneous graphs (user-user and item-item) to obtain the dual representations of users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' To capture information from each modality, DRAGON employs a simple yet effective fusion method, attentive concatenation, to derive the rep- resentations of users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Extensive experiments on three pub- lic datasets and seven baselines show that DRAGON can outperform the strongest baseline by 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='03% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Various ablation studies are conducted on DRAGON to validate its effectiveness in modality fusing and learning dual representations for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Keywords: Multimodal Recommendation · Dyadic Relation · Dual Rep- resentation Learning · Graph Neural Network · Multimodal Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 1 Introduction As society evolves, recommender systems have become indispensable tools to assist users to find products and services of their choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Previous work [1,16,7] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='12097v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='IR] 28 Jan 2023 2 Hongyu Zhou, Xin Zhou, and Zhiqi Shen explores historical user-item interactions that can be considered as a form of dyadic relation to capturing user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' However, these methods show in- ferior performance due to the sparse nature of interactions between users and items in real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' To alleviate the data sparsity problem, recent multimodal recommender sys- tems utilizing multimodal information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', item descriptive texts, product im- ages) to improve recommendation performance have gained considerable atten- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' A line of work [11,6] integrates multimodal features as side information to enhance latent item representations under the classic collaborative filtering framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Inspired by the success of graph neural networks (GNNs) on rec- ommendation [19,7], recent works focus on modeling user-item interactions as a bipartite graph and integrating multimodal information with graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For example, MMGCN [23] builds a user-item bipartite graph for every modal- ity to obtain modal-specific representation to understand user preference better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' GRCN [22] presents a graph refine layer that could identify noisy edges and corrupt false-positive edges to clarify the structure of the user-item interaction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DualGNN [18] and LATTICE [24] introduce either user-user or item-item relations into the user-item interactions and achieve state-of-the-art recommen- dation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Although these models show effective recommendation ac- curacy, we argue the high-order relations in both sides of the dyadic relations can be explored simultaneously to fully address the data sparsity issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Inspired by the dual representation learning mechanism [25], we enhance the representa- tion learning of users and items by incorporating their dual representations to capture both the inter- and intra-relations between users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Furthermore, we experimentally reveal that these methods fail to fuse the modality features effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Specifically, we conduct an ablation study of mul- timodal features on two competitive multimodal models, DualGNN [18] and LATTICE [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The results shown in Table 1 demonstrate that the performance of these models fed with a single modality, especially textual features, outper- forms that with both modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' This finding poses a meaningful question: How can we effectively fuse the multimodal information for recommendation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' To address the above question, we study the performance of different modal- ity fusion methods, including Attentively Sum, Max-pooling, Mean-pooling, and Concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Our experiments show that the late-fusion approach Con- catenation which directly concatenates the textual and visual features as the multimodal representation achieves the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Toward this end, we propose a framework that learns Dual RepresentAtions of both users and items via constructing homogeneous Graphs for multimOdal recommeNdation (DRAGON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' To be specific, DRAGON constructs a heterogeneous user-item bipartite graph for each modality to learn the modality-specific rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' It then utilizes the direct Concatenation fusion method to better exploit the learned modality-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' In order to learn the dual representations, we construct two homogeneous graphs based on the user co- occurrence and the item semantic features to capture the user preference from the neighbor users and the latent item content semantic from the neighbor items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DRAGON 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Performance of DualGNN [18] and LATTICE [24] utilizing features in differ- ent modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' R and N denote evaluation metrics Recall and NDCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' T and V denote textual and visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Dataset Metric DualGNN LATTICE V&T T V V&T T V Baby R@10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0547 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0273 Finally, DRAGON utilizes the learned dual representations of users and items to make recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Extensive experiments are conducted on three public datasets to show the effectiveness of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='1 Multimodal Recommendation Collaborative filtering (CF) based models are widely used [8,19] for recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The CF-based methods leverage the historical interactions between users and items to predict users’ preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' However, they usually suffer from the data sparsity problem since user-item interactions are generally limited for real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' To alleviate the data sparsity problem, massive multimodal content informa- tion has been utilized to improve recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For example, VBPR [6] leverages the visual features from the pre-trained Convolutional Neu- ral Network (CNN) to enhance the matrix factorization by incorporating the visual features with ID embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' VECF [4] models the user’s attention on different regions of images and reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Recently, GNNs have drawn more at- tention when applied to multimodal-based recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' MMGCN [23] improves the quality of learned user and item representations by constructing a modality-specific user-item bipartite graph and adapting the message-passing mechanism of GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Following MMGCN, GRCN [22] presents a graph refine layer that can locate noisy edges and corrupt false-positive edges to refine the structure of the user-item interaction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DualGNN[25] adds an attention mechanism to capture the user’s preference on different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Meanwhile, 4 Hongyu Zhou, Xin Zhou, and Zhiqi Shen it constructs a user-user graph to learn the user preference from neighbor users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' LATTICE [24] builds an item-item graph for each modality, then combines them to create a latent modality-fused item graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' With graph convolution operations, items can share information from highly linked affinities in the graph to enhance their representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' SLMRec [17] incorporates self-supervised learning tasks into the GNNs to uncover the latent patterns from multi-modalities to learn powerful representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='2 Multimodal Fusion Finding a fused multi-modal representation that is complimentary and compre- hensible can greatly improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Technically speaking, multi-modal fusion integrates information from different modalities to create a multimodal representation that can be applied to different tasks, such as link prediction [12] and node classification [26], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' It can be categorized into early fusion, late fusion and hybrid fusion [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Early fusion incorporates the extracted features at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Contrarily, the late fusion integrates after each modality has completed the decision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', classification or regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Hybrid fusion combines the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For example, ACNet [9] adopts an early fusion method based on the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For the late fusion method, NMCL [21] utilizes the cooperative networks for each modality to do feature augmentation with the attention mechanism, then a late fusion is applied to the prediction from vari- ous modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' CELFT [20] designs a hybrid fusion method that combines both early and late fusion together to overcome the shortcomings of the single fusion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Multimodal recommendation models [23,24,25] normally apply mean-pooling, attentive sum or max-pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Our experimental results suggest that those mod- els use a single modality representation leads to better performance than utilizing the multimodal representation learned from these fusion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' It indicates that pooling or attention-based methods may result in information loss when performing multimodal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' To mitigate this problem, we adopt attentive concatenation fusion but without reducing the embedding dimension, which has been demonstrated to be more effective in joining information from different modalities for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 3 Methodology This section introduces the details of our proposed model DRAGON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 1 shows the overall architecture of DRAGON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' There are four main components: (1) graph learning on a modality-specific heterogeneous graph to learn a uni-modal rep- resentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (2) Multimodal representation learning module that captures user preference on different modalities and complementary information from each modality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (3) graph learning on homogeneous graphs to capture the co-occurrence relation between users and the semantic relation between items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (4) a predictor module that ranks the candidate items based on the scores calculated from final user and item representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DRAGON 5 Heterogeneous graph Homogeneous graph Multimodal representation 𝒖𝑣 𝒖𝑡 𝒊𝑣 𝒊𝑡 𝒖𝑓 𝒊𝑓 𝑦𝑢𝑖 𝒛𝑢 𝒛𝑖 Visual Textual Modality-specific user item bipartite graph neighbors of u Layer 1 Layer 3 Layer 2 Normalized sum neighbors of i Layer 1 Layer 2 Normalized sum 𝒊𝟏 𝒊𝟐 𝒊𝟑 𝒖𝟐 𝒖𝟏 𝒖𝑡 𝒊𝑡 𝒖𝑓 𝒊𝑓 Layer 1 Layer 2 Layer 1 Layer 2 𝒉𝑢 𝒉𝑖 Item semantic graph User co-occurrence graph u i Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The diagram depicts the main components of DRAGON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We first learn the modality-specific representations, then the multimodal representation learning mod- ule fused the single-modality representations and utilizes the homogeneous graph to capture the internal relations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='1 Preliminary Given a set of N users u ∈ U, a set of M items i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We model the dyadic relations of user interactions as a user-item bipartite graph G = {U, I, E}, where we regard the historical interactions as the set of edges in the graph denoted by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Besides the user-item interactions, each item is associated with multimodal content information m ∈ {v, t}, where v and t represent the visual and textual features respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We denote the modality feature for an item i as xm i ∈ Rdm, where dm denotes the feature dimension of modality m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' In this paper, we only consider the visual and textual modalities denoted by v and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' However, the proposed framework can be easily extended to scenarios involving more than two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='2 Dual Representation Learning Learning the representations of users and items is critical for the recommendation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' All representation learning-based techniques assume the existence of a common representation with consistent knowledge of different views of items [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Different views of an item contain specific discriminant information in addition to consistent knowledge about this item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We construct the heterogeneous and homogeneous graphs together to learn the dual representations of both user and item, which could capture both the internal association and the relationship between users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 6 Hongyu Zhou, Xin Zhou, and Zhiqi Shen Heterogeneous Graph To learn modality-specific user and item representa- tions, we construct a user-item graph for each modality, which is denoted as Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Following MMGCN [23], we maintain the same graph structure G for dif- ferent Gm, but only keep the node features associated with a specific modality m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We adopt LightGCN [7] to encode Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' As shown in [7], LightGCN simpli- fies the graph convolutional operations by excluding the feature transformation and nonlinear activation modules to improve recommendation performance and meanwhile ease the model optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Specifically, the user and item representations at the (l + 1)-th graph convolution layer of Gm are derived as follows: u(l+1) m = � i∈Nu 1 � |Nu| � |Ni| i(l) m , i(l+1) m = � u∈Ni 1 � |Nu| � |Ni| u(l) m , (1) Where Nu and Ni are the set of first hop neighbors of u and i in Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' u(0) m is randomly initialized and i(0) m is initialized with xm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The symmetric normal- ization 1 √ |Nu|√ |Ni| is used to normalize the modality features learned from each layer which avoids the increase of scale when operates the graph convolutional operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' After L layers of data propagation, we combine the representations from every GCN layer using element-wise summation to derive the modality-specific representations for users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Formally, um = L � l=0 u(l) m , im = L � l=0 i(l) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (2) In such cases, the historical interactions and the modality information have been encoded into the final single-modal representations of users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' And those operations have been applied to each modality by propagating on the modality-specific user-item bipartite graphs to learn the representations for each modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Homogeneous Graph In addition to employing the heterogeneous graph, which encodes the dyadic relation between users and items, we argue that the recommendation performance can be further enhanced by modeling the inter- nal relations between users or items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For the two homogeneous graphs, we pre- established and freeze them to maintain the initial co-occurrence relation and semantic meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' User Co-occurrence Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Based on the assumption that users who have interacted with similar items usually have similar preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We argue that the user’s preference pattern is hidden inside the co-occurrence items and we construct a homogeneous user co-occurrence graph to learn the internal relations between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' However, the numbers of co-occurrence items between users are in a large range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' In a general situation, the user will have a high number of commonly interacted items with a small group of users but with few items with DRAGON 7 other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We only consider those who have more commonly interacted items with the user to capture similar preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' To explicitly model the item co- occurrence patterns of users, we construct a homogeneous user co-occurrence graph ˜G = {U, Pu}, where Pu = {eu,u′|u, u′ ∈ U} denotes the edges between user nodes in ˜G and eu,u′ record the number of items that commonly interacted with u and u’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For every user u ∈ U, we retain its top-k users with the highest number of commonly interacted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Specifically, we keep the edge weight eu,u′ if u′ belongs to the top-k users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Otherwise, the edge weight is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' eu,u′ = � eu,u′ if eu,u′ ∈ top-k(eu), 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (3) After establishing the user co-occurrence graph, we incorporate the atten- tion mechanism when performing the graph propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The weight used for aggregating neighboring nodes for a user is computed using the softmax function to maximize the effect of neighboring users with a higher number of commonly interacted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The representation of u learned from ˜G at layer l+1 is denoted as h(l+1) u , which is derived as follows: h(l+1) u = � u′∈Nu exp(eu,u′) � ˆu∈Nu exp(eu,ˆu)h(l) u′ , (4) Where eu,u′ indicates the number of common interacted items between u and u′ and Nu denotes the neighbors of user u in ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' In this case, the representation of each user can be enhanced based on neighbors in the co-occurrence graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Item Semantic Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Multimodal features offer rich and valuable content information about items, but previous studies [22,18] neglect the significant un- derlying semantic relations of item features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Inspired by [24], we argue that item features are objective and we could establish the modality-specific homoge- neous item graphs based on the raw features to learn the internal relations be- tween items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Specifically, we construct the modality-aware item semantic graph ˆGm = {I, Pi m} for each modality m, where Pi m = {em i,i′|i, i′ ∈ I} denotes the edges between item nodes in ˆGm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For an edge ei,i′, its weight is calculated by the cosine similarity between original modality features of xm i and xm i′ : em i,i′ = (xm i )⊤xm i′ ∥xm i ∥∥xm i′ ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (5) The derived ˆGm is a fully connected graph where edge weights are calculated based on the similarity scores of modality features of connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Next, we make the graph sparse by retaining the top-k similar items of every item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' As the ˆGm is a weighted graph, we convert it into an unweighted graph that captures the fundamental relation structure of the most related items [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Formally, em i,i′ = � 1 if em i,i′ ∈ top-k(em i ), 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (6) 8 Hongyu Zhou, Xin Zhou, and Zhiqi Shen Since we get one item semantic graph for each modality, we combine them by performing weighted summation based on the importance score αm that indicates the contribution of each modality and the summation is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Formally, ˆG = {I, Pi}, where Pi = {ei,i′|i, i′ ∈ I} and ei,i′ = � m∈M αem i,i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' After establishing the item semantic graph, we apply the graph convolution operation on it to capture the item-item relationship: h(l+1) i = � i′∈Ni ei,i′h(l) i′ , (7) Where Ni denotes the neighbors of item i in Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Both h(0) u and h(0) i are initialized with their fused representations uf and if, which are introduced in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='3 Multimodal Fusion An important factor influencing multimodal recommendation accuracy is multi- modal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' As we mentioned in Section 1, previous multimodal recommenda- tion models utilizing single-modal information perform better than multimodal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We speculate their fusion methods may fail to capture modality- specific characteristics and even corrupt the learned single-modality represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We intend to learn the multi-modality that can capture complementary information which can not be contained in the single modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Specifically, to fuse the single modal features derived from modality-specific user-item graphs, we apply the Attentive Concatenation for user multimodal embeddings that capture the user preference on different modalities and direct concatenation for item multimodal embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The attention weight α for users is initialized to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Formally, uf = αuv ∥ (1 − α)ut, if = iv ∥ it, (8) Where ∥ denotes the concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' By performing attentive con- catenation, we assume the single modality representations carry the richest in- formation for each modality and this operation can capture the intact comple- mentary information from each modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='4 Integration with Dual Representations We integrate representations of users and items learned from heterogeneous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', Modality-specific User-Item Graphs) and homogeneous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', User Co-occurrence Graph & Item Semantic Graph) graphs to form their dual representations such that interactions between users and items and their internal relations can be well captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We perform element-wise summation on the outputs learned from the three graphs to generate the dual representations for u and i: zu = uf + hLu u , zi = if + hLi i , (9) Where zu and zi denote the final representations of user u and item i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Lu and Li denote the number of GCN layers for the user co-occurrence graph and item semantic graph respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DRAGON 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='5 Optimization To optimize the parameters of DRAGON for the recommendation task, we leverage the Bayesian Personalized Ranking (BPR) loss [16], which aims to score higher for the positive item than the negative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We construct a triplet set R including the triplet (u, i, j) for each user u with the positive item i and a randomly sampled negative item j that has no interactions with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The loss function Lrec is defined as follows, R = {(u, i, j)|(u, i) ∈ E, (u, j) /∈ E}, Lrec = � (u,i,j)∈R −lnσ(yui − yuj) + λ∥Θ∥2, (10) Where yui = z⊤ u zi calculates the inner product of zu and zi, λ is the L2 regularization weight, and Θ denotes model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='1 Experimental Settings Datasets We conduct experiments on three categories Baby, Sports and Cloth- ing from the Amazon dataset [13], which contains product descriptions and im- ages as textual and visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We retain the 5-core users and items such that each user or item is associated with at least 5 interactions, which is widely used in previous studies [6,7,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We use the pre-trained sentence-transformers [15] to extract text features with a dimension equal to 384 and follow [24] to use the published 4096-dimensional visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The statistics of datasets are summarised in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' In this case, data sparsity is calculated by dividing the number of interactions by the product of the number of items and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Statistics of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Dataset # Users # Items # Interactions Sparsity Baby 19,445 7,050 160,792 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='88% Sports 35,598 18,357 296,337 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='95% Clothing 39,387 23,033 278,677 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='97% Baselines To evaluate the performance of our proposed model DRAGON, we compare it with the following baseline methods which can be divided into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 1) General recommendation models: – BPR [16] optimizes the user and item representations utilizing the matrix factorization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 10 Hongyu Zhou, Xin Zhou, and Zhiqi Shen – LightGCN [7] simplifies the Graph Convolution Network by discarding the feature transformation and nonlinear activation modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 2) Multimodal recommendation models: – VBPR [6] integrates visual features into item representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For fair com- parsion, we combine text and vision features to learn item representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' – DualGNN [18] proposes to use representations learned from modality- specific graphs and fuses the representations of neighbors in the user cor- relation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' – GRCN [22] locates and removes the false-positive edges in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' It then learns representations of items and users by conducting information propagation and aggregation in the refined graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' – LATTICE [24] introduces an item-item graph on each modality and ob- tains the latent item semantic graph by aggregating information from all modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' – SLMRec [17] uses self-supervised learning techniques that supplements the supervised tasks to uncover the hidden signals from the data itself with contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Evaluation Metrics We follow the settings as previous models [23,18,24] to randomly split the historical interactions with the ratio of 8:1:1 as train, valid and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Moreover, we adopt the widely used metrics Recall@K and NDCG@K (denoted by R@K and N@K) to evaluate the top-K recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We empirically set K = 10 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For each metric, we compute the performance of each user in the testing data and report the average performance over all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Implementation Details We implement our proposed model by PyTorch [14] and embed the users and items with a dimensional size of 64 for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We use the Xavier method [5] to initialize the embedding parameters, utilize Adam [10] as the optimizer, and fix the mini-batch size to 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' All models are evaluated on a Tesla V100 32GB GPU card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The optimal hyper-parameters are determined via grid searches on the validation set: we do a grid search on the learning rate in {1e-1, 1e-2, 1e-3, 1e-4, 1e-5,1e-6} and the regularization weight in {1e-1, 1e-2, 1e-3, 1e-4, 1e-5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We fix the number of GCN layers in the heterogeneous graph and homogeneous graph with L = 2 and Lu = Li = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The important score αm for image weight in the item semantic graph is empirically fixed as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The k of top-k in the user co-occurrence graph is set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We set the maximum number of epochs to 1000 and adopt the early stopping strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' That is, the model terminates when R@20 on the validation set does not increase for 20 successive epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='2 Performance Comparison As shown in Table 3, we compare the recommendation performance of the state- of-the-art methods with our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We have the following observations: DRAGON 11 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Performance of baselines in terms of Recall and NDCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Best results are in boldface and the second best is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' “%Imp” denotes the relative improvement of DRAGON over the best baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Dataset Metric General Model Multimodal Model BPR LightGCN VBPR DualGNN GRCN SLMRec LATTICE DRAGON %Imp Baby R@10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0357 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0539 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0662 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='02% R@20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='0663 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', BPR and VBPR), which demonstrates the effectiveness of mod- eling the historical interactions by a graph with graph convolutional opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' – In all evaluation metrics including Recall and NDCG, DRAGON outperforms all the baseline models across all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For example, DRAGON makes an improvement over the strongest baseline in the term of R@10 on the datasets Baby, Sports and Clothing by 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='02%, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='97% and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='11% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The improvement attributes to the dual representations and the multimodal fu- sion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Learning dual representations from the heterogeneous and ho- mogeneous graphs captures both the historical interactions and the internal relations among each set of dyadic objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', users or items).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' A homo- geneous graph helps to learn relevant characteristics from the neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The fusion method helps to learn the complementary information from each single modal to enhance the multimodal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' – The multimodal recommendation models outperform the general recom- mendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' GRCN, LATTICE and SLMRec are multimodal mod- els which outperform all general methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The performance of VBPR that builds upon the BPR framework by introducing modality information out- performs BPR on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' However, there exist multimodal models that have a strong reliance on the representativeness of multimodal characteristics of items, which results in inconsistent performance across various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' For example, DualGNN builds based on LightGCN and outperforms it on the Clothing dataset, but is less effective on Baby and Sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' It is possible that multimodal features are more important to reveal item characteristics in the clothing dataset, but are less informative in the other two datasets, in which DualGNN is inferior to LightGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' In addition, we evaluate the scalability of DRAGON on a larger dataset Electronic of Amazon dataset with around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='7M interactions, 200K users and 63K items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' LATTICE [24] consumes more memory than the other baselines which could not be handled by the 32GB GPU card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' SLMRec [17] needs to find the optimal 12 Hongyu Zhou, Xin Zhou, and Zhiqi Shen parameter by grid search on more than 200 parameter sets which takes a long training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DRAGON is superior to these baselines on the large graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Although SLMRec is the strongest baseline on Electronic, DRAGON is capable of gaining an improvement of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='76% in terms of R@20 compared with SLMRec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='3 Ablation Study In this section, we conduct exhaustive experiments to examine the behaviors of our proposed model under various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Effect of different components of DRAGON We design the following variants of DRAGON based on the homogeneous graphs used and compared with the strongest baselines (LATTICE and SLMRec) to investigate the contribution of different components of DRAGON: – DRAGONUI excludes the homogeneous graphs and only utilizes the heteroge- neous graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' – DRAGONUU incrementally adds the user co-occurrence graph to DRAGONUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' This variant can capture the relations between users, which means only users have dual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Baby Sports Clothing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='11 R@20 SLMRec LATTICE DRAGONUI DRAGONUU DRAGON (a) Comparison of different components Baby Sports Clothing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='10 R@20 DRAGONsum DRAGONmean DRAGONmax DRAGON (b) Comparison of differ- ent fusion methods Baby Sports Clothing Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='10 R@20 DRAGONV DRAGONT DRAGON (c) Comparison of differ- ent modality Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Ablation study of DRAGON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The full version DRAGON adds item-item homogeneous graph to DRAGONUU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 2(a) shows our comparison results in terms of Recall@20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We have the following observations that show all components of DRAGON contribute to the performance: (1) LATTICE and SLMRec are the strongest baselines for multi- modal recommendation, DRAGONUI excludes the dual representation learning and only utilizes our multimodal fusion can reach almost the same or even higher accuracy with the strong baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (2) The performance accuracy comparison result DRAGONUI < DRAGONUU < DRAGON shows that adding user co-occurrence graph gets improvement compared with only utilizing the u-i graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Based on DRAGON 13 DRAGONUU, adding the item semantic graph could reach higher improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We observe that utilizing the user co-occurrence graph does not get much im- provement for Baby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' It suggests that the graph connectivity also influences the construction of the user co-occurrence graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' With more interactions between users and items, more co-occurrence patterns of user relations can be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Effect of different modality fusion method We identify the modality fu- sion issue for previous works and utilize the direct concatenation with attention in our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We compare its performance with the fusion methods mentioned in [18][24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We replace the fusion of DRAGON with weighted sum (de- noted as DRAGONsum), mean (denoted as DRAGONmean) and weighted max (de- noted as DRAGONmax) to demonstrate the superiority of concatenation fusion used in DRAGON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 2(b) shows our comparison results in terms of Recall@20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Clearly, our fusion method outperforms other methods as it can capture the complementary information of each modality and attend information from all modalities for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Effect of single modality vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' multi-modalities In Introduction, we reveal the performance of previous multimodal models might degrade under multimodal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Hence, we compare the performance of DRAGON under uni-modal and multimodal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DRAGONV , DRAGONT and DRAGON denote the models that utilize visual, textual, and both modalities respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 2(c) shows our comparison results in terms of Recall@20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We observe that: (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Models with different single modality information have different performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The textual modality performs better than the visual modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DRAGON with multimodal outperforms that utilizing single modality, it demonstrates that fusing different modalities of information can improve the performance of DRAGON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Thus, the fusion method in our proposed model is indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='4 Hyper-parameter Sensitivity Study To guide hyper-parameters selections of DRAGON, we perform the sensitivity study in terms of Recall@20 and NDCG@20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' We vary the learning rate of DRAGON in {1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1}, and vary the regularization weight λ in {1e-5, 1e-4, 1e-3, 1e-2, 1e-1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 3(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='3(b) show the performance of DRAGON under different combinations of the learning rate and regularization weight on Baby and Clothing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' From the figures, we have the following observa- tions: (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' The suggested learning rate is 1e-4 which obtains the best result across all regularization weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Except for the too-large and too-small learning rates, our model gets stable performance with the middle number of learning rates in {1e-3, 1e-4, 1e-5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' It further demonstrates the efficiency and stability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Compared with the learning rate, the performance is less sensitive to the regularization weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' However, searching the parameter sets of regular- ization weight in DRAGON also improves its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 14 Hongyu Zhou, Xin Zhou, and 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the DRAGON hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' 5 Conclusion In this paper, we aim to solve the modality fusion issue and learn better represen- tations for dyadic-related users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Therefore, we develop a novel model, named DRAGON, to learn the dual representations of users and items by construct- ing homogeneous and heterogeneous graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' In particular, we first construct the modality-specific user-item bipartite graph to learn the modality features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' After getting the representations of each modality, we utilize the late concatenation fusion method to learn the multimodal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Then, we construct the user co- occurrence graph to capture the co-occurrence relations between users and the item semantic graph to capture the semantic relations between items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Therefore, we learn both inter- and intra-relations of the dyadic-related users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Finally, we conduct extensive experiments on three datasets to demonstrate the effectiveness of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Our experimental finding on multimodal fu- sion could shed light on the design of future multimodal recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' DRAGON 15 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Adomavicius, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', Tuzhilin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=': Toward the next generation of recommender sys- tems: A survey of the state-of-the-art and possible extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' IEEE transactions on knowledge and data engineering 17(6), 734–749 (2005) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=' Baltruˇsaitis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', Ahuja, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=', Morency, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFLT4oBgHgl3EQffy-_/content/2301.12097v1.pdf'} +page_content=': Multimodal machine learning: A survey and taxonomy.' metadata={'source': 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b/odFPT4oBgHgl3EQfKzSD/content/tmp_files/2301.13020v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..031d2725c9f211efdd6ffcc4938d21ba1b788cc1 --- /dev/null +++ b/odFPT4oBgHgl3EQfKzSD/content/tmp_files/2301.13020v1.pdf.txt @@ -0,0 +1,2037 @@ +Designing Ad Auctions with Private Constraints for Automated Bidding +Yidan Xing1 , Zhilin Zhang2 , Zhenzhe Zheng1 , Chuan Yu2 , +Jian Xu2 , Fan Wu1 and Guihai Chen1 +1Shanghai Jiao Tong University +2Alibaba Group +{katexing, zhengzhenzhe}@sjtu.edu.cn, {zhangzhilin.pt, yuchuan.yc, xiyu.xj}@alibaba-inc.com, +{fwu, gchen}@cs.sjtu.edu.cn +Abstract +In online advertising, automated bidding (auto- +bidding) has become a widely-used tool for ad- +vertisers to automatically make bids on different +impressions in real time. +Instead of submitting +bids for each single impression, advertisers in auto- +bidding submit their high-level objectives and con- +straints to the auto-bidding tool, and observe the +cumulative advertising performances after all the +auctions within a time period have been finished. +Motivated by the features of automated bidding, we +aim to design auctions with private financial con- +straints for value-maximizing bidders. Specifically, +we consider budget and ROI, the two most com- +mon financial constraints in online advertising, as +the private information of advertisers, and analyse +the conditions of truthfulness. We show that ev- +ery non-decreasing function with budget as input +could be mapped to a truthful auction mechanism +with budget and ROI as input, but this mapping +procedure also introduces complex value group- +ing structures into mechanism design. To achieve +feasible and implementable auctions, we design a +truthful auto-bidding auction mechanism with ad- +justable rank score functions. As the key design to +guarantee truthfulness, our auction utilizes the bid- +der’s budget constraint to compute a critical ROI, +which enables comparisons between the budget and +ROI constraint. We conduct experiments under dif- +ferent auto-bidding settings to validate the perfor- +mance of our proposed auction in terms of revenue +and social welfare. +1 +Introduction +With the rapid development of machine learning techniques, +automated bidding (auto-bidding) has been widely adopted +by various online advertising platforms [Facebook Ads, 2021; +Google Ads, 2021]. In auto-bidding services, advertisers sub- +mit their high-level optimization objectives and constraints to +the platform, and then the bidding agents, powered by ma- +chine learning algorithms, make detailed bidding decisions +in each of ad auctions on behalf of the advertisers. +With +the help of automated bidding tools, advertisers can optimize +their overall advertising objectives with respect to their finan- +cial constraints in a high-level way. +Under automated bidding, we revisit a fundamental prob- +lem in auction theory: +whether the conventional auction +model, where advertisers have private values for items (i.e., +ad impressions) and conduct corresponding strategic bidding +for each single auction, is still appropriate for the new adver- +tising paradigm. As the platform can access historical data +about the interactions between advertisers and users, we can +estimate the potential actions of users (such as clicks and con- +versions), which can be regarded as public values of items for +advertisers. In auto-bidding, the private information from ad- +vertisers are actually their constraints for the whole advertis- +ing campaign. These features require a new ad auction model +to incentivize advertisers to truthfully reveal the high-level +private constraints given the values of items are public. +In this work, we consider a new automated bidding auction +model, where advertisers submit budget as well as return on +investment (ROI) requirement as their (private) constraints, +and aim to maximize the cumulative values of winning im- +pressions from multiple auctions during a certain period. We +analyse the truthfulness conditions with respect to the private +constraints of budget and ROI. Remarkably, we show that +any truthful auction mechanism for this multi-dimensional +setting could be equivalently represented by a series of non- +decreasing functions with budget as input. When these non- +decreasing functions are realized to derive the correspond- +ing auction mechanism, the truthful conditions of budget and +ROI introduce a new value grouping phenomenon: different +budget-ROI types are grouped to share the same cumulative +value, and the grouping pattern is determined by a threshold +ROI function (transformed from the above non-decreasing +function). As the threshold ROI functions are not constrained +in monotonicity, the grouping shape of budget-ROI types is +generally irregular, making it difficult to obtain the closed- +form expression of grouping types and then the auction opti- +mization objectives, such as revenue and social welfare. +Facing these design difficulties, we propose a family of ad +auctions with personalized rank scores to optimize various +design objectives. Our auction adopts rank scores as the cri- +teria to determine item allocations, which shares the similar +ideas with Generalized Second-Price (GSP) auction [Edel- +man et al., 2007]. In guaranteeing truthfulness for the pri- +vate constraints, we design critical ROI to be the largest ROI +arXiv:2301.13020v1 [cs.GT] 30 Jan 2023 + +that can win the most items without breaking the budget con- +straint. It equivalently transforms budget to the same dimen- +sion as ROI, thus allowing us to find a tight constraint limits +the bidder from getting extra utilities and utilize this tight con- +straint to prevent misreporting. We conduct extensive exper- +iments to evaluate the performances of the proposed auction +mechanism under various auto-bidding settings. The evalu- +ation results demonstrate that the designed truthful auction +can achieve more than 90% performance (in terms of rev- +enue and social welfare) of the optimal baselines without the +consideration of truthfulness. The main contributions can be +summarized as follows: +• We consider the unique features within the interaction be- +tween advertisers and online platforms in the context of au- +tomated bidding. Based on these features, we formulate a +new auto-bidding auction model, where value-maximizing +bidders have high-level constraints as private information and +the values of items are public. +• We investigate truthfulness conditions of two-dimensional +private constraints, budget and ROI, under public value set- +ting. We provide full characterizations for the feasible space +of allocation and payment for truthful auctions. +• Based on the derived truthfulness conditions, we design a +family of truthful ad auction mechanisms for automated bid- +ding. With the newly designed rank score functions, the pro- +posed ad auction is simple and flexible to be adapted into var- +ious auto-bidding settings with different optimization goals +such as revenue and social welfare. +• We evaluate our proposed auction mechanisms with vari- +ous experimental settings, and the empirical results validate +the effectiveness of the proposed auctions regarding its per- +formances in terms of social welfare and revenue. +2 +Preliminaries +In this section, we first motivate the considered auction de- +sign problem by the online advertising system with auto- +mated bidding services, and afterwards propose the formal +auction model based on the features of automated bidding. +2.1 +Online Advertising System +The working process of the online advertising auction system +is illustrated in Figure 1. From an advertiser’s perspective, it +can be described as follows: +1) The advertiser sets the bidding configuration in auto- +bidding interface: chooses optimization objectives (e.g., max- +imizing clicks or conversions) and sets cost constraints (e.g., +budget per day, targeted return on investment (ROI) and max- +imum cost per click). The advertiser requires her realized +ROI, defined as the ratio of her gained value and payment, to +be higher than her targeted ROI. +2) Based on the advertiser’s configuration, an auto-bidding +agent represents the advertisers to make bid decisions in mul- +tiple auctions. When each user impression comes, the auto- +bidding agents attend an ad auction to compete for the ad dis- +play opportunities. +3) The auto-bidding agent adopts data-driven algorithms to +predict the value (click through rate or conversion rate) of the +incoming user impression, and bid for each impression while +taking the cost constraints into consideration. +Ad platform +Allocation +Pricing +Objective and +Constraints +Cumulative +Ad Effects +User +Features +Objective and +Constraints +Current +Outcomes +Value & Bid +Calculation +Auto-bidding +Agents +Bids +Outcomes +Auto-bidding +Interface +Budget: 30$ +ROI: 1.5 +Budget: 20$ +ROI: 1.8 +Budget: 40$ +ROI: 1.6 +Budget: 25$ +ROI: 2 +Incoming User Impressions +1 +2 +3 +4 +Figure 1: Ad Auction System Overview +4) During the multiple auctions, the advertiser can check +the cumulative auction outcomes, including spent budget, re- +ceived impressions, clicks, conversions and the average costs. +The advertisers can further adjust their auto-bidding settings +in the interface to achieve their own advertising objectives. +From the above interaction between advertisers and the +platform/auctioneer, we summarize three new features about +the online advertising with auto-bidding services. First, ad- +vertisers only report high-level optimization objectives and +constraints in the bidding configurations for multiple auc- +tions, but not the fine-grained bid for each auction. Second, +advertisers only evaluate cumulative long-term performances +and costs of multiple auctions, instead of the outcome of each +individual auction. Third, since the online platform can ac- +cess all the data produced in advertising, it is reasonable to +claim the monetary value of an incoming impression to a spe- +cific advertiser can be calculated exactly. As the behavior pat- +terns of advertisers and the online platform change distinctly +in automated bidding, we need to investigate the mechanism +design whose formats align with these new features. +2.2 +Auction Model +Based on the motivations of the above online advertising sys- +tem with auto-bidding, we propose the formal auction model +considered in this work. There are n advertisers competing +for m items (user impressions) coming in sequence during a +time period with m time slots, where only one item would ap- +pear in each time slot.1 The advertisers are value-maximizing +bidders [Fadaei and Bichler, 2016; Balseiro et al., 2022a; +Mehta, 2022], who care about the cumulative value of her +allocated impressions across all the time slots when the pay- +ment is within their financial constraints. Each advertiser i +has budget (Bi ≥ 0) and ROI (Ri > 0) constraints, which are +private information and are also called as type ti = (Bi, Ri) +in mechanism design literature. We denote the type profile of +1We would use advertisers with bidders, and items with impres- +sions interchangeably throughout the work. + +all the advertisers as t = (ti)n +i=1 and the space of the type +profile as T = �n +i=1 Ti with t ∈ T and Ti = Bi × Ri. We +denote the reported type of the other bidders except bidder +i as t−i = (t1, . . . , ti−1, ti+1, . . . , tn) and T−i = �n +k̸=i Tk. +We assume advertisers’ valuations on items are public infor- +mation to the platform. We use vi,j > 0 to represent the +advertiser i’s valuation on item j. +After collecting the budget and ROI of all the bidders, the +online platform employs some auction mechanism (A, P) to +decide ad allocations and payments, where A denote a (ran- +domized) allocation rule A : T → [0, 1]n×m and P denote a +(randomized) payment rule P : T → Rn. Specifically, for a +reported type profile t′ ∈ T , the probability of bidder i being +allocated item j is denoted by ai,j(t′) and the expected pay- +ment of bidder i is denoted by pi(t′). For any item j ∈ [m], +the allocation constraint is �n +i=1 ai,j(t′) ≤ 1. Bidder i’s cu- +mulative value in these auctions is +vi(t′) = +m +� +j=1 +vi,jai,j(t′) +and her realized ROI is ROIi(t′) := vi(t′)/pi(t′) (consid- +ered as +∞ if pi(t′) = 0). +The utility for a bidder with budget and ROI constraints +and true type ti when reporting t′ +i is defined as +ui (ti, t′) = +�vi(t′), if pi(t′) ≤ Bi and ROIi(t′) ≥ Ri, +−∞, otherwise, +where the type profile t′ = (t′ +i, t′ +−i). +In this work, we focus on designing truthful auction mech- +anisms satisfying incentive compatible (IC) and individual +rationality (IR) conditions for budget and ROI, which is a +two-dimensional mechanism design problem. Various multi- +dimensional mechanism design problems are shown to be dif- +ficult in both analytical and computational aspects [Pavlov, +2011; Chen et al., 2014; Daskalakis, 2015]. In particular, +we consider dominant-strategy incentive compatible (DSIC) +direct-revelation mechanisms. +Definition 2.1. An auction mechanism is dominant strategy +incentive compatible (DSIC) if ∀i ∈ [n], ti, t′ +i ∈ Ti, t−i ∈ +T−i: ui(ti, (ti, t−i)) ≥ ui(ti, (t′ +i, t−i)). +Similarly, we have the corresponding individual rationality +(IR) condition as follows. +Definition 2.2. An auction mechanism is individual rational- +ity (IR) if ∀t ∈ T , i ∈ [n]: pi(t) ≤ Bi and ROIi(t) ≥ Ri. +The online platform typically has some objectives to max- +imize. Two common design objectives are revenue and so- +cial welfare. Revenue is defined as the sum of payment from +bidders, i.e., � +i pi(t). For social welfare, we need a synony- +mous metric, liquid welfare, defined as the maximum revenue +that can be extracted from bidders without breaking IR con- +straints, to incorporate the existence of financial constraints +[Azar et al., 2017; Aggarwal et al., 2019]. In our context, +liquid welfare can be defined as +LW = +� +i∈[n],ui(ti,t′)≥0 +min +�vi(t′) +Ri +, Bi +� +. +(1) +3 +Characterization of Truthfulness +3.1 +Conditions for Truthfulness +In this section, we investigate the truthfulness conditions for +this multi-dimensional mechanism design problem. Due to +the limitation of space, we present the detailed proofs of our +results in the appendix. +To provide an intuition about the differences between auc- +tion models with private constraint and private valuation, we +start from the analysis of one private constraint. To satisfy the +IC property on budget, we need to guarantee the bidders for +not obtaining a higher utility by reporting a smaller or larger +budget. As reporting a smaller budget will not lead a bidder to +break her original budget constraint, the gained utility should +decrease for reducing the budget. For the bidder misreport- +ing a larger budget and obtaining higher values, we need to +charge the bidder to break her original budget constraint, re- +sulting in negative infinite utility to prevent misreporting. +Theorem 3.1. An auction mechanism is DSIC on budget B +only if ∀i ∈ [n], R ∈ Ri, t−i ∈ T−i: +(1) vi ((B, R), t−i) is non-decreasing in B; +(2) If vi ((B′, R), t−i) > vi ((B, R), t−i) for B′ > B, then +pi ((B′, R), t−i) > B. +We can form a similar statement for ROI by showing that +there is no incentive for a bidder to misreport her ROI. +Theorem 3.2. An auction mechanism is DSIC on ROI R only +if ∀i ∈ [n], B ∈ Bi, t−i ∈ T−i: +(1) vi ((B, R), t−i) is non-increasing in R; +(2) If vi ((B, R′), t−i) > vi ((B, R), t−i) for R′ < R, then +ROIi ((B, R′), t−i) < R. +Although the monotone properties required by conditions (1) +in Theorem 3.1 and 3.2 seem similar to the truthfulness condi- +tions for quasi-linear bidder with private valuations, e.g., My- +erson’s Lemma [Myerson, 1981], conditions (2) reveal their +basic differences: assigning a payment to break at least one +constraint (lead to negative infinite utility) is indispensable to +prevent misreporting for private constraints. +Theorem 3.1 and 3.2 naturally provide necessary condi- +tions on the DSIC of two-dimensional type (B, R). The fol- +lowing result shows that these conditions also provide the suf- +ficient conditions. That is, if a bidder cannot obtain higher +utility through misreporting one of her constraints, misreport- +ing the two constraints can also not obtain higher utility. +Theorem 3.3. An auction mechanism is DSIC on both budget +and ROI if and only if it satisfies Theorem 3.1 and 3.2. +Theorem 3.3 is proved through showing none of the mis- +reporting may bring the bidder higher utility when conditions +in Theorem 3.1 and 3.2 hold, which relies on the mathemat- +ical relationship between payment and ROI. In Theorem 3.3, +the truthful conditions in Theorem 3.1 for budget appear to +be independent with the truthful conditions in Theorem 3.2 +for ROI, and satisfy these conditions simultaneously could +achieve the two-dimensional truthfulness. Nevertheless, the +payment term actually appear in both the conditions (recall +ROI = v/p in condition (2) of Theorem 3.2), i.e., we have +to use the same payment scheme to satisfy these two sets of + +conditions. In order to step toward the full characterization of +truthfulness, we need to further analyse how payment is influ- +enced and constrained by financial constraints and allocation. +To facilitate our discussion, we define an allocation rule A +to be feasible if there exists a payment rule P such that the +mechanism (A, P) is truthful. The set of feasible allocation +rule is the space we can search for truthful auctions. Although +there may exist multiple payment rules that constitute a truth- +ful auction for a feasible allocation rule A, the following the- +orem allows us to focus on the maximum payment rule. +Theorem 3.4. For a feasible allocation rule A, assigning +pi ((Bi, Ri), t−i) = min (vi ((Bi, Ri), t−i) /Ri, Bi) , (2) +for ∀i ∈ [n], (Bi, Ri) ∈ Ti, t−i ∈ T−i constitutes a truthful +auction mechanism. +Theorem 3.4 naturally holds because each type is charged +the maximum payment under her financial constraints. +If +some other payment could constitute a truthful auction with +A, then improving the payment to this maximum amount +would not break the originally held conditions (2) in Theo- +rem 3.1 and Theorem 3.2. By Theorem 3.4, given any feasi- +ble allocation rule A, we could find a corresponding payment +rule to constitute a truthful auction. As the bidders do not +evaluate detailed allocation results of each individual auction, +searching for the truthful allocation rule is equal to designing +cumulative value functions vi((Bi, Ri), t−i) that can be real- +ized. Thus, we could use the corresponding cumulative value +function to represent a feasible allocation rule. +Our next step is to further shrink the consideration space +of comparison in conditions of truthfulness. +Previous +conditions in Theorem 3.3 include comparisons between any +two type sharing the same budget or ROI, e.g., comparing +any (B, R) and (B′, R) in Theorem 3.1, which is still a +large consideration space and leads to difficulties in finding +the cumulative value functions. To enable analysis between +“neighbouring” types, for any allocation rule A and its +cumulative value function v(t), we assume the limitations +limB→B− +i vi ((B, Ri), t−i) and limR→R+ +i vi ((Bi, R), t−i) +exist for any t ∈ T , i ∈ [n]. Through considering these +infinitely close types and substituting the payment terms in +Theorem 3.3 by the payment rule (2), conditions that con- +strain the payment or realized ROI of misreporting another +type to break the original constraints could be converted to +conditions regarding cumulative value of the type itself. +Theorem 3.5. An allocation rule A can derive a truthful auc- +tion if and only if ∀t ∈ T , i ∈ [n]: +(1) The cumulative value vi ((B, R), t−i) is non-decreasing +in B for R = Ri and non-increasing in R for B = Bi; +(2) If vi ((Bi, Ri), t−i) > limB→B− +i vi ((B, Ri), t−i), then +vi ((Bi, Ri), t−i) ≥ Bi × Ri; +(3) If vi ((Bi, Ri), t−i) > limR→R+ +i vi ((Bi, R), t−i), then +vi ((Bi, Ri), t−i) ≤ Bi × Ri. +3.2 +Structures of Feasible Allocation Rule +Theorem 3.5 fully characterizes the conditions of a feasible +allocation rule. In this subsection, we would further exploit +0 +1 +2 +3 +4 +5 +6 +7 +8 +Bi +0 +1 +2 +3 +4 +5 +Ri +thri(Bi) +(B1, R1) +(B2, R2) +Accumulated Value vi(Bi, Ri) +0 +5 +10 +15 +20 +25 +Figure 2: An example of mapping from a threshold ROI function to +cumulative values fixing the others’ types t−i. +the structures of feasible allocation rule indicated by Theorem +3.5 to provide more instructions on truthful auction design. +By Theorem 3.5, the relation between cumulative value v +and the term B × R strictly characterizes whether this type +shares the same cumulative value with its neighbouring types, +which allows us to further clearly represent the structure of +v. Fixing the budget, with the decrease of ROI, the cumula- +tive value assigned to the type should be increasing, notice +the term B × R decreases along with R; however, condi- +tion (3) in Theorem 3.5 requires that if the cumulative value +strictly increases in this process, then its value should not ex- +ceed B ×R. Thus, there must exist some threshold ROI, such +that the increasing cumulative value intersects with the de- +creasing B × R, and the cumulative value could not increase +anymore. Based on the above observation, we define +thri(B, t−i) = sup +R∈Ri +(vi ((B, R), t−i) ≥ B × R) +if the considered set is non-empty and bounded, or otherwise +be 0.2 It turns out that the cumulative value assigned to this +threshold ROI is exactly B × R. +Theorem 3.6. ∀i ∈ [n], t−i ∈ T−i, Bi ∈ Bi, Ri ≤ +thri(Bi, t−i): +vi ((Bi, Ri), t−i) = thri(Bi, t−i)Bi. +We could then combine different budgets together to ob- +tain the complete characterization of feasible allocation in the +two-dimensional type space. Since all types above the thresh- +old ROI has cumulative value v < B × R by the definition of +thr function, they could not satisfy the condition (2) in The- +orem 3.5, and thus are forced to have the same cumulative +value with their neighbouring types with a smaller budget. +Theorem 3.7. ∀i ∈ [n], t−i ∈ T−i, Bi ∈ Bi\{0}, Ri > +thri(Bi, t−i): +vi((Bi, Ri), t−i) = +lim +B→B− +i +vi ((B, Ri), t−i) . +Since the types below (Theorem 3.6) and above (Theo- +rem 3.7) the ROI threshold have both been fully described, +the cumulative values on the entire type space could be fully +defined if thri(Bi, t−i) is given and t−i is fixed. We provide +an example in Figure 2 to illustrate these characterizations. +2This set would be empty only when vi ((B, R), t−i) = 0 for +all R ∈ Ri, and unbounded only when B = 0. + +For types below the threshold ROI curve and have the same +budget B (i.e., types lie in the same vertical line), they share +the same cumulative value thri(B)×B. For each type above +the curve, its cumulative value is equal to that of its neigh- +bouring type with a smaller budget until it reaches some type +on the threshold ROI curve. This is equivalent to search hor- +izontally to the left to find the closest type on its correspond- +ing ROI threshold, where all the types along this horizontal +search share the same cumulative value. +We have shown that each feasible two-dimensional cumu- +lative value function (whose input is (B, R)) for a single +bidder could be represented by a one-dimensional threshold +ROI function (whose input is B) when other bidders’ pro- +file is fixed, but it is unknown what kinds of one-dimensional +threshold ROI functions could represent a feasible cumulative +value function. By condition (1) in Theorem 3.5, the cumula- +tive value when R is infinitely close to 0 should be increasing +for different B, which indicates thri(B, t−i) × B needs to +be non-decreasing in B. Since every cumulative value func- +tion mapped from a non-decreasing thri(B, t−i) × B satis- +fies Theorem 3.5, this becomes the only requirement for a +feasible thri function. For convenience in notations, we de- +note gi(Bi, t−i) = thri(Bi, t−i) × Bi. We denote V as the +space of feasible cumulative value function when i and t−i +are fixed, i.e., vi : Ti → R+, and denote G as the space of +non-decreasing function g : R+ → R+ with g(0) = 0. +Corollary 3.8. There exists a bijective map m : G → V with +(m◦g)(B, R) = +� +� +� +� +� +� +� +g(B) +if R ≤ g(B) +B +, +g +� +sup +B′≤B +{R ≤ g(B′) +B′ +} +� +otherwise, +where we define sup ∅ = 0. +In other words, given any one-dimensional non-decreasing +function with g(0) = 0, we could transform it to a feasible3 +cumulative value function by the above mapping process. +4 +Truthful Auction Design +The derived structure of feasible allocation rules give rise +to a value grouping phenomenon. When reversing the pro- +cess of finding the cumulative value of a certain type given +threshold ROI curve, every type on the threshold ROI curve +shares the same cumulative value with two groups of types: +the types vertically below it, as well as the types locate hor- +izontally right to it and above its corresponding threshold +ROI, as shown in Figure 2. This intricate value grouping +phenomenon are enforced by the truthful conditions instead +of the optimization requirement. As we can observe in Fig- +ure 2, since the horizontal search terminates when meeting +a type on the threshold ROI curve, the grouping shape is +determined by the relative rank of thri(Bi, t−i) for differ- +ent Bi. However, though thri(Bi, t−i) is required to keep +g(B) non-decreasing, it is not required to be monotone it- +self, which leads the terminate of horizontal search and the +3Our feasibility is with respective to truthfulness, and does not +specify how to realize the allocations from detailed items. +Algorithm 1: A Family of Simple Truthful Auctions +Input: Bidder’s reported type (Bi, Ri), and +non-increasing rank score functions fi,j. +Output: Bidder’s allocated items Ai and payment pi. +1 Initialize Ai = {} for i ∈ [n]; +2 Compute virtual bids bi,j ← vi,j × fi,j(Ri); +3 for each item j ∈ [m] do +4 +Find bidder i0 with the highest virtual bid bi0,j; +5 +Record the second highest bid cj ← maxi̸=i0 bi,j; +6 +Add item j into Ai0 and set ai0,j = 1; +7 for each bidder i ∈ [n] do +8 +Compute ROI ri,j = f −1 +i,j ( cj +vi,j ) required for +winning the item j ∈ Ai; +9 +Find a largest Rc +i ∈ Ri s.t. +� +j∈Ai I (Rc +i ≤ ri,j) × vi,j +Rc +i ≥ Bi, and let +d ← the difference between left-hand side and +right-hand side of the above inequality; +10 +if d > 0 then +11 +Find the items j′ with ri,j′ = Rc +i, and remove +the items with value d/Rc +i; +12 +if all of these items are removed then +13 +Round up Rc +i to the smallest ri,j > Rc +i; +14 +if Ri ≤ Rc +i then +15 +Remove the items with ri,j < Rc +i from Ai; +16 +pi ← min +� � +j∈Ai vi,j·ai,j +Ri +, Bi +� +; +17 return (Ai, pi) for i ∈ [n]. +grouping shapes to be irregular. This also brings difficulties +in deriving general closed-form conditions for the group of +types to share the same allocation. Furthermore, although the +cumulative values of the grouping types are forced to be the +same, their payment are calculated by Theorem 3.4, leading +to varying payments between grouping types and the non- +linear revenue with respective to the threshold ROI. +The above characteristics of value grouping phenomenon, +i.e., irregularity in grouping shape and non-linear payment, +make any analytical or mathematical programming approach +difficult to adopt. To reach feasible and implementable auc- +tion design, we propose a family of simple truthful auctions +based on newly designed rank score functions. +The detailed auction mechanism is presented in Algo- +rithm 1. For each item, bidders are ranked based on their +virtual bids bi,j, which is defined as vi,j multiplying a rank +score fi,j(Ri) with some pre-defined non-increasing function +fi,j (Line 2). The items are then temporarily allocated to each +bidder with the highest virtual bids (Line 4-6). After the can- +didate allocation set Ai for each bidder i has been determined, +we compute the the corresponding ROI requirement ri,j for +the bidder i to rank the first in item j ∈ Ai, i.e., bidder i needs +to report Ri ≤ ri,j in order to maintain ranked first for item j +(Line 8). The critical ROI in Line 9 is our key design to guar- +antee truthfulness, which computes the largest ROI a bidder +could report to win enough items in Ai and spend out her +budget. Intuitively, critical ROI simulates the best-response + +ROI of the bidder given her budget constraint and keeps in- +dependent of the bidder’s true ROI constraint. We naturally +utilize this critical ROI, which transforms the budget to the +same dimension of ROI, as the threshold ROI function of our +auction, and design the remaining parts based on the truthful +conditions in Section 3. Line 10-13 are designed to guarantee +the value of remaining items in Ai would exactly equal to the +budget of the bidder times the computed critical ROI (The- +orem 3.6). Line 14-15 are designed to guarantee the types +with ROI larger than the corresponding critical ROI would be +allocated based on their reported ROI (Theorem 3.7) and get +all items with ri,j ≥ Ri. Line 16 determines the payment for +each bidder i based on Theorem 3.4. +Theorem 4.1. If the pre-determined functions fi,j(R) are +non-increasing with R, then the auction mechanism corre- +sponding to Algorithm 1 is truthful. +Our proof is conducted through figuring out and verifying +the corresponding g(B) and threshold ROI function of Algo- +rithm 1. We can design personalized fi,j(R) for bidders and +items according to the auction objective and prior knowledge +before the advertising campaign, such as the type distribution +of bidders and the information of incoming items. As the cost +of bypassing the design hardness brought by value grouping +of multiple bidders, our mechanism may not allocate all the +items if the corresponding highest ranking bidder has fully +consumed her budget. Nevertheless, the auctioneer could take +multiple approaches to alleviate this phenomenon. While ad- +justing the functions fi,j(R) with the pre-known information +is the most direct approach, dividing advertisers into groups +and circulating the unsold items could also be effective. +5 +Experiments +In this section, we conduct experiments with synthetic data to +validate the performance of our proposed auctions. +5.1 +Experimental Setup +There are two sets of experiments with i.i.d. and non-i.i.d. +bidders, respectively. We vary the number of bidders and +items to simulate the changes in demand and supply. The +presented results are averaged by 50 runs. We choose distri- +bution parameters based on the common fluctuations in ad- +vertising markets, and normalize the lower bound of vi,j. +Symmetric bidders +Bidders are symmetric with vi,j ∼ +U[1, 4], Bi ∼ U[40, 80] and Ri ∼ U[1, 3], where U[a, b] +is the uniform distribution within the range [a, b].4 +Mixed bidders +There are low and high distributions for +vi,j, Bi and Ri as bidders’ possible types. +specifically, +vi,j ∼ U[1, 2] or vi,j ∼ U[2, 3], Bi ∼ U[20, 40] or Bi ∼ +U[80, 100], and Ri ∼ U[1, 2] or Ri ∼ U[2, 3]. Combinations +of these categories result in 8 groups of bidders. +Baseline Auctions Since no existing mechanism guarantees +the IC properties of both budget and ROI, we consider com- +mon repeated auction formats: first-price and second-price +auctions. Due to their non-truthfulness (examples provided +in the appendix), we involve misreporting for these auctions. +4We have tested several other parameters under uniform distri- +bution, and the trends of results are the same. +• Repeated first-price auctions: +The auctioneer holds first- +price auctions for every single item, where bidder i bids +vi,j/Ri for item j. The auctioneer allocates the item to the +highest ranking bidder who has remaining budget to afford, +and charge the first-price payment. We aim to simulate bid- +ders’ misreporting behaviors by the classical best-response +dynamics with the true profile as starting points. In repeated +first-price auctions, the bidders have no incentive to misre- +port their budget. We calculate the best response ROI as the +smallest ROI that wins the most items in historical auctions +without breaking the original financial constraints. +• Repeated second-price auction: This auction format is sim- +ilar to repeated first-price auction with second-price auction +used in each round instead. We still simulate bidders’ misre- +porting behaviors to be the best-response dynamics as above, +with payment changed to be the second price, where the best- +response ROI here may be larger or smaller than the true ROI. +• Non-truthful Optimal Baseline: The optimal offline revenue +could be computed through linear programming, where we +set each bidder i’s payment to item j as exactly vi,j ×ai,j/Ri, +and her total payment is constrained to not exceed Bi. +Evaluation Metrics We consider revenue and liquid welfare +as the optimization goal. Since the payment formula (2) in +Theorem 3.4 aligns with the definition of liquid welfare in +equation (1) when no bidder receives negative infinite util- +ity, the metrics of revenue and liquid welfare would be the +same for truthful auctions, and we will thus only report rev- +enue in our results. Besides, as advertisers may have fluctu- +ating auction performances across time slots, fairness should +also be considered in order to preserve bidders’ willingness +to attend the auctions. +Using liquid welfare to substitute +the traditional valuation function [Bez´akov´a and Dani, 2005; +Chakrabarty et al., 2009], fairness is defined as +fairness = min +i∈[n] min +�vi(t′) +Ri +, Bi +� +. +Rank Score Function +We adopt rank scores in the form +fi,j(R) = αi,j × e−βR, where αi,j is drawn from a recti- +fied normal distribution N(µi, σ2 +i ), and β, µi, σi are pre-set +parameters. In f(R), β is used to adjust the impact of ROI on +equivalent bids, and α maintains the rank scores of various +bidders in a comparable range. Since items may be discarded +when some bidders win excessive items in the proposed auc- +tion (Line 15 in Algorithm 1), in order to avoid loss in welfare +and revenue, we should give other bidders non-trivial win- +ning opportunities when the supply is sufficient, which is pro- +vided by the randomness in α ∼ N(µi, σ2 +i ). For each auto- +mated bidding environment with different setting of bidders +and items, we set rank score function parameters to be the +same for bidders following the same distribution, and choose +the parameters with better revenue in this environment. +5.2 +Experimental Results +Symmetric bidders The revenue of our truthful auction (re- +ferred as DSIC) and the baseline auctions with different num- +ber of bidders is reported in Figure 3a, and revenue when +number of items changes is reported in Figure 3b. +Our +truthful auction achieves better revenue than first-price and +second-price auctions, and achieves more than 90% revenue + + 200 + 250 + 300 + 350 + 400 + 450 + 500 + 550 + 600 + 650 + 5 + 15 + 25 + 35 + 45 + 55 + 65 +Revenue +Number of bidders +OPT +DSIC +First-Price +Second-Price +(a) + 400 + 600 + 800 + 1000 + 1200 + 1400 + 1600 + 1800 + 2000 + 2200 + 2400 + 200 400 600 800 1000 1200 1400 1600 +Revenue +Number of items +OPT +DSIC +First-Price +Second-Price +(b) + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 200 400 600 800 1000 1200 1400 1600 +Fairness +Number of items +OPT +DSIC +First-Price +Second-Price +(c) + 860 + 870 + 880 + 890 + 900 + 910 + 920 + 930 + 940 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 +β=0 +β=0.01 +β=0.02 +β=0.03 +β=0.04 +β=0.05 +β=0.06 +β=0.07 +Revenue +β=0 +β=0.01 +β=0.02 +β=0.03 +β=0.04 +β=0.05 +β=0.06 +β=0.07 +Revenue +Fairness +(d) +Figure 3: Revenue and fairness in different experiment setting for symmetric bidders: (a) Revenue for different number of i.i.d bidders with +200 items; (b) Revenue for different number of items with 40 i.i.d bidders; (c) Fairness for different number of items with 40 i.i.d bidders; (d) +Fairness of our DSIC auction with 40 i.i.d bidders and 400 items using different parameter β in rank score functions; + 0 + 400 + 800 + 1200 + 1600 + 2000 + 2400 +Setting 1 +Setting 2 +Setting 3 +Revenue +Number of items +OPT +First-Price +Second-Price +DSIC-1 +DSIC-2 +DSIC-3 +Figure 4: Revenue with 40 mixed bidders in Setting 1: 200 items; +Setting 2: 1000 items; Setting 3: 1600 items; with three different +groups of rank score functions for our DSIC auction +of the near optimal baseline. +Remarkably, first-price auc- +tions displays evident decrease in revenue when the items are +excessive, which also leads to decreases in fairness (Figure +3c). This is because the bidders exceedingly increase their +reported ROI when they face some unsold items with low +competing prices and have to pay proportionally for them. +In contrast, for second-price auctions, the payment of unsold +items only consume a negligible part of budget when com- +puting best-response ROI, and the reported ROI of bidders +maintains relatively small. +In Figure 3c, we report the fairness results under different +number of items a fixed number of 40 bidders as in Figure 3b. +As designed in the rank score function, our auction does not +allocate excessive items to a certain bidder to avoid waste of +items, and thus achieves better fairness then the optimal base- +line when the number of items is in the range of 600 to 1200. +When items are in shortage (200/400 items in Figure 3c), our +truthful auction harms the fairness to some extent due to the +pursuit of revenue. We can control the tradeoff between fair- +ness and revenue with different rank score functions through +the parameter β. Figure 3d reports a set of revenue and fair- +ness performances with different β in the same setting as 400 +items in Figure 3c. To achieve higher revenue, it is inevitable +to have discriminating allocation among different bidders and +harms the fairness when items are not abundant. +Mixed bidders Instead of presenting trends similar to i.i.d. +setting, we demonstrate the importance of choosing appro- +priate rank score functions in non-i.i.d. settings. We choose +three groups of rank score functions respectively with good +performance in three typical settings: +40 mixed bidders +with 200/1000/1600 items, and test their performances in +the other two settings. The results are presented in Figure +4, where DSIC-n represents the truthful auction with rank +score functions performing well in the Setting n. The DSIC- +n mechanism only perform well in Setting n, and the reason +should come from the distinct allocation approaches needed +to achieve high revenue for different number of items. For +example, to achieve high revenue when items are in short- +age (Setting 1), DSIC-1 auction allocates most items to the +bidders with high value and low ROI, leading to allocating +excessive items to these bidders and neglecting other bidders +in Setting 2 and 3. +6 +Related Work +Auctions in online advertising has attracted much attention +from game theory field. Initiated by [Varian, 2007; Edelman +et al., 2007], the early work studied the properties of keyword +auctions in sponsored search, where bidders set a fixed bid for +all the queries with a certain keyword. An extensive survey +on this topic can be found at [Qin et al., 2015]. +Due to the distinct features compared to traditional ad +auctions, auction design and game theoretical analysis for +automated bidding come into researchers’ view in recent +years. Starting from [Aggarwal et al., 2019], a line of work +takes the uniform pacing assumption to analyse the equi- +librium and efficiency between automated bidding agents, +where the agent applies a same pacing parameter to the +true value of each item as their bids. [Deng et al., 2021; +Balseiro et al., 2021a] studied the usage of boosting and re- +serve price to increase the efficiency and revenue of auto- +bidding auctions. [Chen et al., 2021; Conitzer et al., 2021; +Conitzer et al., 2022] analysed the existence and computa- +tional complexity of pacing equilibrium under different set- +tings. [Mehta, 2022] designed a truthful auction with ran- +domization to improve the price of anarchy in auto-bidding +settings. [Deng et al., 2022] studied the efficiency of first- +price auctions under a more generalized setting with no uni- +form pacing assumption, and [Liaw et al., 2022] designed +a non-truthful auction to further improve the price of an- +archy. Some recent work assumed agents adopting regret- + +minimizing learning algorithms to adjust the bid or pacing +parameter, and analysed the convergence as well as game +properties [Gaitonde et al., 2022; Fikioris and Tardos, 2022; +Kolumbus and Nisan, 2022]. +Our work could be classified as considering the incentive of +bidders when setting automated bidding parameters. Under +this topic, [Li et al., 2020] provided distributed bidding algo- +rithms for IC issues assuming public values and private ROI. +[Balseiro et al., 2021b] studied the optimal mechanism design +for bidders with ROI constraint under different information +structure. [Kolumbus and Nisan, 2021] considered the incen- +tive of parameter setting for agents in general games. As one +of the most relevant work, [Balseiro et al., 2022b] considered +the mechanism design with private ROI constraint and public +budget, and derived the optimal auction for all the two-bidder +cases and some specific multi-bidder cases based on elabo- +rate ironing techniques. Distinct from the above work, our +work considers both budget and ROI as private constraints, +which aligns with the real automated bidding and reveals the +complex value grouping phenomenon for the first time. +Another closely relevant line of work is mechanism de- +sign for financially-constrained bidders. Since [Laffont and +Robert, 1996], intensive research has been conducted for bid- +ders with budget constraint and progress gradually [Borgs et +al., 2005; Bhattacharya et al., 2010; Dobzinski et al., 2012; +Pai and Vohra, 2014]. +[Szymanski and Lee, 2006] anal- +ysed the impact of ROI constraints on bidding and revenue of +several common auction forms, and [Golrezaei et al., 2021] +firstly considered auction design for bidders with ROI con- +straints. Differing from those existing work studying quasi- +linear bidders with private values and one financial constraint, +our work consider value-maximizing bidders with multiple +private constraints in the automated bidding environment. +From the computational perspective of multi-dimensional +mechanism design, the representing series of work [Cai et al., +2012; Alaei et al., 2012; Cai et al., 2021] provided a black- +box linear programming framework to the classical multi- +item auction design problem under Bayesian Incentive Com- +patibility (BIC). However, due to the “if, then” structure in +our truthfulness conditions (Theorem 3.5) and the resulting +value grouping phenomenon, those existing linear program- +ming techniques could not be simply adapted to our problem. +7 +Conclusion +In this work, we have considered truthful auto-bidding auc- +tion design for bidders with private budget and ROI con- +straints across multiple impressions, while the values of im- +pressions are public to the auctioneer. We have characterized +the truthfulness conditions in this new auto-bidding auction +model, which involves irregular grouping constraints on bid- +ders’ cumulative utilities. We have proposed a series of sim- +ple truthful auction mechanisms with flexible rank score func- +tions as a solution to this automated bidding auction design +problem. 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Citeseer, +2006. +[Varian, 2007] Hal R Varian. +Position auctions. +Interna- +tional Journal of Industrial Organization, 25(6):1163– +1178, 2007. + +A +Detailed Proofs for Results +During the proofs of results, for ease of notations, we may +drop the term of other bidders’ reported types t−i and the +subscript i in u, v, and p, e.g., vi((Bi, Ri), t−i) to v(Bi, Ri) +when the context is clear. With dropped t−i and i, we imply +that all the mentioned statements should hold for every pos- +sible t−i ∈ T−i for bidder i. We omit the proofs of Theorem +3.1, 3.2 and 3.7 since we have detailedly explained how to +derive them in the main text. +A.1 +Proof of Theorem 3.3 +To help prove Theorem 3.3, we consider the following +Lemma A.1, which provides additional insight on the struc- +ture of feasible allocations. It describes the following fact: if +some type A has both a higher budget and higher ROI than +another type B (indicating we cannot simply distinguish who +would have higher utility from their types), and they turn out +to share the same cumulative value, then the corresponding +lower utility diagonal type, i.e., the type who has the lower +budget and higher ROI, must have the same cumulative value. +Lemma A.1. For any auction satisfying Theorem 3.1 and 3.2, +if vi((B′ +i, Ri), t−i) = vi((Bi, R′ +i), t−i) = c +for some c ≥ 0, i ∈ [n], t−i ∈ T−i, B′ +i > Bi and R′ +i < Ri, +then +vi((Bi, Ri), t−i) = c. +Proof. By Theorem 3.1 and 3.2, vi(Bi, Ri) ≤ c. +Sup- +pose vi(Bi, Ri) < c, then v(B′ +i, Ri) > v(Bi, Ri), indicat- +ing p(B′ +i, Ri) > Bi and ROI(B′ +i, Ri) ≥ Ri. +Similarly, +v(Bi, R′ +i) > v(Bi, Ri), indicating ROI(Bi, R′ +i) < Ri and +p(Bi, R′ +i) ≤ Bi. Thus, we have p(Bi, R′ +i) ≤ Bi < p(B′ +i, Ri) +and ROI(Bi, R′ +i) < Ri ≤ ROI(B′ +i, Ri). As vi = pi × ROIi, +v(Bi, R′ +i) < v(B′ +i, Ri), leading to a contradiction. +Proof. (Theorem 3.3) The only if direction directly follows +from Theorem 3.1 and 3.2. For the if direction, we divide the +possible misreporting into four categories according to their +relationship with the true type ti = (Bi.Ri), and verify none +of them could achieve higher utility when conditions in Theo- +rem 3.1 and 3.2 hold. Suppose the bidder misreports (B′ +i, R′ +i). +The cases B′ +i = Bi or R′ +i = Ri are covered directly. +• B′ +i > Bi and R′ +i > Ri. +As R′ +i > Ri, v(Bi, Ri) ≥ +v(Bi, R′ +i). +If v(B′ +i, R′ +i) > v(Bi, Ri), then v(B′ +i, R′ +i) > +v(Bi, R′ +i), indicating p(B′ +i, R′ +i) > Bi. Therefore, if misre- +porting could bring higher cumulative value, then the pay- +ment would break the bidder’s budget constraint. +• B′ +i > Bi and R′ +i < Ri. We would have v(B′ +i, R′ +i) ≥ +v(Bi, R′ +i) ≥ v(Bi, Ri), and v(B′ +i, R′ +i) ≥ v(B′ +i, Ri) ≥ +v(Bi, Ri). If v(B′ +i, R′ +i) > v(Bi, Ri), at least one inequal- +ity strictly holds respectively in the above two inequality +formulas. +If v(B′ +i, R′ +i) > v(Bi, R′ +i), then p(B′ +i, R′ +i) > +Bi, which breaks the bidder’s budget constraint. +Thus, +v(B′ +i, R′ +i) = v(Bi, R′ +i) > v(Bi, Ri) should hold. +Simi- +larly, v(B′ +i, R′ +i) = v(B′ +i, Ri) > v(Bi, Ri). We then have +v(B′ +i, R′ +i) = v(B′ +i, Ri) = v(Bi, R′ +i) > v(Bi, Ri), which +contradicts with Lemma A.1. +• B′ +i < Bi and R′ +i > Ri. According to condition (1), the +bidder gets lower allocated value. +• B′ +i < Bi and R′ +i < Ri. This case is symmetric to B′ +i > Bi +and R′ +i > Ri, and we omit the proof here. +A.2 +Proof of Theorem 3.4 +Proof. Since A is feasible, there exists corresponding P′ +that results in DSIC and IR mechanism (A, P′). +Notice +pi(Bi, Ri) = min (vi(Bi, Ri)/Ri, Bi) is the possible largest +payment that can be extracted from bidder i with type +(Bi, Ri) to satisfy the bidder’s ROI and budget requirement. +That is to say, p′ +i(Bi, Ri) ≤ pi(Bi, Ri). On the other hand, +increasing payment of a type will not break originally satis- +fied conditions in Theorem 3.1 and 3.2. As modifying from +p′ +i to pi(Bi, Ri) = min (vi(Bi, Ri)/Ri, Bi) satisfies the bud- +get/ROI requirement of the bidder and maintaining the DSIC +conditions, (A, P) is a truthful mechanism. +A.3 +Proof of Theorem 3.5 +Proof. The if direction could be verified through applying +Theorem 3.3 and 3.4. For the only if direction, condition (1) +directly follows from Theorem 3.3. To show condition (2), if +v(Bi, Ri) < BiRi, then p(Bi, Ri) ≤ v(Bi, Ri)/Ri < Bi, +leading to contradiction with Theorem 3.3 when B′ +i → B− +i +(denoting some B′ +i < Bi infinitely close to Bi). The analysis +is similar for condition (3). +A.4 +Proof of Theorem 3.6 +Proof. When thri(Bi) = 0, the statement is vacuously true. +When thri(Bi) > 0, +lim +R→thri(Bi)− vi(Bi, R) ≥ thri(Bi) × Bi +from definition of thr function. +By the monotone prop- +erty, for any Ri < thri(Bi), vi(Bi, Ri) > Bi × Ri, in- +dicating vi(Bi, Ri) = limR→R+ +i vi(Bi, R). +As a result, +vi(Bi, Ri) = vi(Bi, thri(Bi)) for any Ri ≤ thri(Bi). If +vi(Bi, thri(Bi)) > thri(Bi) × Bi, by Theorem 3.5, +lim +R→thri(Bi)+ vi(Bi, R) = vi(Bi, thri(Bi)) > thri(Bi) × Bi. +If vi(Bi, thri(Bi)) < thri(Bi) × Bi, there would exist some +R → thri(Bi)− whose vi(Bi, R) < Bi×R. Both cases con- +tradicts of the definition of supremum in thr function. Thus, +vi(Bi, thri(Bi)) = thri(Bi) × Bi. +A.5 +Proof of Corollary 3.8 +Proof. We consider the map m as follows. +(m◦g)(B, R) = +� +� +� +� +� +� +� +g(B) +if R ≤ g(B) +B +g +� +sup +B′≤B +{R ≤ g(B′) +B′ +} +� +otherwise, +where we define sup ∅ = 0. Given a non-decreasing func- +tion g, we could derive the corresponding unique cumulative +value function v based on the construction of m, and the con- +structed cumulative value function must be feasible since ev- +ery type satisfies conditions in Theorem 3.5. Inversely, given +any possible cumulative value function vi, we can find the +unique threshold ROI function by definition +thri(B, t−i) = sup +R∈Ri +(vi ((B, R), t−i) ≥ B × R) , + +and compute the corresponding +gi(Bi, t−i) = thri(Bi, t−i) × Bi. +Due to the the truthfulness conditions in Theorem 3.5, such +derived gi(Bi, t−i) must be non-decreasing in Bi since the +cumulative value when R is infinitely close to 0 should be +non-decreasing for different Bi. +A.6 +Proof of Theorem 4.1 +Proof. We provide the proof through figuring out the function +g(B) for each bidder i when t−i is fixed. Fixing bidder i, +given cj and vi,j, we rank the items by ri,j = f −1 +i,j ( cj +vi,j ) in +an decreasing order. Without loss of generality, we renumber +these items from 1 to m with r1 +i > r2 +i > . . . > rm +i and denote +the corresponding value as vj +i . +We would then verify that the allocation assigned by our +algorithm aligns with the cumulative values implied by fol- +lowing function g(B) with p = 1, 2, . . . , m − 1. The corre- +sponding threshold ROI function is thr(B) = g(B)/B. +g(B) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +r1 +i · B +if B < v1 +i +r1 +i +, +p +� +j=1 +vj +i +if B ∈ [ +�p +j=1 vj +i +rp +i +, +�p +j=1 vj +i +rp+1 +i +), +rp+1 +i +· B +if B ∈ [ +�p +j=1 vj +i +rp+1 +i +, +�p+1 +j=1 vj +i +rp+1 +i +), +m +� +j=1 +vj +i +if B ≥ +�m +j=1 vj +i +rm +i +. +For convenience in notations, we use Ai(R) to denote the +winning set Ai containing all items the bidder has the high- +est virtual bid when she reports R (Line 4-6), namely all the +items with rj +i ≥ R, and use Ai(0) to denote the whole set of +items. We use Rc +i(B; Ai(R)) to denote the critical ROI Rc +i +computed after Line 9 when bidder i reports (B, R), and use +d(B; Ai(R)) to represent the corresponding d. +We would show the following properties to finish the proof. +(a) Rc +i(B; Ai(0)) aligns with the threshold ROI function in- +dicated by g(B); +(b) When R ≤ Rc +i(B; Ai(0)), +Rc +i(B; Ai(0)) = Rc +i(B; Ai(R)), +d(B; Ai(0)) = d(B; Ai(R)), +and when R > Rc +i(B; Ai(0)), R > Rc +i(B; Ai(R)). +(c) When R ≤ Rc +i(B; Ai(R)), R ≤ g(B) and the cumula- +tive values assigned to the bidder align with those defined +by g(B); +(d) When R > Rc +i(B; Ai(R)), R > g(B) and the bidder +gets all the items in Ai(R); +(e) The corresponding allocation indicated by (d) align with +those indicated by g(B). +To show property (a), we analyse the detailed calculation in +Line 9. The calculation in Line 9 exactly locates budget B in +the segmentation of above function g(B). The left-hand side +of Line 9 is strictly increasing when Rc +i decreases, and is con- +tinuous almost everywhere except at points Rc +i = {rj +i }j∈[m]. +In detail, when Rc +i decreases from (rp+1 +i +)+ to (rp+1 +i +), the +left-hand side of Line 9 jumps from B = +� �p +j=1 vj +i +rp+1 +i +�− +to +�p+1 +j=1 vj +i +rp+1 +i +, where (·)+ or (·)− represents the one-sided limita- +tion. Therefore, we have +Rc +i(B; Ai(0)) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +r1 +i +if B < v1 +i +r1 +i +, +� p +� +j=1 +vj +i +� +/B +if B ∈ [ +�p +j=1 vj +i +rp +i +, +�p +j=1 vj +i +rp+1 +i +), +rp+1 +i +if B ∈ [ +�p +j=1 vj +i +rp+1 +i +, +�p+1 +j=1 vj +i +rp+1 +i +), +� m +� +j=1 +vj +i +� +/B +if B ≥ +�m +j=1 vj +i +rm +i +, +and +d(B; Ai(0)) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +v1 +i /r1 +i − B +if B < v1 +i +r1 +i +, +0 +if B ∈ [ +�p +j=1 vj +i +rp +i +, +�p +j=1 vj +i +rp+1 +i +), +p+1 +� +j=1 +vj +i /rp+1 +i +− B +if B ∈ [ +�p +j=1 vj +i +rp+1 +i +, +�p+1 +j=1 vj +i +rp+1 +i +), +0 +if B ≥ +�m +j=1 vj +i +rm +i +. +Comparing thr(B) = g(B)/B with Rc +i(B; Ai(0)) gives us +Rc +i(B; Ai(0)) = thr(B), thus proving property (a). +To show property (b), we should notice the calculation +of Rc +i(B; A(R)) and d(B; A(R)) relies on the items with +rj +i ≥ Rc +i(B; A(R)), and the existence of other items in A(R) +would not affect the result according to the way we compute +Rc +i. This provides the proof for the first half of property (b). +To show the second half, note that a larger set Ai could only +result in an equal or larger Rc +i, and Ai(R) ⊂ Ai(0). +When R ≤ Rc +i(B; Ai(R)), R ≤ g(B) follows from prop- +erty (b). The algorithm goes into Line 15, thus items with +rj +i < Rc +i(B; Ai(R)) would be removed. After the operations +to remove items with value d/Rc +i in Line 10-13, the values +of remaining items would finally become � +j,rj +i ≥Rc +i (B) vj +i − +d(B)/Rc +i(B), where all the set used to compute Rc +i and d +here could be regarded as Ai(0) by property (b). It can be +verified that the above values align with our defined g(B), +thus proving property (c). +When R > Rc +i(B; Ai(R)), R > g(B) follows from prop- +erty (b), and the algorithm would not go into Line 15 to re- +move parts of the items. This also means B ≥ +�ni +j=1 vj +i +r +ni +i +, +where we use ni to denote the item with smallest rj +i in Ai(R), + +and further indicates d(B; Ai(R)) = 0. Therefore, the algo- +rithm would not go into Line 10-13, and no operation about +removing the items would be conducted. The bidder would +get all items in Ai(R) after Line 6, which proves property (d). +By property (d), we naturally find that the cumulative val- +ues are the same for types with the same ROI and locate +above the threshold ROI function. This structure aligns with +the non-increasing threshold ROI function indicated by the +defined g(B). +Based on Corollary 3.8, in order to prove +property (e), we only need to show the type which has the +largest budget among all the types with ROI R on the thresh- +old ROI function, i.e., the type (B∗(R), R) with B∗(R) = +supB′{B′| thr(B′) = R}, would get an cumulative value +equal to the valuations of items in Ai(R) (as characterized +in property (d)). Since we have shown the cumulative values +for types below and on the threshold ROI function are aligned +between our algorithm and defined g(B) in properties (a)-(c), +we could use the defined g(B) and corresponding thr(B) to +verify the above statement. We could find +B∗(R) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +�p +j=1 vj +i +rp +i +if R = rp +i , +p +� +j=1 +vj +i /R +if R ∈ (rp+1 +i +, rp +i ), +m +� +j=1 +vj +i /R +if R ≤ rm +i , +for p = 1, . . . , m − 1. We do not define B∗(R) for R > r1 +i +since both our algorithm and the defined g(B) determine to +not allocate to these types. We could verify g (B∗(R)) = +� +j,rj +i ≥R vj +i , thus finish the proof for property (e). +By the above properties (a)-(e), the cumulative value func- +tion indicated by our algorithm aligns with the defined g(B). +Since this g(B) is non-decreasing in B, the mechanism is +truthful and preserves the bidders’ financial constraints with +the payment in Line 16. Finally, our mechanism does not +over-allocate items because each item may only be allocated +to the bidder with highest virtual bid (breaking possible ties +with arbitrary rules). +B +Non-Truthfulness of Repeated First-Price +and Second-Price Auctions +We provide counter-examples to show the non-truthfulness +of repeated first-price and second-price auctions. For sim- +plicity of illustration and understanding, we consider small +cases with several bidders and items that do not fit with real +situations in automated bidding, while the ideas behind the +non-truthfulness are the same. +B.1 +Repeated First-Price Auctions +In repeated first-price auction, the realized ROI of a bidder is +exactly her reported ROI. Thus, bidders have no incentive to +misreport a smaller ROI. However, reporting a larger ROI can +possibly bring the bidder higher utility since it could reduce +the payment for per unit gained value if the bidder could still +win enough items. +Example 1. Consider a simple case with two bidders and two +items shown in the following table. +Item 1 +Item 2 +ROI +Budget +Bidder 1 +v1,1 = 4 +v1,2 = 4 +R1 = 2 +B1 = 3 +Bidder 2 +v2,1 = 1 +v2,2 = 1 +R2 = 1.5 +B2 = 6 +Assume bidder 2 reports her real type, and we consider the +strategy of bidder 1. If bidder 1 reports her real type, she will +only win item 1 at price of 2, leading the remaining budget to +be 1, thus cannot afford the second item. However, if bidder 1 +reports her ROI higher as R1 = 4, she could win both items +at total price 2 and realized ROI 4, getting a higher utility. +B.2 +Repeated Second-Price Auctions +In repeated second-price auctions, both misreporting a larger +or smaller ROI might bring the bidder larger utility. +Example 2. Consider a simple case with two bidders and two +items shown in the following table. +Item 1 +Item 2 +ROI +Budget +Bidder 1 +v1,1 = 4 +v1,2 = 8 +R1 = 1 +B1 = 3 +Bidder 2 +v2,1 = 4 +v2,2 = 4 +R2 = 1.5 +B2 = 6 +Assume bidder 2 reports her real type, and we consider the +strategy of bidder 1. If bidder 1 reports her real type, she will +win item 1 at price 8/3, leading the remaining budget to be +1/3, thus cannot afford the remaining higher value item 2. +However, if bidder 1 reports her ROI higher as R1 = 2, then +she will lose item 1, and win item 2 at price 8/3 with realized +ROI 3, getting a higher utility. +Reporting a smaller ROI may help the bidder win more items +and balance the realized ROI between items (items with real- +ized ROI higher or lower than the real ROI both exist) without +breaking her true ROI. +Example 3. Consider a simple case with two bidders and two +items shown in the following table. +Item 1 +Item 2 +ROI +Budget +Bidder 1 +v1,1 = 4 +v1,2 = 3 +R1 = 2 +B1 = 6 +Bidder 2 +v2,1 = 1 +v2,2 = 4 +R2 = 2 +B2 = 6 +Assume bidder 2 reports her real type, and we consider the +strategy of bidder 1. If bidder 1 reports her real type, she will +win item 1 at price of 0.5 and lose item 2 since bidder 2 has a +higher v/R for it. However, if bidder 1 reports her ROI lower +as R1 = 1.2, then she could win both items at total price 2.5 +and realized ROI 2.8, getting a higher utility. + diff --git a/odFPT4oBgHgl3EQfKzSD/content/tmp_files/load_file.txt b/odFPT4oBgHgl3EQfKzSD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab0c6861479441294b6b1b0604bd6201ec3db950 --- /dev/null +++ b/odFPT4oBgHgl3EQfKzSD/content/tmp_files/load_file.txt @@ -0,0 +1,900 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf,len=899 +page_content='Designing Ad Auctions with Private Constraints for Automated Bidding Yidan Xing1 , Zhilin Zhang2 , Zhenzhe Zheng1 , Chuan Yu2 , Jian Xu2 , Fan Wu1 and Guihai Chen1 1Shanghai Jiao Tong University 2Alibaba Group {katexing, zhengzhenzhe}@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='cn, {zhangzhilin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='pt, yuchuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='yc, xiyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='xj}@alibaba-inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='com, {fwu, gchen}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='cn Abstract In online advertising, automated bidding (auto- bidding) has become a widely-used tool for ad- vertisers to automatically make bids on different impressions in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Instead of submitting bids for each single impression, advertisers in auto- bidding submit their high-level objectives and con- straints to the auto-bidding tool, and observe the cumulative advertising performances after all the auctions within a time period have been finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Motivated by the features of automated bidding, we aim to design auctions with private financial con- straints for value-maximizing bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Specifically, we consider budget and ROI, the two most com- mon financial constraints in online advertising, as the private information of advertisers, and analyse the conditions of truthfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We show that ev- ery non-decreasing function with budget as input could be mapped to a truthful auction mechanism with budget and ROI as input, but this mapping procedure also introduces complex value group- ing structures into mechanism design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To achieve feasible and implementable auctions, we design a truthful auto-bidding auction mechanism with ad- justable rank score functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As the key design to guarantee truthfulness, our auction utilizes the bid- der’s budget constraint to compute a critical ROI, which enables comparisons between the budget and ROI constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We conduct experiments under dif- ferent auto-bidding settings to validate the perfor- mance of our proposed auction in terms of revenue and social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 1 Introduction With the rapid development of machine learning techniques, automated bidding (auto-bidding) has been widely adopted by various online advertising platforms [Facebook Ads, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Google Ads, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In auto-bidding services, advertisers sub- mit their high-level optimization objectives and constraints to the platform, and then the bidding agents, powered by ma- chine learning algorithms, make detailed bidding decisions in each of ad auctions on behalf of the advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' With the help of automated bidding tools, advertisers can optimize their overall advertising objectives with respect to their finan- cial constraints in a high-level way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Under automated bidding, we revisit a fundamental prob- lem in auction theory: whether the conventional auction model, where advertisers have private values for items (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', ad impressions) and conduct corresponding strategic bidding for each single auction, is still appropriate for the new adver- tising paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As the platform can access historical data about the interactions between advertisers and users, we can estimate the potential actions of users (such as clicks and con- versions), which can be regarded as public values of items for advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In auto-bidding, the private information from ad- vertisers are actually their constraints for the whole advertis- ing campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' These features require a new ad auction model to incentivize advertisers to truthfully reveal the high-level private constraints given the values of items are public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In this work, we consider a new automated bidding auction model, where advertisers submit budget as well as return on investment (ROI) requirement as their (private) constraints, and aim to maximize the cumulative values of winning im- pressions from multiple auctions during a certain period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We analyse the truthfulness conditions with respect to the private constraints of budget and ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Remarkably, we show that any truthful auction mechanism for this multi-dimensional setting could be equivalently represented by a series of non- decreasing functions with budget as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' When these non- decreasing functions are realized to derive the correspond- ing auction mechanism, the truthful conditions of budget and ROI introduce a new value grouping phenomenon: different budget-ROI types are grouped to share the same cumulative value, and the grouping pattern is determined by a threshold ROI function (transformed from the above non-decreasing function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As the threshold ROI functions are not constrained in monotonicity, the grouping shape of budget-ROI types is generally irregular, making it difficult to obtain the closed- form expression of grouping types and then the auction opti- mization objectives, such as revenue and social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Facing these design difficulties, we propose a family of ad auctions with personalized rank scores to optimize various design objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Our auction adopts rank scores as the cri- teria to determine item allocations, which shares the similar ideas with Generalized Second-Price (GSP) auction [Edel- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In guaranteeing truthfulness for the pri- vate constraints, we design critical ROI to be the largest ROI arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='13020v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='GT] 30 Jan 2023 that can win the most items without breaking the budget con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' It equivalently transforms budget to the same dimen- sion as ROI, thus allowing us to find a tight constraint limits the bidder from getting extra utilities and utilize this tight con- straint to prevent misreporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We conduct extensive exper- iments to evaluate the performances of the proposed auction mechanism under various auto-bidding settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The evalu- ation results demonstrate that the designed truthful auction can achieve more than 90% performance (in terms of rev- enue and social welfare) of the optimal baselines without the consideration of truthfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The main contributions can be summarized as follows: We consider the unique features within the interaction be- tween advertisers and online platforms in the context of au- tomated bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Based on these features, we formulate a new auto-bidding auction model, where value-maximizing bidders have high-level constraints as private information and the values of items are public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We investigate truthfulness conditions of two-dimensional private constraints, budget and ROI, under public value set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We provide full characterizations for the feasible space of allocation and payment for truthful auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Based on the derived truthfulness conditions, we design a family of truthful ad auction mechanisms for automated bid- ding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' With the newly designed rank score functions, the pro- posed ad auction is simple and flexible to be adapted into var- ious auto-bidding settings with different optimization goals such as revenue and social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We evaluate our proposed auction mechanisms with vari- ous experimental settings, and the empirical results validate the effectiveness of the proposed auctions regarding its per- formances in terms of social welfare and revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 2 Preliminaries In this section, we first motivate the considered auction de- sign problem by the online advertising system with auto- mated bidding services, and afterwards propose the formal auction model based on the features of automated bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 Online Advertising System The working process of the online advertising auction system is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' From an advertiser’s perspective, it can be described as follows: 1) The advertiser sets the bidding configuration in auto- bidding interface: chooses optimization objectives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', max- imizing clicks or conversions) and sets cost constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', budget per day, targeted return on investment (ROI) and max- imum cost per click).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The advertiser requires her realized ROI, defined as the ratio of her gained value and payment, to be higher than her targeted ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 2) Based on the advertiser’s configuration, an auto-bidding agent represents the advertisers to make bid decisions in mul- tiple auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' When each user impression comes, the auto- bidding agents attend an ad auction to compete for the ad dis- play opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 3) The auto-bidding agent adopts data-driven algorithms to predict the value (click through rate or conversion rate) of the incoming user impression, and bid for each impression while taking the cost constraints into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ad platform Allocation Pricing Objective and Constraints Cumulative Ad Effects User Features Objective and Constraints Current Outcomes Value & Bid Calculation Auto-bidding Agents Bids Outcomes Auto-bidding Interface Budget: 30$ ROI: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 Budget: 20$ ROI: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='8 Budget: 40$ ROI: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='6 Budget: 25$ ROI: 2 Incoming User Impressions 1 2 3 4 Figure 1: Ad Auction System Overview 4) During the multiple auctions, the advertiser can check the cumulative auction outcomes, including spent budget, re- ceived impressions, clicks, conversions and the average costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The advertisers can further adjust their auto-bidding settings in the interface to achieve their own advertising objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' From the above interaction between advertisers and the platform/auctioneer, we summarize three new features about the online advertising with auto-bidding services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' First, ad- vertisers only report high-level optimization objectives and constraints in the bidding configurations for multiple auc- tions, but not the fine-grained bid for each auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Second, advertisers only evaluate cumulative long-term performances and costs of multiple auctions, instead of the outcome of each individual auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Third, since the online platform can ac- cess all the data produced in advertising, it is reasonable to claim the monetary value of an incoming impression to a spe- cific advertiser can be calculated exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As the behavior pat- terns of advertisers and the online platform change distinctly in automated bidding, we need to investigate the mechanism design whose formats align with these new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 Auction Model Based on the motivations of the above online advertising sys- tem with auto-bidding, we propose the formal auction model considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' There are n advertisers competing for m items (user impressions) coming in sequence during a time period with m time slots, where only one item would ap- pear in each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 The advertisers are value-maximizing bidders [Fadaei and Bichler, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Mehta, 2022], who care about the cumulative value of her allocated impressions across all the time slots when the pay- ment is within their financial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Each advertiser i has budget (Bi ≥ 0) and ROI (Ri > 0) constraints, which are private information and are also called as type ti = (Bi, Ri) in mechanism design literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We denote the type profile of 1We would use advertisers with bidders, and items with impres- sions interchangeably throughout the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' all the advertisers as t = (ti)n i=1 and the space of the type profile as T = �n i=1 Ti with t ∈ T and Ti = Bi × Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We denote the reported type of the other bidders except bidder i as t−i = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' , ti−1, ti+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' , tn) and T−i = �n k̸=i Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We assume advertisers’ valuations on items are public infor- mation to the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We use vi,j > 0 to represent the advertiser i’s valuation on item j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' After collecting the budget and ROI of all the bidders, the online platform employs some auction mechanism (A, P) to decide ad allocations and payments, where A denote a (ran- domized) allocation rule A : T → [0, 1]n×m and P denote a (randomized) payment rule P : T → Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Specifically, for a reported type profile t′ ∈ T , the probability of bidder i being allocated item j is denoted by ai,j(t′) and the expected pay- ment of bidder i is denoted by pi(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For any item j ∈ [m], the allocation constraint is �n i=1 ai,j(t′) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Bidder i’s cu- mulative value in these auctions is vi(t′) = m � j=1 vi,jai,j(t′) and her realized ROI is ROIi(t′) := vi(t′)/pi(t′) (consid- ered as +∞ if pi(t′) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The utility for a bidder with budget and ROI constraints and true type ti when reporting t′ i is defined as ui (ti, t′) = �vi(t′), if pi(t′) ≤ Bi and ROIi(t′) ≥ Ri, −∞, otherwise, where the type profile t′ = (t′ i, t′ −i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In this work, we focus on designing truthful auction mech- anisms satisfying incentive compatible (IC) and individual rationality (IR) conditions for budget and ROI, which is a two-dimensional mechanism design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Various multi- dimensional mechanism design problems are shown to be dif- ficult in both analytical and computational aspects [Pavlov, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Daskalakis, 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In particular, we consider dominant-strategy incentive compatible (DSIC) direct-revelation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' An auction mechanism is dominant strategy incentive compatible (DSIC) if ∀i ∈ [n], ti, t′ i ∈ Ti, t−i ∈ T−i: ui(ti, (ti, t−i)) ≥ ui(ti, (t′ i, t−i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Similarly, we have the corresponding individual rationality (IR) condition as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' An auction mechanism is individual rational- ity (IR) if ∀t ∈ T , i ∈ [n]: pi(t) ≤ Bi and ROIi(t) ≥ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The online platform typically has some objectives to max- imize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Two common design objectives are revenue and so- cial welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Revenue is defined as the sum of payment from bidders, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', � i pi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For social welfare, we need a synony- mous metric, liquid welfare, defined as the maximum revenue that can be extracted from bidders without breaking IR con- straints, to incorporate the existence of financial constraints [Azar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In our context, liquid welfare can be defined as LW = � i∈[n],ui(ti,t′)≥0 min �vi(t′) Ri , Bi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (1) 3 Characterization of Truthfulness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 Conditions for Truthfulness In this section, we investigate the truthfulness conditions for this multi-dimensional mechanism design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Due to the limitation of space, we present the detailed proofs of our results in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To provide an intuition about the differences between auc- tion models with private constraint and private valuation, we start from the analysis of one private constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To satisfy the IC property on budget, we need to guarantee the bidders for not obtaining a higher utility by reporting a smaller or larger budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As reporting a smaller budget will not lead a bidder to break her original budget constraint, the gained utility should decrease for reducing the budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For the bidder misreport- ing a larger budget and obtaining higher values, we need to charge the bidder to break her original budget constraint, re- sulting in negative infinite utility to prevent misreporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' An auction mechanism is DSIC on budget B only if ∀i ∈ [n], R ∈ Ri, t−i ∈ T−i: (1) vi ((B, R), t−i) is non-decreasing in B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (2) If vi ((B′, R), t−i) > vi ((B, R), t−i) for B′ > B, then pi ((B′, R), t−i) > B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We can form a similar statement for ROI by showing that there is no incentive for a bidder to misreport her ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' An auction mechanism is DSIC on ROI R only if ∀i ∈ [n], B ∈ Bi, t−i ∈ T−i: (1) vi ((B, R), t−i) is non-increasing in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (2) If vi ((B, R′), t−i) > vi ((B, R), t−i) for R′ < R, then ROIi ((B, R′), t−i) < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Although the monotone properties required by conditions (1) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 seem similar to the truthfulness condi- tions for quasi-linear bidder with private valuations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', My- erson’s Lemma [Myerson, 1981], conditions (2) reveal their basic differences: assigning a payment to break at least one constraint (lead to negative infinite utility) is indispensable to prevent misreporting for private constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 naturally provide necessary condi- tions on the DSIC of two-dimensional type (B, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The fol- lowing result shows that these conditions also provide the suf- ficient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' That is, if a bidder cannot obtain higher utility through misreporting one of her constraints, misreport- ing the two constraints can also not obtain higher utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' An auction mechanism is DSIC on both budget and ROI if and only if it satisfies Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3 is proved through showing none of the mis- reporting may bring the bidder higher utility when conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 hold, which relies on the mathemat- ical relationship between payment and ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3, the truthful conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 for budget appear to be independent with the truthful conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 for ROI, and satisfy these conditions simultaneously could achieve the two-dimensional truthfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Nevertheless, the payment term actually appear in both the conditions (recall ROI = v/p in condition (2) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', we have to use the same payment scheme to satisfy these two sets of conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In order to step toward the full characterization of truthfulness, we need to further analyse how payment is influ- enced and constrained by financial constraints and allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To facilitate our discussion, we define an allocation rule A to be feasible if there exists a payment rule P such that the mechanism (A, P) is truthful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The set of feasible allocation rule is the space we can search for truthful auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Although there may exist multiple payment rules that constitute a truth- ful auction for a feasible allocation rule A, the following the- orem allows us to focus on the maximum payment rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For a feasible allocation rule A, assigning pi ((Bi, Ri), t−i) = min (vi ((Bi, Ri), t−i) /Ri, Bi) , (2) for ∀i ∈ [n], (Bi, Ri) ∈ Ti, t−i ∈ T−i constitutes a truthful auction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4 naturally holds because each type is charged the maximum payment under her financial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If some other payment could constitute a truthful auction with A, then improving the payment to this maximum amount would not break the originally held conditions (2) in Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4, given any feasi- ble allocation rule A, we could find a corresponding payment rule to constitute a truthful auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As the bidders do not evaluate detailed allocation results of each individual auction, searching for the truthful allocation rule is equal to designing cumulative value functions vi((Bi, Ri), t−i) that can be real- ized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Thus, we could use the corresponding cumulative value function to represent a feasible allocation rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Our next step is to further shrink the consideration space of comparison in conditions of truthfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Previous conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3 include comparisons between any two type sharing the same budget or ROI, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', comparing any (B, R) and (B′, R) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1, which is still a large consideration space and leads to difficulties in finding the cumulative value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To enable analysis between “neighbouring” types, for any allocation rule A and its cumulative value function v(t), we assume the limitations limB→B− i vi ((B, Ri), t−i) and limR→R+ i vi ((Bi, R), t−i) exist for any t ∈ T , i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Through considering these infinitely close types and substituting the payment terms in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3 by the payment rule (2), conditions that con- strain the payment or realized ROI of misreporting another type to break the original constraints could be converted to conditions regarding cumulative value of the type itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' An allocation rule A can derive a truthful auc- tion if and only if ∀t ∈ T , i ∈ [n]: (1) The cumulative value vi ((B, R), t−i) is non-decreasing in B for R = Ri and non-increasing in R for B = Bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (2) If vi ((Bi, Ri), t−i) > limB→B− i vi ((B, Ri), t−i), then vi ((Bi, Ri), t−i) ≥ Bi × Ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (3) If vi ((Bi, Ri), t−i) > limR→R+ i vi ((Bi, R), t−i), then vi ((Bi, Ri), t−i) ≤ Bi × Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 Structures of Feasible Allocation Rule Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 fully characterizes the conditions of a feasible allocation rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In this subsection, we would further exploit 0 1 2 3 4 5 6 7 8 Bi 0 1 2 3 4 5 Ri thri(Bi) (B1, R1) (B2, R2) Accumulated Value vi(Bi, Ri) 0 5 10 15 20 25 Figure 2: An example of mapping from a threshold ROI function to cumulative values fixing the others’ types t−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' the structures of feasible allocation rule indicated by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 to provide more instructions on truthful auction design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5, the relation between cumulative value v and the term B × R strictly characterizes whether this type shares the same cumulative value with its neighbouring types, which allows us to further clearly represent the structure of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Fixing the budget, with the decrease of ROI, the cumula- tive value assigned to the type should be increasing, notice the term B × R decreases along with R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' however, condi- tion (3) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 requires that if the cumulative value strictly increases in this process, then its value should not ex- ceed B ×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Thus, there must exist some threshold ROI, such that the increasing cumulative value intersects with the de- creasing B × R, and the cumulative value could not increase anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Based on the above observation, we define thri(B, t−i) = sup R∈Ri (vi ((B, R), t−i) ≥ B × R) if the considered set is non-empty and bounded, or otherwise be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 It turns out that the cumulative value assigned to this threshold ROI is exactly B × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' ∀i ∈ [n], t−i ∈ T−i, Bi ∈ Bi, Ri ≤ thri(Bi, t−i): vi ((Bi, Ri), t−i) = thri(Bi, t−i)Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We could then combine different budgets together to ob- tain the complete characterization of feasible allocation in the two-dimensional type space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since all types above the thresh- old ROI has cumulative value v < B × R by the definition of thr function, they could not satisfy the condition (2) in The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5, and thus are forced to have the same cumulative value with their neighbouring types with a smaller budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' ∀i ∈ [n], t−i ∈ T−i, Bi ∈ Bi\\{0}, Ri > thri(Bi, t−i): vi((Bi, Ri), t−i) = lim B→B− i vi ((B, Ri), t−i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since the types below (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='6) and above (Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='7) the ROI threshold have both been fully described, the cumulative values on the entire type space could be fully defined if thri(Bi, t−i) is given and t−i is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We provide an example in Figure 2 to illustrate these characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 2This set would be empty only when vi ((B, R), t−i) = 0 for all R ∈ Ri, and unbounded only when B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For types below the threshold ROI curve and have the same budget B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', types lie in the same vertical line), they share the same cumulative value thri(B)×B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For each type above the curve, its cumulative value is equal to that of its neigh- bouring type with a smaller budget until it reaches some type on the threshold ROI curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' This is equivalent to search hor- izontally to the left to find the closest type on its correspond- ing ROI threshold, where all the types along this horizontal search share the same cumulative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We have shown that each feasible two-dimensional cumu- lative value function (whose input is (B, R)) for a single bidder could be represented by a one-dimensional threshold ROI function (whose input is B) when other bidders’ pro- file is fixed, but it is unknown what kinds of one-dimensional threshold ROI functions could represent a feasible cumulative value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' By condition (1) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5, the cumula- tive value when R is infinitely close to 0 should be increasing for different B, which indicates thri(B, t−i) × B needs to be non-decreasing in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since every cumulative value func- tion mapped from a non-decreasing thri(B, t−i) × B satis- fies Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5, this becomes the only requirement for a feasible thri function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For convenience in notations, we de- note gi(Bi, t−i) = thri(Bi, t−i) × Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We denote V as the space of feasible cumulative value function when i and t−i are fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', vi : Ti → R+, and denote G as the space of non-decreasing function g : R+ → R+ with g(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' There exists a bijective map m : G → V with (m◦g)(B, R) = � � � � � � � g(B) if R ≤ g(B) B , g � sup B′≤B {R ≤ g(B′) B′ } � otherwise, where we define sup ∅ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In other words, given any one-dimensional non-decreasing function with g(0) = 0, we could transform it to a feasible3 cumulative value function by the above mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 4 Truthful Auction Design The derived structure of feasible allocation rules give rise to a value grouping phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' When reversing the pro- cess of finding the cumulative value of a certain type given threshold ROI curve, every type on the threshold ROI curve shares the same cumulative value with two groups of types: the types vertically below it, as well as the types locate hor- izontally right to it and above its corresponding threshold ROI, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' This intricate value grouping phenomenon are enforced by the truthful conditions instead of the optimization requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As we can observe in Fig- ure 2, since the horizontal search terminates when meeting a type on the threshold ROI curve, the grouping shape is determined by the relative rank of thri(Bi, t−i) for differ- ent Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' However, though thri(Bi, t−i) is required to keep g(B) non-decreasing, it is not required to be monotone it- self, which leads the terminate of horizontal search and the 3Our feasibility is with respective to truthfulness, and does not specify how to realize the allocations from detailed items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Algorithm 1: A Family of Simple Truthful Auctions Input: Bidder’s reported type (Bi, Ri), and non-increasing rank score functions fi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Output: Bidder’s allocated items Ai and payment pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 1 Initialize Ai = {} for i ∈ [n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 2 Compute virtual bids bi,j ← vi,j × fi,j(Ri);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 3 for each item j ∈ [m] do 4 Find bidder i0 with the highest virtual bid bi0,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 5 Record the second highest bid cj ← maxi̸=i0 bi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 6 Add item j into Ai0 and set ai0,j = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 7 for each bidder i ∈ [n] do 8 Compute ROI ri,j = f −1 i,j ( cj vi,j ) required for winning the item j ∈ Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 9 Find a largest Rc i ∈ Ri s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' � j∈Ai I (Rc i ≤ ri,j) × vi,j Rc i ≥ Bi, and let d ← the difference between left-hand side and right-hand side of the above inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 10 if d > 0 then 11 Find the items j′ with ri,j′ = Rc i, and remove the items with value d/Rc i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 12 if all of these items are removed then 13 Round up Rc i to the smallest ri,j > Rc i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 14 if Ri ≤ Rc i then 15 Remove the items with ri,j < Rc i from Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 16 pi ← min � � j∈Ai vi,j·ai,j Ri , Bi � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 17 return (Ai, pi) for i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' grouping shapes to be irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' This also brings difficulties in deriving general closed-form conditions for the group of types to share the same allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Furthermore, although the cumulative values of the grouping types are forced to be the same, their payment are calculated by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4, leading to varying payments between grouping types and the non- linear revenue with respective to the threshold ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The above characteristics of value grouping phenomenon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', irregularity in grouping shape and non-linear payment, make any analytical or mathematical programming approach difficult to adopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To reach feasible and implementable auc- tion design, we propose a family of simple truthful auctions based on newly designed rank score functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The detailed auction mechanism is presented in Algo- rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For each item, bidders are ranked based on their virtual bids bi,j, which is defined as vi,j multiplying a rank score fi,j(Ri) with some pre-defined non-increasing function fi,j (Line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The items are then temporarily allocated to each bidder with the highest virtual bids (Line 4-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' After the can- didate allocation set Ai for each bidder i has been determined, we compute the the corresponding ROI requirement ri,j for the bidder i to rank the first in item j ∈ Ai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', bidder i needs to report Ri ≤ ri,j in order to maintain ranked first for item j (Line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The critical ROI in Line 9 is our key design to guar- antee truthfulness, which computes the largest ROI a bidder could report to win enough items in Ai and spend out her budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Intuitively, critical ROI simulates the best-response ROI of the bidder given her budget constraint and keeps in- dependent of the bidder’s true ROI constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We naturally utilize this critical ROI, which transforms the budget to the same dimension of ROI, as the threshold ROI function of our auction, and design the remaining parts based on the truthful conditions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Line 10-13 are designed to guarantee the value of remaining items in Ai would exactly equal to the budget of the bidder times the computed critical ROI (The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Line 14-15 are designed to guarantee the types with ROI larger than the corresponding critical ROI would be allocated based on their reported ROI (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='7) and get all items with ri,j ≥ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Line 16 determines the payment for each bidder i based on Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If the pre-determined functions fi,j(R) are non-increasing with R, then the auction mechanism corre- sponding to Algorithm 1 is truthful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Our proof is conducted through figuring out and verifying the corresponding g(B) and threshold ROI function of Algo- rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We can design personalized fi,j(R) for bidders and items according to the auction objective and prior knowledge before the advertising campaign, such as the type distribution of bidders and the information of incoming items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As the cost of bypassing the design hardness brought by value grouping of multiple bidders, our mechanism may not allocate all the items if the corresponding highest ranking bidder has fully consumed her budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Nevertheless, the auctioneer could take multiple approaches to alleviate this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' While ad- justing the functions fi,j(R) with the pre-known information is the most direct approach, dividing advertisers into groups and circulating the unsold items could also be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 5 Experiments In this section, we conduct experiments with synthetic data to validate the performance of our proposed auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 Experimental Setup There are two sets of experiments with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' and non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' bidders, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We vary the number of bidders and items to simulate the changes in demand and supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The presented results are averaged by 50 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We choose distri- bution parameters based on the common fluctuations in ad- vertising markets, and normalize the lower bound of vi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Symmetric bidders Bidders are symmetric with vi,j ∼ U[1, 4], Bi ∼ U[40, 80] and Ri ∼ U[1, 3], where U[a, b] is the uniform distribution within the range [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4 Mixed bidders There are low and high distributions for vi,j, Bi and Ri as bidders’ possible types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' specifically, vi,j ∼ U[1, 2] or vi,j ∼ U[2, 3], Bi ∼ U[20, 40] or Bi ∼ U[80, 100], and Ri ∼ U[1, 2] or Ri ∼ U[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Combinations of these categories result in 8 groups of bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Baseline Auctions Since no existing mechanism guarantees the IC properties of both budget and ROI, we consider com- mon repeated auction formats: first-price and second-price auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Due to their non-truthfulness (examples provided in the appendix), we involve misreporting for these auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 4We have tested several other parameters under uniform distri- bution, and the trends of results are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Repeated first-price auctions: The auctioneer holds first- price auctions for every single item, where bidder i bids vi,j/Ri for item j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The auctioneer allocates the item to the highest ranking bidder who has remaining budget to afford, and charge the first-price payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We aim to simulate bid- ders’ misreporting behaviors by the classical best-response dynamics with the true profile as starting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In repeated first-price auctions, the bidders have no incentive to misre- port their budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We calculate the best response ROI as the smallest ROI that wins the most items in historical auctions without breaking the original financial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Repeated second-price auction: This auction format is sim- ilar to repeated first-price auction with second-price auction used in each round instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We still simulate bidders’ misre- porting behaviors to be the best-response dynamics as above, with payment changed to be the second price, where the best- response ROI here may be larger or smaller than the true ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Non-truthful Optimal Baseline: The optimal offline revenue could be computed through linear programming, where we set each bidder i’s payment to item j as exactly vi,j ×ai,j/Ri, and her total payment is constrained to not exceed Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Evaluation Metrics We consider revenue and liquid welfare as the optimization goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since the payment formula (2) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4 aligns with the definition of liquid welfare in equation (1) when no bidder receives negative infinite util- ity, the metrics of revenue and liquid welfare would be the same for truthful auctions, and we will thus only report rev- enue in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Besides, as advertisers may have fluctu- ating auction performances across time slots, fairness should also be considered in order to preserve bidders’ willingness to attend the auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Using liquid welfare to substitute the traditional valuation function [Bez´akov´a and Dani, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Chakrabarty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2009], fairness is defined as fairness = min i∈[n] min �vi(t′) Ri , Bi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Rank Score Function We adopt rank scores in the form fi,j(R) = αi,j × e−βR, where αi,j is drawn from a recti- fied normal distribution N(µi, σ2 i ), and β, µi, σi are pre-set parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In f(R), β is used to adjust the impact of ROI on equivalent bids, and α maintains the rank scores of various bidders in a comparable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since items may be discarded when some bidders win excessive items in the proposed auc- tion (Line 15 in Algorithm 1), in order to avoid loss in welfare and revenue, we should give other bidders non-trivial win- ning opportunities when the supply is sufficient, which is pro- vided by the randomness in α ∼ N(µi, σ2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For each auto- mated bidding environment with different setting of bidders and items, we set rank score function parameters to be the same for bidders following the same distribution, and choose the parameters with better revenue in this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 Experimental Results Symmetric bidders The revenue of our truthful auction (re- ferred as DSIC) and the baseline auctions with different num- ber of bidders is reported in Figure 3a, and revenue when number of items changes is reported in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Our truthful auction achieves better revenue than first-price and second-price auctions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' and achieves more than 90% revenue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='450 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='550 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='650 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Revenue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Number of bidders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='OPT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='DSIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='First-Price ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Second-Price ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1000 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1000 1200 1400 1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Revenue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Number of items ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='OPT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='DSIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='First-Price ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Second-Price ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1000 1200 1400 1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Fairness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Number of items ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='OPT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='DSIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='First-Price ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Second-Price ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='860 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='870 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='880 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='890 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='910 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='920 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='930 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='β=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='01 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='02 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='03 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='04 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='05 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='06 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='07 Revenue β=0 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='01 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='02 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='03 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='04 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='05 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='06 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='07 Revenue Fairness (d) Figure 3: Revenue and fairness in different experiment setting for symmetric bidders: (a) Revenue for different number of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='d bidders with 200 items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (b) Revenue for different number of items with 40 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='d bidders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (c) Fairness for different number of items with 40 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='d bidders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (d) Fairness of our DSIC auction with 40 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='d bidders and 400 items using different parameter β in rank score functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 0 400 800 1200 1600 2000 2400 Setting 1 Setting 2 Setting 3 Revenue Number of items OPT First-Price Second-Price DSIC-1 DSIC-2 DSIC-3 Figure 4: Revenue with 40 mixed bidders in Setting 1: 200 items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Setting 2: 1000 items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Setting 3: 1600 items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' with three different groups of rank score functions for our DSIC auction of the near optimal baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Remarkably, first-price auc- tions displays evident decrease in revenue when the items are excessive, which also leads to decreases in fairness (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' This is because the bidders exceedingly increase their reported ROI when they face some unsold items with low competing prices and have to pay proportionally for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In contrast, for second-price auctions, the payment of unsold items only consume a negligible part of budget when com- puting best-response ROI, and the reported ROI of bidders maintains relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In Figure 3c, we report the fairness results under different number of items a fixed number of 40 bidders as in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As designed in the rank score function, our auction does not allocate excessive items to a certain bidder to avoid waste of items, and thus achieves better fairness then the optimal base- line when the number of items is in the range of 600 to 1200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' When items are in shortage (200/400 items in Figure 3c), our truthful auction harms the fairness to some extent due to the pursuit of revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We can control the tradeoff between fair- ness and revenue with different rank score functions through the parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Figure 3d reports a set of revenue and fair- ness performances with different β in the same setting as 400 items in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To achieve higher revenue, it is inevitable to have discriminating allocation among different bidders and harms the fairness when items are not abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Mixed bidders Instead of presenting trends similar to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' setting, we demonstrate the importance of choosing appro- priate rank score functions in non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We choose three groups of rank score functions respectively with good performance in three typical settings: 40 mixed bidders with 200/1000/1600 items, and test their performances in the other two settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The results are presented in Figure 4, where DSIC-n represents the truthful auction with rank score functions performing well in the Setting n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The DSIC- n mechanism only perform well in Setting n, and the reason should come from the distinct allocation approaches needed to achieve high revenue for different number of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For example, to achieve high revenue when items are in short- age (Setting 1), DSIC-1 auction allocates most items to the bidders with high value and low ROI, leading to allocating excessive items to these bidders and neglecting other bidders in Setting 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 6 Related Work Auctions in online advertising has attracted much attention from game theory field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Initiated by [Varian, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Edelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2007], the early work studied the properties of keyword auctions in sponsored search, where bidders set a fixed bid for all the queries with a certain keyword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' An extensive survey on this topic can be found at [Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Due to the distinct features compared to traditional ad auctions, auction design and game theoretical analysis for automated bidding come into researchers’ view in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Starting from [Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2019], a line of work takes the uniform pacing assumption to analyse the equi- librium and efficiency between automated bidding agents, where the agent applies a same pacing parameter to the true value of each item as their bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2021a] studied the usage of boosting and re- serve price to increase the efficiency and revenue of auto- bidding auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2022] analysed the existence and computa- tional complexity of pacing equilibrium under different set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Mehta, 2022] designed a truthful auction with ran- domization to improve the price of anarchy in auto-bidding settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2022] studied the efficiency of first- price auctions under a more generalized setting with no uni- form pacing assumption, and [Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2022] designed a non-truthful auction to further improve the price of an- archy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Some recent work assumed agents adopting regret- minimizing learning algorithms to adjust the bid or pacing parameter, and analysed the convergence as well as game properties [Gaitonde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Fikioris and Tardos, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Kolumbus and Nisan, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Our work could be classified as considering the incentive of bidders when setting automated bidding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Under this topic, [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2020] provided distributed bidding algo- rithms for IC issues assuming public values and private ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2021b] studied the optimal mechanism design for bidders with ROI constraint under different information structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Kolumbus and Nisan, 2021] considered the incen- tive of parameter setting for agents in general games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As one of the most relevant work, [Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2022b] considered the mechanism design with private ROI constraint and public budget, and derived the optimal auction for all the two-bidder cases and some specific multi-bidder cases based on elabo- rate ironing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Distinct from the above work, our work considers both budget and ROI as private constraints, which aligns with the real automated bidding and reveals the complex value grouping phenomenon for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Another closely relevant line of work is mechanism de- sign for financially-constrained bidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since [Laffont and Robert, 1996], intensive research has been conducted for bid- ders with budget constraint and progress gradually [Borgs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Dobzinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Pai and Vohra, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Szymanski and Lee, 2006] anal- ysed the impact of ROI constraints on bidding and revenue of several common auction forms, and [Golrezaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2021] firstly considered auction design for bidders with ROI con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Differing from those existing work studying quasi- linear bidders with private values and one financial constraint, our work consider value-maximizing bidders with multiple private constraints in the automated bidding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' From the computational perspective of multi-dimensional mechanism design, the representing series of work [Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Alaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2021] provided a black- box linear programming framework to the classical multi- item auction design problem under Bayesian Incentive Com- patibility (BIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' However, due to the “if, then” structure in our truthfulness conditions (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5) and the resulting value grouping phenomenon, those existing linear program- ming techniques could not be simply adapted to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' 7 Conclusion In this work, we have considered truthful auto-bidding auc- tion design for bidders with private budget and ROI con- straints across multiple impressions, while the values of im- pressions are public to the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We have characterized the truthfulness conditions in this new auto-bidding auction model, which involves irregular grouping constraints on bid- ders’ cumulative utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We have proposed a series of sim- ple truthful auction mechanisms with flexible rank score func- tions as a solution to this automated bidding auction design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The experimental results validate the performances and flexibility of the proposed auction mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' References [Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', 2019] Gagan Aggarwal, Ashwinkumar Badanidiyuru, and Aranyak Mehta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} 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advances and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' ACM Transactions on Intelligent Systems and Technology (TIST), 5(4):1–34, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Szymanski and Lee, 2006] Boleslaw K Szymanski and Juong-Sik Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Impact of roi on bidding and revenue in sponsored search advertisement auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In Second Work- shop on Sponsored Search Auctions, volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Citeseer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' [Varian, 2007] Hal R Varian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Position auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Interna- tional Journal of Industrial Organization, 25(6):1163– 1178, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A Detailed Proofs for Results During the proofs of results, for ease of notations, we may drop the term of other bidders’ reported types t−i and the subscript i in u, v, and p, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', vi((Bi, Ri), t−i) to v(Bi, Ri) when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' With dropped t−i and i, we imply that all the mentioned statements should hold for every pos- sible t−i ∈ T−i for bidder i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We omit the proofs of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='7 since we have detailedly explained how to derive them in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3 To help prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3, we consider the following Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1, which provides additional insight on the struc- ture of feasible allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' It describes the following fact: if some type A has both a higher budget and higher ROI than another type B (indicating we cannot simply distinguish who would have higher utility from their types), and they turn out to share the same cumulative value, then the corresponding lower utility diagonal type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', the type who has the lower budget and higher ROI, must have the same cumulative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For any auction satisfying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2, if vi((B′ i, Ri), t−i) = vi((Bi, R′ i), t−i) = c for some c ≥ 0, i ∈ [n], t−i ∈ T−i, B′ i > Bi and R′ i < Ri, then vi((Bi, Ri), t−i) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2, vi(Bi, Ri) ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Sup- pose vi(Bi, Ri) < c, then v(B′ i, Ri) > v(Bi, Ri), indicat- ing p(B′ i, Ri) > Bi and ROI(B′ i, Ri) ≥ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Similarly, v(Bi, R′ i) > v(Bi, Ri), indicating ROI(Bi, R′ i) < Ri and p(Bi, R′ i) ≤ Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Thus, we have p(Bi, R′ i) ≤ Bi < p(B′ i, Ri) and ROI(Bi, R′ i) < Ri ≤ ROI(B′ i, Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As vi = pi × ROIi, v(Bi, R′ i) < v(B′ i, Ri), leading to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3) The only if direction directly follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For the if direction, we divide the possible misreporting into four categories according to their relationship with the true type ti = (Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='Ri), and verify none of them could achieve higher utility when conditions in Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Suppose the bidder misreports (B′ i, R′ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The cases B′ i = Bi or R′ i = Ri are covered directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' B′ i > Bi and R′ i > Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As R′ i > Ri, v(Bi, Ri) ≥ v(Bi, R′ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If v(B′ i, R′ i) > v(Bi, Ri), then v(B′ i, R′ i) > v(Bi, R′ i), indicating p(B′ i, R′ i) > Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Therefore, if misre- porting could bring higher cumulative value, then the pay- ment would break the bidder’s budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' B′ i > Bi and R′ i < Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We would have v(B′ i, R′ i) ≥ v(Bi, R′ i) ≥ v(Bi, Ri), and v(B′ i, R′ i) ≥ v(B′ i, Ri) ≥ v(Bi, Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If v(B′ i, R′ i) > v(Bi, Ri), at least one inequal- ity strictly holds respectively in the above two inequality formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If v(B′ i, R′ i) > v(Bi, R′ i), then p(B′ i, R′ i) > Bi, which breaks the bidder’s budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Thus, v(B′ i, R′ i) = v(Bi, R′ i) > v(Bi, Ri) should hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Simi- larly, v(B′ i, R′ i) = v(B′ i, Ri) > v(Bi, Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We then have v(B′ i, R′ i) = v(B′ i, Ri) = v(Bi, R′ i) > v(Bi, Ri), which contradicts with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' B′ i < Bi and R′ i > Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' According to condition (1), the bidder gets lower allocated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' B′ i < Bi and R′ i < Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' This case is symmetric to B′ i > Bi and R′ i > Ri, and we omit the proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since A is feasible, there exists corresponding P′ that results in DSIC and IR mechanism (A, P′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Notice pi(Bi, Ri) = min (vi(Bi, Ri)/Ri, Bi) is the possible largest payment that can be extracted from bidder i with type (Bi, Ri) to satisfy the bidder’s ROI and budget requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' That is to say, p′ i(Bi, Ri) ≤ pi(Bi, Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' On the other hand, increasing payment of a type will not break originally satis- fied conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As modifying from p′ i to pi(Bi, Ri) = min (vi(Bi, Ri)/Ri, Bi) satisfies the bud- get/ROI requirement of the bidder and maintaining the DSIC conditions, (A, P) is a truthful mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The if direction could be verified through applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For the only if direction, condition (1) directly follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To show condition (2), if v(Bi, Ri) < BiRi, then p(Bi, Ri) ≤ v(Bi, Ri)/Ri < Bi, leading to contradiction with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='3 when B′ i → B− i (denoting some B′ i < Bi infinitely close to Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The analysis is similar for condition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='4 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' When thri(Bi) = 0, the statement is vacuously true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' When thri(Bi) > 0, lim R→thri(Bi)− vi(Bi, R) ≥ thri(Bi) × Bi from definition of thr function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' By the monotone prop- erty, for any Ri < thri(Bi), vi(Bi, Ri) > Bi × Ri, in- dicating vi(Bi, Ri) = limR→R+ i vi(Bi, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' As a result, vi(Bi, Ri) = vi(Bi, thri(Bi)) for any Ri ≤ thri(Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If vi(Bi, thri(Bi)) > thri(Bi) × Bi, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5, lim R→thri(Bi)+ vi(Bi, R) = vi(Bi, thri(Bi)) > thri(Bi) × Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If vi(Bi, thri(Bi)) < thri(Bi) × Bi, there would exist some R → thri(Bi)− whose vi(Bi, R) < Bi×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Both cases con- tradicts of the definition of supremum in thr function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Thus, vi(Bi, thri(Bi)) = thri(Bi) × Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We consider the map m as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (m◦g)(B, R) = � � � � � � � g(B) if R ≤ g(B) B g � sup B′≤B {R ≤ g(B′) B′ } � otherwise, where we define sup ∅ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Given a non-decreasing func- tion g, we could derive the corresponding unique cumulative value function v based on the construction of m, and the con- structed cumulative value function must be feasible since ev- ery type satisfies conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Inversely, given any possible cumulative value function vi, we can find the unique threshold ROI function by definition thri(B, t−i) = sup R∈Ri (vi ((B, R), t−i) ≥ B × R) , and compute the corresponding gi(Bi, t−i) = thri(Bi, t−i) × Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Due to the the truthfulness conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5, such derived gi(Bi, t−i) must be non-decreasing in Bi since the cumulative value when R is infinitely close to 0 should be non-decreasing for different Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='6 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We provide the proof through figuring out the function g(B) for each bidder i when t−i is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Fixing bidder i, given cj and vi,j, we rank the items by ri,j = f −1 i,j ( cj vi,j ) in an decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Without loss of generality, we renumber these items from 1 to m with r1 i > r2 i > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' > rm i and denote the corresponding value as vj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We would then verify that the allocation assigned by our algorithm aligns with the cumulative values implied by fol- lowing function g(B) with p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The corre- sponding threshold ROI function is thr(B) = g(B)/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' g(B) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � r1 i · B if B < v1 i r1 i , p � j=1 vj i if B ∈ [ �p j=1 vj i rp i , �p j=1 vj i rp+1 i ), rp+1 i B if B ∈ [ �p j=1 vj i rp+1 i , �p+1 j=1 vj i rp+1 i ), m � j=1 vj i if B ≥ �m j=1 vj i rm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For convenience in notations, we use Ai(R) to denote the winning set Ai containing all items the bidder has the high- est virtual bid when she reports R (Line 4-6), namely all the items with rj i ≥ R, and use Ai(0) to denote the whole set of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We use Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)) to denote the critical ROI Rc i computed after Line 9 when bidder i reports (B, R), and use d(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)) to represent the corresponding d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We would show the following properties to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (a) Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)) aligns with the threshold ROI function in- dicated by g(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (b) When R ≤ Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)), Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)) = Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)), d(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)) = d(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)), and when R > Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)), R > Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (c) When R ≤ Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)), R ≤ g(B) and the cumula- tive values assigned to the bidder align with those defined by g(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (d) When R > Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)), R > g(B) and the bidder gets all the items in Ai(R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' (e) The corresponding allocation indicated by (d) align with those indicated by g(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To show property (a), we analyse the detailed calculation in Line 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The calculation in Line 9 exactly locates budget B in the segmentation of above function g(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The left-hand side of Line 9 is strictly increasing when Rc i decreases, and is con- tinuous almost everywhere except at points Rc i = {rj i }j∈[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' In detail, when Rc i decreases from (rp+1 i )+ to (rp+1 i ), the left-hand side of Line 9 jumps from B = � �p j=1 vj i rp+1 i �− to �p+1 j=1 vj i rp+1 i , where (·)+ or (·)− represents the one-sided limita- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Therefore, we have Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � r1 i if B < v1 i r1 i , � p � j=1 vj i � /B if B ∈ [ �p j=1 vj i rp i , �p j=1 vj i rp+1 i ), rp+1 i if B ∈ [ �p j=1 vj i rp+1 i , �p+1 j=1 vj i rp+1 i ), � m � j=1 vj i � /B if B ≥ �m j=1 vj i rm i , and d(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � v1 i /r1 i − B if B < v1 i r1 i , 0 if B ∈ [ �p j=1 vj i rp i , �p j=1 vj i rp+1 i ), p+1 � j=1 vj i /rp+1 i − B if B ∈ [ �p j=1 vj i rp+1 i , �p+1 j=1 vj i rp+1 i ), 0 if B ≥ �m j=1 vj i rm i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Comparing thr(B) = g(B)/B with Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)) gives us Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(0)) = thr(B), thus proving property (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To show property (b), we should notice the calculation of Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A(R)) and d(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A(R)) relies on the items with rj i ≥ Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' A(R)), and the existence of other items in A(R) would not affect the result according to the way we compute Rc i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' This provides the proof for the first half of property (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' To show the second half, note that a larger set Ai could only result in an equal or larger Rc i, and Ai(R) ⊂ Ai(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' When R ≤ Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)), R ≤ g(B) follows from prop- erty (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The algorithm goes into Line 15, thus items with rj i < Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)) would be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' After the operations to remove items with value d/Rc i in Line 10-13, the values of remaining items would finally become � j,rj i ≥Rc i (B) vj i − d(B)/Rc i(B), where all the set used to compute Rc i and d here could be regarded as Ai(0) by property (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' It can be verified that the above values align with our defined g(B), thus proving property (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' When R > Rc i(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)), R > g(B) follows from prop- erty (b), and the algorithm would not go into Line 15 to re- move parts of the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' This also means B ≥ �ni j=1 vj i r ni i , where we use ni to denote the item with smallest rj i in Ai(R), and further indicates d(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Ai(R)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Therefore, the algo- rithm would not go into Line 10-13, and no operation about removing the items would be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' The bidder would get all items in Ai(R) after Line 6, which proves property (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' By property (d), we naturally find that the cumulative val- ues are the same for types with the same ROI and locate above the threshold ROI function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' This structure aligns with the non-increasing threshold ROI function indicated by the defined g(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Based on Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='8, in order to prove property (e), we only need to show the type which has the largest budget among all the types with ROI R on the thresh- old ROI function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=', the type (B∗(R), R) with B∗(R) = supB′{B′| thr(B′) = R}, would get an cumulative value equal to the valuations of items in Ai(R) (as characterized in property (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since we have shown the cumulative values for types below and on the threshold ROI function are aligned between our algorithm and defined g(B) in properties (a)-(c), we could use the defined g(B) and corresponding thr(B) to verify the above statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We could find B∗(R) = � � � � � � � � � � � � � � � � � � � � � � � �p j=1 vj i rp i if R = rp i , p � j=1 vj i /R if R ∈ (rp+1 i , rp i ), m � j=1 vj i /R if R ≤ rm i , for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We do not define B∗(R) for R > r1 i since both our algorithm and the defined g(B) determine to not allocate to these types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' We could verify g (B∗(R)) = � j,rj i ≥R vj i , thus finish the proof for property (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' By the above properties (a)-(e), the cumulative value func- tion indicated by our algorithm aligns with the defined g(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Since this g(B) is non-decreasing in B, the mechanism is truthful and preserves the bidders’ financial constraints with the payment in Line 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Finally, our mechanism does not over-allocate items because each item may only be allocated to the bidder with highest virtual bid (breaking possible ties with arbitrary rules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' B Non-Truthfulness of Repeated First-Price and Second-Price Auctions We provide counter-examples to show the non-truthfulness of repeated first-price and second-price auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' For sim- plicity of illustration and understanding, we consider small cases with several bidders and items that do not fit with real situations in automated bidding, while the ideas behind the non-truthfulness are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='1 Repeated First-Price Auctions In repeated first-price auction, the realized ROI of a bidder is exactly her reported ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Thus, bidders have no incentive to misreport a smaller ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' However, reporting a larger ROI can possibly bring the bidder higher utility since it could reduce the payment for per unit gained value if the bidder could still win enough items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Consider a simple case with two bidders and two items shown in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Item 1 Item 2 ROI Budget Bidder 1 v1,1 = 4 v1,2 = 4 R1 = 2 B1 = 3 Bidder 2 v2,1 = 1 v2,2 = 1 R2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 B2 = 6 Assume bidder 2 reports her real type, and we consider the strategy of bidder 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If bidder 1 reports her real type, she will only win item 1 at price of 2, leading the remaining budget to be 1, thus cannot afford the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' However, if bidder 1 reports her ROI higher as R1 = 4, she could win both items at total price 2 and realized ROI 4, getting a higher utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2 Repeated Second-Price Auctions In repeated second-price auctions, both misreporting a larger or smaller ROI might bring the bidder larger utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Consider a simple case with two bidders and two items shown in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Item 1 Item 2 ROI Budget Bidder 1 v1,1 = 4 v1,2 = 8 R1 = 1 B1 = 3 Bidder 2 v2,1 = 4 v2,2 = 4 R2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 B2 = 6 Assume bidder 2 reports her real type, and we consider the strategy of bidder 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If bidder 1 reports her real type, she will win item 1 at price 8/3, leading the remaining budget to be 1/3, thus cannot afford the remaining higher value item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' However, if bidder 1 reports her ROI higher as R1 = 2, then she will lose item 1, and win item 2 at price 8/3 with realized ROI 3, getting a higher utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Reporting a smaller ROI may help the bidder win more items and balance the realized ROI between items (items with real- ized ROI higher or lower than the real ROI both exist) without breaking her true ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Consider a simple case with two bidders and two items shown in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' Item 1 Item 2 ROI Budget Bidder 1 v1,1 = 4 v1,2 = 3 R1 = 2 B1 = 6 Bidder 2 v2,1 = 1 v2,2 = 4 R2 = 2 B2 = 6 Assume bidder 2 reports her real type, and we consider the strategy of bidder 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' If bidder 1 reports her real type, she will win item 1 at price of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 and lose item 2 since bidder 2 has a higher v/R for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content=' However, if bidder 1 reports her ROI lower as R1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='2, then she could win both items at total price 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='5 and realized ROI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} +page_content='8, getting a higher utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odFPT4oBgHgl3EQfKzSD/content/2301.13020v1.pdf'} diff --git a/p9FRT4oBgHgl3EQfdzfW/content/2301.13569v1.pdf b/p9FRT4oBgHgl3EQfdzfW/content/2301.13569v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b95212862f26783728d65af83b27f8f7a0afea9f --- /dev/null +++ b/p9FRT4oBgHgl3EQfdzfW/content/2301.13569v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5f681c568a1f0be5c3f37903bab6d68ab824519de04d90d284b1ec2675a64a8e +size 1095147 diff --git a/p9FRT4oBgHgl3EQfdzfW/vector_store/index.pkl b/p9FRT4oBgHgl3EQfdzfW/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..46ca0910c2d36fb58784f86aad08becaf54248c0 --- /dev/null +++ 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sha256:dcd6b131c7b3d8d1686fd2c51c56f23104cd93133573d34b58806da87c985cd1 +size 124086 diff --git a/qNAyT4oBgHgl3EQfl_iN/content/tmp_files/2301.00463v1.pdf.txt b/qNAyT4oBgHgl3EQfl_iN/content/tmp_files/2301.00463v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3952d7d3fb6055ce826ff143a5ad432782590fb6 --- /dev/null +++ b/qNAyT4oBgHgl3EQfl_iN/content/tmp_files/2301.00463v1.pdf.txt @@ -0,0 +1,3905 @@ +arXiv:2301.00463v1 [math.CA] 1 Jan 2023 +Multi-linear forms, graphs, and Lp-improving measures in Fd +q +Pablo Bhowmick +Alex Iosevich +Doowon Koh +Thang Pham +January 3, 2023 +Contents +1 +Introduction +2 +2 +The spherical averaging problem +8 +3 +Sharp mapping properties for the K2-Operator +10 +4 +Boundedness problem for the K3-Operator +13 +4.1 +Boundedness results for ∆ on Fd +q +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +4.2 +Sharp restricted strong-type estimates in two dimensions . . . . . . . . . . . . . . . +16 +5 +Boundedness results for the P2-Operator +19 +6 +Mapping properties for the (C4 + t)-Operator +23 +6.1 +Boundedness results for ♦t on Fd +q . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +6.2 +Sharp boundedness results up to endpoints for ♦t on F2 +q . . . . . . . . . . . . . . . +26 +7 +Boundedness problem for the C4-Operator +27 +7.1 +Boundedness results for the C4-Operator ✸ on Fd +q +. . . . . . . . . . . . . . . . . . +28 +7.2 +Sharp boundedness results for the C4-Operator ✸ on F2 +q . . . . . . . . . . . . . . . +31 +8 +Boundedness problem for the P3-Operator +32 +8.1 +Boundedness results for the P3-Operator ⊓ on Fd +q . . . . . . . . . . . . . . . . . . . +33 +9 +Operators associated to the graph K3 + tail (a kite) +38 +9.1 +Sufficient conditions for the boundedness of ✂ on Fd +q +. . . . . . . . . . . . . . . . . +39 +9.2 +Boundedness of ✂ in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +10 Boundedness problems for the Y -shaped graph +44 +10.1 Sufficient conditions for the boundedness of Y on Fd +q . . . . . . . . . . . . . . . . . +45 +11 Acknowledgements +48 +12 Appendix +48 +1 + +Abstract +The purpose of this paper is to introduce and study the following graph theoretic paradigm. +Let +TKf(x) = +� +K(x, y)f(y)dµ(y), +where f : X → R, X a set, finite or infinite, and K and µ denote a suitable kernel and a measure, +respectively. Given a connected ordered graph G on n vertices, consider the multi-linear form +ΛG(f1, f2, . . . , fn) = +� +x1,...,xn∈X +� +(i,j)∈E(G) +K(xi, xj) +n +� +l=1 +fl(xl)dµ(xl), +where E(G) is the edge set of G. Define ΛG(p1, . . . , pn) as the smallest constant C > 0 such +that the inequality +ΛG(f1, . . . , fn) ≤ C +n +� +i=1 +||fi||Lpi(X,µ) +(0.1) +holds for all non-negative real-valued functions fi, 1 ≤ i ≤ n, on X. The basic question is, how +does the structure of G and the mapping properties of the operator TK influence the sharp +exponents in (0.1). In this paper, this question is investigated mainly in the case X = Fd +q, the +d-dimensional vector space over the field with q elements, and K(xi, xj) is the indicator function +of the sphere evaluated at xi − xj. This provides a connection with the study of Lp-improving +measures and distance set problems. +1 +Introduction +One of the fundamental objects in harmonic analysis is the operator of the form +TKf(x) = +� +Rd K(x, y)f(y)dy, +(1.1) +where K : Rd × Rd → R is a suitable kernel and f is a locally integrable function. See [21] and +the references contained therein for a variety of manifestations of operators of this type and their +bounds. +The purpose of this paper is to study operators from (1.1) in the context of vector spaces over +finite fields. Let Fq denote the finite field with q elements and Fd +q the d-dimensional vector space +over this field. Let K : Fd +q × Fd +q → C be a suitable kernel, and define +TKf(x) = +� +y∈Fdq +K(x, y)f(y). +Operators of this type have been studied before ([3, 12, 14, 15]). In particular, the operator TK +with K(x, y) = St(x − y), where St is the indicator function of the sphere +St = {x ∈ Fd +q : ||x|| = t}, +||x|| = x2 +1 + x2 +2 + · · · + x2 +d comes up naturally in the study of the Erd˝os-Falconer distance problem +2 + +in vector spaces over finite fields, namely the question of how large a subset E ⊂ Fd +q needs to be +to ensure that if +∆(E) = {||x − y|| : x, y ∈ E}, +then |∆(E)| ≥ q +2. Here and throughout, |S|, with S a finite set, denotes the number of elements +in this set. See, for example ([2, 4, 10, 11, 18]). +If one is interested in studying more complicated geometric objects than distances, an interesting +modification of the spherical averaging operator needs to be made. +Indeed, let E ⊂ Fd +q, and +suppose that we want to know how many equilateral triangles of side-length 1 it determines. The +quantity that counts such triangles is given by +� +x,y,z∈Fdq +K(x, y)K(x, z)K(y, z)E(x)E(y)E(z), +(1.2) +where K(x, y) = S1(x − y). +Let us interpret the quantity (1.2) in the following way. Let us view x, y, z as vertices, and let us +view the presence of K(x, y) as determining the edge connecting x and y, and so on. In this way, +the quantity (1.2) is associated with the graph K3, the complete graph on three vertices (Figure +1b). +Another natural example is the following. Let K(x, y) = S1(x−y), and consider the quantity that +counts rhombi of side-length 1, i.e. +� +x,y,z,w∈Fdq +K(x, y)K(y, z)K(z, w)K(w, x)E(x)E(y)E(z)E(w). +(1.3) +Arguing as above, we associate this form with the graph C4, the cycle on four vertices (Figure 1e). +In general, let K be a kernel function, and let G be a connected ordered graph on n vertices. +Define +ΛG(f1, f2, . . . , fn) = +1 +N(G) +� +x1,...,xn∈Fdq +� +(i,j)∈E(G) +K(xi, xj) +n +� +l=1 +fl(xl), +(1.4) +where E(G) is the edge set of G and N(G) is the normalizing factor defined as the number of +distinct embeddings of G in Fd +q. +We note in passing that the paradigm we just introduced extends readily to the setting of hy- +pergraphs. If we replace our basic object, the linear operator TK, by an m-linear operator MK, +the problem transforms to the setting where the edges dictated by the kernel K are replaced by +hyperedges induced by the multi-linear kernel K(x1, . . . , xm+1). We shall address this formulation +of the problem in the sequel. +The norm ||f||p, 1 ≤ p < ∞, is defined to be associated with normalizing counting measure on Fd +q. +3 + +More precisely, given a function f on Fd +q, we define +||f||p := + +q−d � +x∈Fdq +|f(x)|p + + +1 +p +(1 ≤ p < ∞), +and +||f||∞ := max +x∈Fdq +|f(x)|. +Definition 1.1. Let 1 ≤ pi ≤ ∞, i = 1, . . . , n, be integers. +We define ΛG(p1, . . . , pn) as the +smallest constant C > 0 such that the following inequality +ΛG(f1, . . . , fn) ≤ C +n +� +i=1 +||fi||pi +(1.5) +holds for all non-negative real-valued functions fi, 1 ≤ i ≤ n, on Fd +q. +If there is a uniform constant C ≥ 1 independent of q such that ΛG(p1, . . . , pn) ≤ C, then we use +the notation ΛG(p1, . . . , pn) ≲ 1. +For the remainder of this paper, the kernel function K(x, y) is assumed to be St(x − y) with t ̸= 0. +In addition, when the dimension d is two, we assume that the number 3 in Fq is a square number +so that we can exclude the trivial case in which the shape of an equilateral triangle in F2 +q does not +occur. +The main purpose of this paper is to determine all numbers 1 ≤ pi ≤ ∞, i = 1, 2, . . . , n, such that +ΛG(p1, . . . , pn) ≲ 1. +This problem will be called the boundedness problem for the operator ΛG on Fd +q. +In this paper we shall mainly confine ourselves to the graphs G which possess one of the following +specific structures: K2 (the graph with two vertices and one edge), K3 (the cycle with three vertices +and three edges), K3 + tail (a kite), P2 (the path of length 2), P3 (the path of length 3), C4 (the +cycle with 4 vertices and 4 edges), C4 + diagonal, Y -shape (a space station). It is worth noting +that the study of the distribution of these graphs in a given set has been received much attention +from people in Discrete Geometry and Geometric Measure Theory during the last decade, see for +example [5, 9, 7, 8, 13, 17, 19, 20]. +When the graph G is the K2, the complete answer to the boundedness problem will be given in +all dimensions. To deduce the result, we will invoke the spherical averaging estimates over finite +fields (see Theorem 3.5). +When the number of the vertices of the graph G is 3 or 4, we will obtain reasonably good bound- +edness results in two dimensions. In particular, in the case when the degree of each vertex is +at least two (K3, C4+ a diagonal, and C4), we shall prove sharp results (up to the endpoints) +for the operators on F2 +q (see Theorems 4.11, 6.6, and 7.6). For three and higher dimensions, the +boundedness problem is not simple and we will address partial results. +4 + +(a) G = K2 +(b) G = K3 +(c) G = P2 +(d) G = C4+ diagonal +(e) G = C4 +(f) G = P3 +(g) G = K3+ tail +(h) G = Y -shape +It is very natural to ask whether or not one can prove a general theorem that addresses all connected +graphs on n vertices. Unfortunately, such a result is beyond the scope of this paper. The main +difficulties arise when the maximal degree is large or the edge set is dense, or if the graph contains +a cycle or not. All of these issues will be illustrated in the proofs of our results. +Notation: Let 1 ≤ pi ≤ ∞ and fi, 1 ≤ i ≤ n, be non-negative real-valued functions on Fd +q. +• When n = 2, we denote L(f1, f2) := ΛG(f1, f2) and L(p1, p2) := ΛG(p1, p2). +• (Case of n = 3) +1. If G is the graph K3, we denote ∆(f1, f2, f3) := ΛG(f1, f2, f3) and ∆(p1, p2, p3) := +ΛG(p1, p2, p3). +2. If G is the graph P2, we denote Λ(f1, f2, f3) := ΛG(f1, f2, f3) and Λ(p1, p2, p3) := +ΛG(p1, p2, p3). +• (Case of n = 4) +1. If G is the C4 + diagonal, we denote ♦t(f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and +♦t(p1, p2, p3, p4) := ΛG(p1, p2, p3, p4). +2. If G is the C4, we denote ✸(f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and ✸(p1, p2, p3, p4) := +ΛG(p1, p2, p3, p4). +3. If G is the P3, we denote ⊓(f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and ⊓(p1, p2, p3, p4) := +5 + +ΛG(p1, p2, p3, p4). +4. If G is the K3 + tail (a kite), we denote ✂(f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and +✂(p1, p2, p3, p4) := ΛG(p1, p2, p3, p4). +5. If G is the Y -shape (a space station), we denote Y (f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) +and Y (p1, p2, p3, p4) := ΛG(p1, p2, p3, p4). +• We denote ΛG(p1, . . . , pn) ⪅ 1 if the inequality (1.5) holds true for all characteristic functions +on Fd +q. +• By F∗ +q, we mean the set of all non-zero elements in Fq. +• For t ∈ F∗ +q, we denote by Sn−1 +t +the sphere of radius t centered at the origin in Fn +q : +Sn−1 +t +:= {x ∈ Fn +q : ||x|| = t}. +Unless otherwise specified in this paper, d represents the general dimension of Fd +q, d ≥ 2. +When n = d, we write St instead of Sd−1 +t +for simplicity. +• We identify the set St with its indicator function 1St, namely, St(x) = 1St(x). +• We write δ0 for the indicator function of the set of the zero vector in Fd +q. +• For positive numbers A, B > 0, we write A ≲ B if A ≤ CB for some constant C > 0 +independent of q, the size of the underlying finite field Fq. The notation A ∼ B means that +A ≲ B and B ≲ A. +We also study the boundedness relation between the operators associated with a graph G and its +subgraph G′ with n-vertices. Throughout the paper, we always assume that the graph G and its +subgraph G′ are connected ordered graphs with |G| = |G′| in Fd +q, and two vertices x, y in G is +connected if ||x − y|| = t ̸= 0. +In Theorem 5.6, we will see that any exponents 1 ≤ p1, p2, p3 ≤ ∞ with ∆(p1, p2, p3) ≲ 1 satisfy +that Λ(p1, p2, p3) ≲ 1. Notice that P2 can be considered as a subgraph of K3, and the operators Λ +and ∆ are related to the graphs P2 and K3, respectively. Hence, in view of Theorem 5.6, one may +have a question that, “Compared to a graph G, does the operator associated with its subgraph +yield less restricted mapping exponents?”. More precisely, one may pose the following question. +Question 1.2. Suppose that G′ is a subgraph of the graph G with n vertices in Fd +q. Let 1 ≤ pi ≤ +∞, 1 ≤ i ≤ n. If ΛG(p1, . . . , pn) ≲ 1, is it true that ΛG′(p1, . . . , pn) ≲ 1? +Somewhat surprisingly, the answer turns out to be no! When G = K3 and G′ = P2, the answer +to Question 1.2 is positive as Theorem 5.6 shows. However, it turns out that there exist a graph +G and its subgraph G′ yielding a negative answer, although the answers are positive for the most +graphs which we consider in this paper. For example, the answer to Question 1.2 is negative when +G is the C4 + diagonal and G′ is the C4 (see Theorem 7.7). +6 + +Since the general answer to Question 1.2 is not always positive, we pose the following natural +question. +Problem 1.3. Find general properties of the graph G and its subgraph G′ which yield a positive +answer to Question 1.2. +The main goal of this paper is to address a conjecture on this problem and to confirm it in two +dimensions. To precisely state our conjecture on the problem, let us review the standard definition +and notation for the minimal degree of a graph. +Definition 1.4. The Minimum Degree of a graph G, denoted by δ(G), is defined as the degree of +the vertex with the least number of edges incident to it. +We propose the following conjecture which can be a solution of Problem 1.3. +Conjecture 1.5. Let G′ be a subgraph of the graph G in Fd +q, d ≥ 2, with n vertices, and let +1 ≤ pi ≤ ∞, 1 ≤ i ≤ n. In addition, assume that +min {δ(G), d} > δ(G′). +(1.6) +Then if ΛG(p1, . . . , pn) ≲ 1, we have ΛG′(p1, . . . , pn) ≲ 1. +Note that the condition (1.6) in Conjecture 1.5 is equivalent to the following: +(i) δ(G) > δ(G′) +and +(ii) d > δ(G′). +(1.7) +We have some comments and further questions below, regarding the above conjecture and our +main theorems which we will state and prove in the body of this paper. +• Our results in this paper confirm Conjecture 1.5, possibly up to endpoints, for all graphs G +and their subgraphs G′ on n = 3, 4 vertices in F2 +q (see Theorem 10.8). Also note that when +d = 2 and n = 3, 4, the condition (1.6) is equivalent to the first condition (i) in (1.7) since +δ(G) ≤ 2 (see the figures above). +• It will be shown that the conclusion of Conjecture 1.5 cannot be reversed at least for n = 3, 4 +in two dimensions (see Remark 5.7 for n = 3, and see Remarks 8.10, 8.13, 9.12, 10.6 for +n = 4.) +It is worth investigating whether the key hypothesis (1.6) of Conjecture 1.5 can be relaxed. +• The conclusion of Conjecture 1.5 does not hold in general if the > in the assumption (1.6) +is replaced by ≥. To see this, consider G = C4+diagonal and G′ = C4 on 4 vertices in Fd +q, +with d = 2. It is obvious that G′ is a subgraph of G and min{δ(G), d} = δ(G′) = 2. However, +Theorem 7.7 (ii) implies that the conclusion of Conjecture 1.5 is not true. +• We are not sure what can we say about the conclusion of Conjecture 1.5 if the main hypothesis +(1.6) of Conjecture 1.5 is relaxed by the second one of the conditions (1.7). To be precise, +when (1.6) is replaced by the second statement of (1.7), that is, d > δ(G′), we do not have +7 + +a definitive answer even for n = 4 in F2 +q. For instance, let G = Y -shape and G′ = K3+ +a tail on F2 +q. Then it is clear that d = 2 > δ(G′) = 1 and 1 = δ(G) = δ(G′) and so this +provides an example that does not satisfy the assumption (1.6) of Conjecture 1.5 but satisfy +the second statement of (1.7). Unfortunately, in this paper we have not found any inclusive +bounedness relations between the operators corresponding to such graphs. In order to exclude +this uncertain case, both conditions in (1.7) were taken as the hypothesis for Conjecture 1.5, +namely the condition (1.6). +The rest of this paper is organized as follows: In Section 2, we recall known results on the spherical +averaging operator, which functions as a fundamental tool to prove our theorems. Sections 3 to 10 +are devoted to the presentation and proofs of our main results associated to the graphs mentioned +above. The appendix contains some technical lemmas on the number of intersection points of two +spheres in Fd +q. +2 +The spherical averaging problem +In the finite field setting, Carbery, Stones, and Wiright [3] initially formulated and studied the +averaging problem over the varieties defined by vector valued polynomials. This problem for general +varieties was studied by Chun-Yen Shen and the third listed author [14]. Here, we introduce the +standard results on the averaging problem over the spheres. We adopt the notation in [14]. +Let dx be the normalizing counting measure on Fd +q. For each non-zero t, we endow the sphere St +with the normalizing surface measure dσt. We recall that +dσt(x) = qd +|St|1St(x)dx +so that we can identify the measure dσt with the function +qd +|St|1St on Fd +q. +The spherical averaging operator ASt is defined by +AStf(x) = f ∗ dσt(x) = +� +St +f(x − y)dσt(y) = +1 +|St| +� +y∈St +f(x − y), +(2.1) +where f is a function on Fd +q. By a change of variables, we also have +AStf(x) = +1 +|St| +� +y∈Fdq +St(x − y)f(y). +(2.2) +For 1 ≤ p, r ≤ ∞, we define ASt(p → r) to be the smallest number such that the averaging estimate +||f ∗ dσt||Lr(Fdq,dx) ≤ ASt(p → r)||f||Lp(Fdq,dx) +(2.3) +holds for all test functions f on Fd +q. +8 + +Problem 2.1 (Spherical averaging problem). Determine all exponents 1 ≤ p, r ≤ ∞ such that +ASt(p → r) ≲ 1. +Notation 2.2. From now on, we simply write A for the spherical averaging operator ASt. +By testing (2.3) with f = δ0 and by using the duality of the averaging operator, it is not hard to +notice that the necessary conditions for the boundedness of A(p → r) are as follows: (1/p, 1/r) is +contained in the convex hull of points (0, 0), (0, 1), (1, 1), and ( +d +d+1, +1 +d+1). +Using the Fourier decay estimate on St and its cardinality, it can be shown that these necessary +conditions are sufficient. For the reader’s convenience, we give a detail proof although the argument +is standard, as is well known in the literature such as [3, 14]. +Theorem 2.3. Let 1 ≤ p, r ≤ ∞ be numbers such that (1/p, 1/r) lies on the convex hull of points +(0, 0), (0, 1), (1, 1), and ( +d +d+1, +1 +d+1). Then we have A(p → r) ≲ 1. +Proof. Since both dσt and dx have total mass 1, it follows from Young’s inequality for convolution +functions that if 1 ≤ r ≤ p ≤ ∞, then +∥f ∗ dσt∥Lr(Fdq,dx) ≤ ∥f∥Lp(Fdq,dx). +(2.4) +We notice that these results do not hold for the Euclidean Averaging problem. +By the interpolation and the duality, we only need to establish the following critical estimate: +A +�d + 1 +d +→ d + 1 +� +≲ 1. +It is well known that for non-zero t, +|(dσt)∨(m)| := +����� +1 +|St| +� +x∈St +χ(m · x) +����� ≲ q− (d−1) +2 +for all m ̸= (0, . . . , 0), +where χ denotes a non-trivial additive character of Fq (see the proof of Lemma 2.2 in [11]). +Since |St| ∼ qd−1, we complete the proof by combining this Fourier decay estimate with the +following lemma. +Lemma 2.4. Let dσ be the normalized surface measure on an variety S in Fd +q with |S| ∼ qd−1. If +|(dσ)∨(m)| ≲ q− k +2 for all m ∈ Fd +q \ (0, . . . , 0) and for some k > 0, then we have +A +�k + 2 +k + 1 → k + 2 +� +≲ 1. +Proof of Lemma 2.4 Define K = (dσ)∨ − δ0. We want to prove that for every function f on +9 + +(Fd +q, dx), +∥f ∗ dσ∥Lk+2(Fdq,dx) ≲ ∥f∥ +L +k+2 +k+1 (Fdq,dx), +where dx is the normalized counting measure on Fd +q. +Since dσ = �K + �δ0 = �K + 1 and ∥f ∗ 1∥Lk+2(Fdq,dx) ≲ ∥f∥ +L +k+2 +k+1 (Fdq,dx), it suffices to show that for +every f on (Fd +q, dx), +∥f ∗ �K∥Lk+2(Fdq,dx) ≲ ∥f∥ +L +k+2 +k+1 (Fdq,dx). +(2.5) +Notice that this inequality can be obtained by interpolating the following two estimates: +∥f ∗ �K∥L2(Fdq,dx) ≲ q− k +2 ∥f∥L2(Fdq,dx), +(2.6) +and +∥f ∗ �K∥L∞(Fdq,dx) ≲ q∥f∥L1(Fdq,dx). +(2.7) +It remains to prove the above two estimates. The inequality (2.6) follows from the Plancherel +theorem, the size assumption of |(dσ)∨|, and the definition of K. On the other hand, the inequality +(2.7) follows from Young’s inequality and the observation that ∥ �K∥L∞(Fdq,dx) ≲ q. +The boundary points of the convex hull play an important role in the application of Theorem 2.3. +More precisely, we will apply the following result. +Lemma 2.5. Let 1 ≤ p, r ≤ ∞ and let A denote the averaging operator over the sphere St, t ̸= 0, +in Fd +q, d ≥ 2. +(i) If 1 ≤ 1 +p ≤ +d +d+1 and 1 +r = +1 +dp, then A(p → r) ≲ 1. +(ii) If +d +d+1 ≤ 1 +p ≤ 1 and 1 +r = d +p − d + 1, then A(p → r) ≲ 1. +Proof. Let L1 and L2 denote the line segments connecting the two points (0, 0), (d/(d+1), 1/(d+1)) +and the two points (d/(d + 1), 1/(d + 1)), (1, 1), respectively. Theorem 2.3 implies that for any +point (1/p, 1/r) lying on either L1 or L2, we have A(p → r) ≲ 1. Hence, the first part of the +lemma follows since L1 = {(1/p, 1/r) : 0 ≤ 1/p ≤ d/(d + 1), 1/r = 1/(dp)}. Similarly, since +L2 = {(1/p, 1/r) : d/(d + 1) ≤ 1/p ≤ 1, 1/r = d/p − d + 1}, the second part of the lemma is +obtained, as required. +3 +Sharp mapping properties for the K2-Operator +In this section, we provide the sharp mapping properties of the operator associated to the graph +K2. To this end, as described below, we relate the problem to the spherical averaging problem. +As usual, the inner product of the non-negative real-valued functions f, g on Fd +q is defined as +< f, g >:= ||fg||1 = 1 +qd +� +x∈Fdq +f(x)g(x). +10 + +Let t ∈ F∗ +q and let f1, f2 be non-negative real-valued functions on Fd +q, d ≥ 2. Then the operator L +associated with the graph K2 is defined as +L(f1, f2) = +1 +qd|St| +� +x1,x2∈Fdq +St(x1 − x2)f1(x1)f2(x2), +(3.1) +and it is called the K2-Operator on Fd +q. Here, the quantity qd|St| represents the normalizing factor +N(G) in (1.4) when the graph G is K2. +By a change of variables, we can write +L(f1, f2) = 1 +qd +� +x1∈Fdq +f1(x1) + + 1 +|St| +� +x2∈Fdq +f2(x1 − x2)St(x2) + + =< f1, Af2 >, +(3.2) +where A denotes the averaging operator related to the sphere St. Likewise, we also obtain that +L(f1, f2) =< Af1, f2 > . +In a usual way, we define the operator norm of the K2-Operator L as follows. +Definition 3.1. Let 1 ≤ p1, p2 ≤ ∞. We define L(p1, p2) as the smallest constant such that the +following estimate holds for all functions f1, f2 on Fd +q : +L(f1, f2) ≤ L(p1, p2)||f1||p1||f2||p2. +The main goal of this section is to address all numbers 1 ≤ p1, p2 ≤ ∞ satisfying L(p1, p2) ≲ 1. +We begin with the necessary conditions for the boundedness of the K2-Operator L on Fd +q. +Proposition 3.2. Let 1 ≤ p1, p2 ≤ ∞. Suppose that L(p1, p2) ≲ 1. Then we have +1 +p1 ++ d +p2 +≤ d +and +d +p1 ++ 1 +p2 +≤ d. +Proof. By symmetry, it is clear that L(p1, p2) ≲ 1 ⇐⇒ L(p2, p1) ≲ 1. Hence, it suffices to prove +the first listed conclusion that +1 +p1 + d +p2 ≤ d. +From (3.2) and our assumption that L(p1, p2) ≲ 1, we must have +L(f1, f2) := 1 +qd +� +x1∈Fdq +f1(x1) + + 1 +|St| +� +x2∈Fdq +f2(x1 − x2)St(x2) + + ≲ ||f1||p1||f2||p2. +We test this inequality with f1 = 1St and f2 = δ0. Then +L(f1, f2) = 1 +qd +� +x1∈St +1 +|St| = q−d, +and +||f1||p1||f2||p2 ∼ (q−d|St|)1/p1(q−d)1/p2 ∼ q− 1 +p1 − d +p2 . +11 + +By a direct comparison, we get the desired result. +Remark 3.3. For 1 ≤ p1, p2 ≤ ∞, one can note that +1 +p1 + d +p2 ≤ d and +d +p1 + 1 +p2 ≤ d if and only if +(1/p1, 1/p2) ∈ [0, 1] × [0, 1] lies on the convex hull of points (0, 0), (0, 1), ( +d +d+1 , +d +d+1), (1, 0). +Let us move to the sufficient conditions on the exponents 1 ≤ p1, p2 ≤ ∞ such that L(p1, p2) ≲ 1. +We will invoke the following lemma which demonstrates that the boundedness of the K2-Operator +L can be directly determined by the spherical averaging estimates over the finite fields. +For 1 ≤ p ≤ ∞, we denote by p′ the H¨older conjugate of p, namely, 1/p + 1/p′ = 1. +Lemma 3.4. Suppose that A(p2 → p′ +1) ≲ 1 with 1 ≤ p1, p2 ≤ ∞. Then we have +L(p1, p2) ≲ 1. +Proof. Since L(f1, f2) =< f1, Af2 >, it follows by H¨older’s inequality that +L(f1, f2) ≤ ||f1||p1||Af||p′ +1 ≲ ||f1||p1||f2||p2, +where the averaging assumption was used in the last inequality. +We now show that the necessary conditions are in fact sufficient conditions for L(p1, p2) ≲ 1. +Theorem 3.5 (Sharp boundedness result for the K2-Operator on Fd +q). Let 1 ≤ p1, p2 ≤ ∞. Then +we have +L(p1, p2) ≲ 1 +if and only if +1 +p1 ++ d +p2 +≤ d, +d +p1 ++ 1 +p2 +≤ d. +Proof. By Proposition 3.2, it will be enough to prove that L(p1, p2) ≲ 1 for all 1 ≤ p1, p2 ≤ ∞ +satisfying +1 +p1 ++ d +p2 +≤ d, +d +p1 ++ 1 +p2 +≤ d. +By the interpolation theorem and the nesting property of the norm, it suffices to establish the +estimates on the critical end-points (1/p1, 1/p2) ∈ [0, 1] × [0, 1], which are (0, 1), (1, 0), and (d/(d + +1), d/(d + 1)). In other words, it remains to prove the following estimates: +L(∞, 1) ≲ 1, L(1, ∞) ≲ 1, L +�d + 1 +d +, d + 1 +d +� +≲ 1. +Using Lemma 3.4, matters are reduced to establishing the following averaging estimates: +A(1 → 1) ≲ 1, +A(∞ → ∞) ≲ 1, +A +�d + 1 +d +→ d + 1 +� +≲ 1. +However, these averaging estimates are clearly valid by Theorem 2.3. Hence, the proof is complete. +The following result is a special case of Theorem 3.5, but it is very useful in practice. +12 + +Corollary 3.6. For any dimensions d ≥ 2, we have L +� d+1 +d , d+1 +d +� +≲ 1. +Proof. Notice that if p1 = p2 = d+1 +d , then it satisfies that +1 +p1 + d +p2 ≤ d and +d +p1 + 1 +p2 ≤ d. Hence, the +statement follows immediately from Theorem 3.5. +4 +Boundedness problem for the K3-Operator +Let t ∈ F∗ +q. The operator ∆ related to the graph K3 can be defined as +∆(f1, f2, f3) = 1 +qd +1 +|St||Sd−2 +t +| +� +x1,x2,x3∈Fdq +St(x1 − x2)St(x2 − x3)St(x3 − x1) +3 +� +i=1 +fi(xi), +(4.1) +where each fi, i = 1, 2, 3, is a non-negative real-valued function on Fd +q, and the quantity qd|St||Sd−2 +t +| +stands for the normalizing factor N(G) in (1.4) when G = K3. We name the operator ∆ as the +K3-Operator on Fd +q. +Definition 4.1. Let 1 ≤ p1, p2, p3 ≤ ∞. We define ∆(p1, p2, p3) as the best constant such that the +following estimate holds for all non-negative real-valued functions fi, i = 1, 2, 3, on Fd +q : +∆(f1, f2, f3) ≤ ∆(p1, p2, p3)||f1||p1||f2||p2||f3||p3. +The purpose of this section is to find the numbers 1 ≤ p1, p2, p3 ≤ ∞ such that ∆(p1, p2, p3) ≲ 1. +When the dimension d is two, we will settle this problem up to the endpoint estimate. To this end, +we relate our problem to the estimate of the Bilinear Averaging Operator (see (4.2)) for which we +establish the sharp bound. +On the other hand, as we shall see, in three and higher dimensions d ≥ 3, it is not easy to deduce +the sharp results. +However, when one of the exponents p1, p2, p3 is ∞, we will able to obtain +the optimal results. This will be done by applying Theorem 3.5, the boundedness result for the +K2-Operator L on Fd +q. +We begin by deducing necessary conditions for our problem in Fd +q, d ≥ 2. Recall that for d = 2, we +pose an additional restriction that 3 ∈ Fq is a square number. +Proposition 4.2 (Necessary conditions for the boundedness of ∆). Let 1 ≤ p1, p2, p3 ≤ ∞. +Suppose that ∆(p1, p2, p3) ≲ 1. Then we have +d +p1 ++ 1 +p2 ++ 1 +p3 +≤ d, +1 +p1 ++ d +p2 ++ 1 +p3 +≤ d, +1 +p1 ++ 1 +p2 ++ d +p3 +≤ d. +In particular, when d = 2, it can be shown by Polymake1[1, 6] that (1/p1, 1/p2, 1/p3) is contained in +the convex hull of the points: (0, 0, 1), (0, 1, 0), (2/3, 2/3, 0), (1/2, 1/2, 1/2), (2/3, 0, 2/3), (1, 0, 0), +(0, 0, 0), (0, 2/3, 2/3). +1Polymake is software for the algorithmic treatment of convex polyhedra. +13 + +Proof. We only prove the first inequality in the conclusion since we can establish other inequalities +by symmetric property of ∆(f1, f2, f3). We will use the simple fact that x ∈ St if and only if +−x ∈ St. In the definition (4.1), taking f1 = δ0, f2 = 1St, and f3 = 1St, we see that +||f1||p1||f2||p2|||f3||p3 ∼ q− d +p1 q− 1 +p2 q− 1 +p3 , +∆(f1, f2, f3) = 1 +qd +1 +|St||Sd−2 +t +| + + +� +x2,x3∈St:||x2−x3||=t +1 + + ∼ q−d, +where the last similarity above follows from Corollary 12.4 in Appendix with our assumption that +3 ∈ Fq is a square number for d = 2. +By the direct comparison of these estimates, we obtain the required necessary condition. +Remark 4.3. In order to prove that the necessary conditions in Proposition 4.2 are sufficient con- +ditions for d = 2, we only need to establish the following critical endpoint estimates: ∆(2, 2, 2) ≲ +1, ∆(∞, ∞, ∞) ≲ 1, ∆(1, ∞, ∞) ≲ 1, ∆(∞, 1, ∞) ≲ 1, ∆(∞, ∞, 1) ≲ 1, ∆ +�3 +2, 3 +2, ∞ +� +≲ 1, +∆ +� 3 +2, ∞, 3 +2 +� +≲ 1, ∆ +� +∞, 3 +2, 3 +2 +� +≲ 1. In fact, this claim follows by interpolating the critical points +given in the second part of Proposition 4.2. +4.1 +Boundedness results for ∆ on Fd +q +The graph K2 can be obtained by removing any one of three vertices in the graph K3. Therefore, +the boundedness of L(p1, p2) can determine the boundedness of ∆(p1, p2, ∞). More precisely, we +have the following result. +Lemma 4.4. Let 1 ≤ a, b ≤ ∞. If L(a, b) ≲ 1, then +∆(a, b, ∞) ≲ 1, +∆(a, ∞, b) ≲ 1, +and +∆(∞, a, b) ≲ 1. +Proof. By symmetry, to complete the proof, it suffices to prove that ∆(a, b, ∞) ≲ 1. +Recall from (4.1) and (3.1) that +∆(f1, f2, f3) = 1 +qd +1 +|St||Sd−2 +t +| +� +x1,x2,x3∈Fdq +St(x1 − x2)St(x2 − x3)St(x3 − x1) +3 +� +i=1 +fi(xi), +and +L(f1, f2) = +1 +qd|St| +� +x1,x2∈Fdq +St(x1 − x2)f1(x1)f2(x2). +Since fi, i = 1, 2, 3, are non-negative real-number functions on Fd +q, we have +∆(f1, f2, f3) ≤ +1 +|Sd−2 +t +| + + +max +x1,x2∈Fdq:||x1−x2||=t +� +x3∈Fdq:||x2−x3||=t=||x3−x1|| +f3(x3) + + L(f1, f2). +14 + +Since |Sd−2 +t +| ∼ qd−2 and L(f1, f2) ≲ ||f1||a||f2||b by our assumption, it suffices to prove that the +maximum value in the above parenthesis is ≲ qd−2||f3||∞. Let us denote by I the maximum above. +By a change of variables, x = x1, y = x1 − x2, we see that +I ≤ + + +max +x∈Fdq,y∈St +� +x3∈Fdq:||x−y−x3||=t=||x3−x|| +1 + + ||f3||∞. +By another change of variables by putting z = x − x3, we get +I ≤ + + +max +x∈Fdq,y∈St +� +z∈St:||z−y||=t +1 + + ||f3||∞ = + +max +y∈St +� +z∈St:||z−y||=t +1 + + ||f3||∞. +Now applying Corollary 12.4 in Appendix, we conclude that I ≲ qd−2||f3||∞ as required. +When one of p1, p2, p3 is ∞, we are able to obtain sharp boundedness results for ∆(p1, p2, p3). +Theorem 4.5. Let 1 ≤ a, b ≤ ∞ satisfy that 1 +a+ d +b ≤ d and d +a+ 1 +b ≤ d. Then we have ∆(a, b, ∞) ≲ 1, +∆(a, ∞, b) ≲ 1 and ∆(∞, a, b) ≲ 1. +Proof. The statement follows immediately by combining Theorem 3.5 and Lemma 4.4. +It is not hard to see from Proposition 4.2 that the above theorem cannot be improved in the case +when one of the exponents p1, p2, p3 is ∞. In particular, we have the following critical endpoint +estimates. +Corollary 4.6. We have ∆ +� d+1 +d , d+1 +d , ∞ +� +≲ 1, ∆ +� d+1 +d , ∞, d+1 +d +� +≲ 1, and ∆ +� +∞, d+1 +d , d+1 +d +� +≲ 1. +Proof. Put a = b = d+1 +d . Then we see that 1 +a + d +b = d = d +a + 1 +b, Hence, the statement of the corollary +follows directly from Theorem 4.5. +Theorem 4.5 also implies the following result. +Corollary 4.7. We have ∆(∞, ∞, ∞) ≲ 1, ∆(1, ∞, ∞) ≲ 1, ∆(∞, 1, ∞) ≲ 1, ∆(∞, ∞, 1) ≲ 1. +Proof. By taking a = b = ∞ in Theorem 4.5, we obtain the estimate that ∆(∞, ∞, ∞) ≲ 1. Now, +by symmetry, it will be enough to show that ∆(1, ∞, ∞) ≲ 1. However, this is easily shown by +taking a = 1, b = ∞ in Theorem 4.5. Thus, the proof is complete. +Remark 4.8. From Corollary 4.6, Corollary 4.7, and Remark 4.3, we see that to completely solve +the problem on the boundedness of ∆(p1, p2, p3) for d = 2, we only need to establish the following +critical endpoint estimate +∆ +�d + 2 +d +, d + 2 +d +, d + 2 +d +� += ∆(2, 2, 2) ≲ 1. +15 + +4.2 +Sharp restricted strong-type estimates in two dimensions +Although Theorem 4.5 is valid for all dimensions d ≥ 2, it is not sharp, compared to the necessary +conditions given in Proposition 4.2. In this subsection, we will deduce the sharp boundedness +results up to the endpoints for ∆(p1, p2, p3) in two dimensions. To this end, we need the following +theorem. +Theorem 4.9. Let ∆ be the K3-Operator on F2 +q. Then, for all subsets E, F, H of F2 +q, the following +estimate holds: ∆(E, F, H) ≲ ||E||2||F||2||H||2. +For 1 ≤ p1, p2, p3 ≤ ∞, we say that the restricted strong-type ∆(p1, p2, p3) estimate holds if the +estimate +∆(E, F, H) ≲ ||E||p1||F||p2||H||p3 +is valid for all subsets E, F, H of F2 +q. In this case, we write ∆(p1, p2, p3) ⪅ 1. +Proof. The proof proceeds with some reduction. When d = 2, by a change of variables by letting +x = x3, y = x3 − x1, z = x3 − x2, (4.1) becomes +∆(f1, f2, f3) = 1 +q2 +� +x∈F2q +f3(x) + + 1 +|St| +� +y,z∈F2q +St(z − y)St(z)St(y)f1(x − y)f2(x − z) + + . +We define B(f1, f2)(x) as the value in the bracket above, namely, +B(f1, f2)(x) := +1 +|St| +� +y,z∈St:||z−y||=t +f1(x − y)f2(x − z). +(4.2) +We refer to this operator B as “the bilinear averaging operator”. It is clear that +∆(f1, f2, f3) = 1 +q2 +� +x∈F2q +B(f1, f2)(x)f3(x) =< B(f1, f2), f3 > . +By H¨older’s inequality, we have +∆(f1, f2, f3) ≤ ||B(f1, f2)||2||f3||2. +Thus, Theorem 4.9 follows immediately from the reduction lemma below +Lemma 4.10. Let B(f1, f2) be the bilinear averaging operator defined as in (4.2). Then, for all +subsets E, F of F2 +q, we have +||B(E, F)||2 ≲ ||E||2||F||2. +Proof. We begin by representing the bilinear averaging operator B(f1, f2). From (4.2), note that +B(f1, f2)(x) = +1 +|St| +� +y∈St +f1(x − y) + + +� +z∈St:||z−y||=t +f2(x − z) + + . +16 + +For each y ∈ St, let Θ(y) := {z ∈ St : ||z − y|| = t}. With this notation, the bilinear averaging +operator is written as +B(f1, f2)(x) = +1 +|St| +� +y∈St +f1(x − y) + + � +z∈Θ(y) +f2(x − z) + + . +Let η denote the quadratic character of F∗ +q. Recall that η(s) = 1 for a square number s in F∗ +q, and +η(s) = −1 otherwise. Notice from Corollary 12.4 in Appendix that Θ(y) is the empty set for all +y ∈ St if d = 2 and η(3) = −1. In this case, the problem is trivial since B(f1, f2)(x) = 0 for all +x ∈ F2 +q. Therefore, when d = 2, we always assume that η(3) = 1. +Notice that |Θ(y)| = 2 for all y ∈ St, which follows from the last statement of Corollary 12.4 in +Appendix. More precisely, for each y ∈ St, we can write +Θ(y) = {θy, θ−1y}, +where θy denotes the rotation of y by “60 degrees,” and ||y − θy|| = t = ||y − θ−1y||. +From these observations, the bilinear averaging operator B(f1, f2) can be represented as follows: +B(f1, f2)(x) = Bθ(f1, f2)(x) + Bθ−1(f1, f2)(x). +(4.3) +Here, we define +Bθ(f1, f2)(x) := +1 +|St| +� +y∈St +f1(x − y)f2(x − θy), +(4.4) +and +Bθ−1(f1, f2) := +1 +|St| +� +y∈St +f1(x − y)f2(x − θ−1y). +In order to complete the proof of the lemma, it suffices to establish the following two estimates: +for all subsets E, F of Fd +q, +||Bθ(E, F)||2 ≲ ||E||2||F||2, +(4.5) +and +||Bθ−1(E, F)||2 ≲ ||E||2||F||2. +(4.6) +We will only provide the proof of the estimate (4.5) since the proof of (4.6) is the same. +Now we start proving the estimate (4.5). Since ||E||2 +2 = q−2|E| and ||F||2 +2 = q−2|F|, it is enough +to prove that +||Bθ(E, F)||2 +2 ≲ q−4|E||F|. +(4.7) +Without loss of generality, we may assume that |E| ≤ |F|. By the definition, it follows that +||Bθ(E, F)||2 +2 = q−2|St|−2 � +x∈F2q +� +y,y′∈St +E(x − y)E(x − y′)F(x − θy)F(x − θy′) = I + II, +17 + +where the first term I is the value corresponding to the case where y = y′, whereas the second +term II is corresponding to the case where y ̸= y′. We have +I = q−2|St|−2 � +y∈St +� +x∈F2q +E(x − y)F(x − θy). +Applying a change of variables by replacing x with x + y, we see that +I = q−2|St|−2 � +y∈St +� +x∈F2q +E(x)F(x + y − θy) = q−2|St|−2 � +x∈E + +� +y∈St +F(x + y − θy) + + . +Observe that y − θy ̸= y′ − θy′ for all y, y′ in St with y ̸= y′. Then we see that the value in the +parentheses above is bounded above by |St ∩ F| ≤ |F|. Therefore, we obtain the desired estimate: +I ≤ q−2|St|−2|E||F| ∼ q−4|E||F|. +Next, it remains to show that II ≲ q−4|E||F|. Since we have assumed that |E| ≤ |F|, it suffices to +show that II ≲ q−4|E|2. +By the definition of II, it follows that +II = q−2|St|−2 � +x∈F2q +� +y,y′∈St:y̸=y′ +E(x − y)E(x − y′)F(x − θy)F(x − θy′). +It is obvious that +II ≤ q−2|St|−2 +� +y,y′∈St:y̸=y′ +� +x∈F2q +E(x − y)E(x − y′). +We use a change of variables by replacing x with x + y. Then we have +II ≤ q−2|St|−2 � +x∈F2q +E(x) + + +� +y,y′∈St:y̸=y′ +E(x + y − y′) + + += q−2|St|−2 � +x∈F2q +E(x) + + � +0̸=u∈F2q +E(x + u)W(u) + + , +where W(u) denotes the number of pairs (y, y′) ∈ St × St such that u = y − y′ and y ̸= y′. It is +not hard to see that for any non-zero vector u ∈ F2 +q, we have W(u) ≤ 2. So we obtain that +II ≲ q−2|St|−2 � +x∈F2q +E(x) + + � +0̸=u∈F2q +E(x + u) + + ≲ q−4|E|2, +as required. +In two dimensions, we are able to obtain the optimal boundedness of ∆(p1, p2, p3) except for one +endpoint. Indeed, we have the following result. +18 + +Theorem 4.11. Let 1 ≤ p1, p2, p3 ≤ ∞ and let ∆ be the K3-Operator on F2 +q. +(i) If ∆(p1, p2, p3) ≲ 1, then +2 +p1 ++ 1 +p2 ++ 1 +p3 +≤ 2, +1 +p1 ++ 2 +p2 ++ 1 +p3 +≤ 2, +1 +p1 ++ 1 +p2 ++ 2 +p3 +≤ 2. +(4.8) +(ii) Conversely, if (p1, p2, p3) satisfies all three inequalities (4.8), then ∆(p1, p2, p3) ≲ 1 for +(p1, p2, p3) ̸= (2, 2, 2), and we have ∆(2, 2, 2) ⪅ 1. +Proof. The first part of the theorem is the special case of Proposition 4.2 with d = 2. Now we +prove the second part. As stated in Proposition 4.2, one can notice by using Polymake [1, 6] that +all the points +� +1 +p1, 1 +p2, 1 +p3 +� +∈ [0, 1]3 satisfying all three inequalities (4.8) are contained in the convex +hull of the critical points +(1/2, 1/2, 1/2), (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), +�2 +3, 2 +3, 0 +� +, +�2 +3, 0, 2 +3 +� +, +� +0, 2 +3, 2 +3 +� +. +Notice from Theorem 4.9 with d = 2 that the restricted strong-type estimate for the operator +∆ holds for the point (1/p1, 1/p2, 1/p3) = (1/2, 1/2, 1/2). In addition, notice from Corollary 4.6 +and Corollary 4.7 with d = 2 that ∆(p1, p2, p3) ≲ 1 for the above critical points (1/p1, 1/p2, 1/p3) +except for (1/2, 1/2, 1/2). Hence, the statement of the second part follows immediately by invoking +the interpolation theorem. +5 +Boundedness results for the P2-Operator +For t ∈ F∗ +q and functions fi, i = 1, 2, 3, on Fd +q, the operator Λ associated with the graph P2, called +the P2-Operator on Fd +q, is defined as +Λ(f1, f2, f3) = +1 +qd|St|2 +� +x1,x2,x3∈Fdq +St(x1 − x2)St(x2 − x3)f1(x1)f2(x2)f3(x3), +(5.1) +where the quantity qd|St|2 stands for the normalizing factor N(G) in (1.4) when G = P2. Note +that this can be written as +Λ(f1, f2, f3) =< f2, Af1 · Af3 > . +(5.2) +Definition 5.1. Let 1 ≤ p1, p2, p3 ≤ ∞. We define Λ(p1, p2, p3) as the smallest constant such that +the following estimate holds for all non-negative real-valued functions f1, f2, f3 on Fd +q : +Λ(f1, f2, f3) ≤ Λ(p1, p2, p3)||f1||p1||f2||p2||f3||p3. +In this section, we study the problem determining all numbers 1 ≤ p1, p2, p3 ≤ ∞ satisfying +Λ(p1, p2, p3) ≲ 1. Compared to the K3-Operator ∆, this problem is much hard to find the optimal +answers. Based on the formula (5.2) with the averaging estimates in Lemma 2.5, we are able to +19 + +address partial results on this problem (see Theorem 5.4). +Proposition 5.2 (Necessary conditions for the boundedness of Λ). Let 1 ≤ p1, p2, p3 ≤ ∞. Suppose +that Λ(p1, p2, p3) ≲ 1. Then we have +1 +p1 ++ d +p2 ++ 1 +p3 +≤ d, +d +p1 ++ 1 +p2 +≤ d, +1 +p2 ++ d +p3 +≤ d, +d +p1 ++ 1 +p2 ++ d +p3 +≤ 2d − 1. +Also, under this assumption when d = 2, it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3) +is contained in the convex hull of points: (0, 1, 0), (1/2, 0, 1), (1, 0, 1/2),(1, 0, 0), (5/6, 1/3, 1/2), +(1/2, 1/3, 5/6), (2/3, 2/3, 0), (0, 2/3, 2/3), (0, 0, 0), (0, 0, 1). +Proof. Suppose that Λ(p1, p2, p3) ≲ 1. Then, for all functions fi, i = 1, 2, 3, on Fd +q, we have +Λ(f1, f2, f3) = +1 +qd|St|2 +� +x1,x2,x3∈Fdq +St(x1 − x2)St(x2 − x3)f1(x1)f2(x2)f3(x3) ≲ ||f1||p1||f2||p2||f3||p3. +We test the above inequality with f2 = δ0, f1 = f3 = 1St. It is plain to note that +||f1||p1||f2||p2||f3||p3 = (q−d|St|) +1 +p1 q− d +p2 (q−d|St|) +1 +p3 ∼ q− 1 +p1 − d +p2 − 1 +p3 , +and +Λ(f1, f2, f3) = q−d. +Therefore, we obtain that q−d ≲ q− 1 +p1 − d +p2 − 1 +p3 . This implies the first inequality in the conclusion +that +1 +p1 + d +p2 + 1 +p3 ≤ d. +To obtain the second inequality in the conclusion, we choose f1 = δ0, f2 = 1St, and f3 = 1Fdq. Then +it is easy to check that +||f1||p1||f2||p2||f3||p3 ∼ q− d +p1 − 1 +p2 , +and +Λ(f1, f2, f3) = +1 +qd|St|2 +� +x2∈St +� +x3∈Fdq +St(x2 − x3) = q−d. +Comparing these estimates gives the second inequality in the conclusion. +The third inequality in the conclusion can be easily obtained by switching the roles of f1 and f3 +in the proof of the second one. +To deduce the last inequality in the conclusion, we take f1 = f3 = δ0 and f2 = 1St. Then +||f1||p1||f2||p2||f3||p3 ∼ q− d +p1 − 1 +p2 − d +p3 , +and +Λ(f1, f2, f3) = +1 +qd|St| ∼ q−2d+1. +From these, we have the required result that +d +p1 + 1 +p2 + d +p3 ≤ 2d − 1. +20 + +By symmetry, it is not hard to note that Λ(p1, p2, p3) ≲ 1 ⇐⇒ Λ(p3, p2, p1) ≲ 1. In the following +lemma, we prove that the boundedness question for the P2-Operator Λ is closely related to the +spherical averaging problem over finite fields. +Lemma 5.3. Suppose that +1 +r1 + +1 +p2 + 1 +r3 = 1, A(p1 → r1) ≲ 1, and A(p3 → r3) ≲ 1 for some +1 ≤ p1, p2, p3, r1, r3 ≤ ∞. Then we have Λ(p1, p2, p3) ≲ 1. +Proof. Since Λ(f1, f2, f3) =< f2, Af1 · Af3 >, we obtain by H¨older’s inequality with the first +assumption that +Λ(f1, f2, f3) ≤ ||Af1||r1||f2||p2||Af3||r3 ≲ ||f1||p1||f2||p2||f3||p3, +where the averaging assumption was used for the last inequality. +Hence, Λ(p1, p2, p3) ≲ 1, as +required. +Now we state and prove our boundedness results of Λ(p1, p2, p3) on Fd +q. +Theorem 5.4. Let 1 ≤ p1, p2, p3 ≤ ∞. Then, for the P2-Operator Λ on Fd +q, the following four +statements hold: +(i) If 0 ≤ 1 +p1, 1 +p3 ≤ +d +d+1 and +1 +p1 + d +p2 + 1 +p3 ≤ d, then Λ(p1, p2, p3) ≲ 1. +(ii) If 0 ≤ 1 +p1 ≤ +d +d+1 ≤ 1 +p3 ≤ 1 and +1 +dp1 + 1 +p2 + d +p3 ≤ d, then Λ(p1, p2, p3) ≲ 1. +(iii) If 0 ≤ 1 +p3 ≤ +d +d+1 ≤ 1 +p1 ≤ 1 and +d +p1 + 1 +p2 + +1 +dp3 ≤ d, then Λ(p1, p2, p3) ≲ 1. +(iv) If +d +d+1 ≤ 1 +p1, 1 +p3 ≤ 1 and +d +p1 + 1 +p2 + d +p3 ≤ 2d − 1, then Λ(p1, p2, p3) ≲ 1. +Proof. We proceed as follows. +(i) By the nesting property of the norm, it suffices to prove it in the case when 0 ≤ +1 +p1, 1 +p3 ≤ +d +d+1 +and 1 +p1 + d +p2 + 1 +p3 = d. This equation can be rewritten as +1 +dp1 + 1 +p2 + +1 +dp3 = 1. Since 0 ≤ 1 +p1, 1 +p3 ≤ +d +d+1, we see from Lemma 2.5 (i) that letting +1 +r1 = +1 +dp1 , 1 +r3 = +1 +dp3, we have A(p1 → r1) ≲ 1 and +A(p3 → r3) ≲ 1. +Since +1 +r1 + 1 +p2 + 1 +r3 = 1, applying Lemma 5.3 gives the required result. +(ii) As in the proof of the first part of the theorem, it will be enough to prove Λ(p1, p2, p3) ≲ 1 in +the case when 0 ≤ +1 +p1 ≤ +d +d+1 ≤ +1 +p3 ≤ 1 and +1 +dp1 + 1 +p2 + d +p3 = d. Let 1 +r1 = +1 +dp1 and 1 +r3 = d +p3 −d+1. +Then by Lemma 2.5 it follows that A(p1 → r1) ≲ 1 and A(p3 → r3) ≲ 1. Also notice that +1 +r1 + 1 +p2 + 1 +r3 = 1. Hence, Theorem 5.4 (ii) follows from Lemma 5.3. +(iii) Switching the roles of p1, p2, the proof is exactly the same as that of the second part of this +theorem. +(iv) As before, it suffices to prove the case when +d +d+1 ≤ +1 +p1, 1 +p3 ≤ 1 and +d +p1 + 1 +p2 + d +p3 = 2d − 1. +Put +1 +rk = +d +pk − d + 1 for k = 1, 3. Then we see from Lemma 2.5 (ii) that A(pk → rk) ≲ 1 for +k = 1, 3. Notice that +1 +r1 + 1 +p2 + 1 +r3 = 1. Therefore, using Lemma 5.3 we finish the proof. +21 + +As a special case of Theorem 5.4, we obtain the following. +Corollary 5.5. For any dimensions d ≥ 2, we have Λ +� +d+1 +d , d+1 +d−1, d+1 +d +� +≲ 1. +Proof. This clearly follows from Theorem 5.4 by taking p1 = p3 = d+1 +d , and p2 = d+1 +d−1. +While we do not know whether Theorem 5.4 is optimal or not, the result will play a crucial role +in proving the following theorem which implies that Conjecture 1.5 is true for the graph K3 and +its subgraph P2 in all dimensions d ≥ 2 (see Corollary 5.8 below). +Theorem 5.6. Let ∆ and Λ be the operators associated with K3 and P2, respectively, on Fd +q. Then +if ∆(p1, p2, p3) ≲ 1 for 1 ≤ p1, p2, p3 ≤ ∞, we have Λ(p1, p2, p3) ≲ 1. +Proof. Suppose that ∆(p1, p2, p3) ≲ 1 for 1 ≤ p1, p2, p3 ≤ ∞. Then, by Proposition 4.2, the +exponents p1, p2, p3 satisfy the following three inequalities: +d +p1 ++ 1 +p2 ++ 1 +p3 +≤ d, +1 +p1 ++ d +p2 ++ 1 +p3 +≤ d, +1 +p1 ++ 1 +p2 ++ d +p3 +≤ d. +(5.3) +To complete the proof, it remains to show that Λ(p1, p2, p3) ≲ 1. We will prove this by considering +the four cases depending on the sizes of p1 and p3. +Case 1: Suppose that 0 ≤ +1 +p1, 1 +p3 ≤ +d +d+1. The condition (5.3) clearly implies that +1 +p1 + d +p2 + 1 +p3 ≤ d. +Thus, by Theorem 5.4 (i), we obtain the required conclusion that Λ(p1, p2, p3) ≲ 1. +Case 2: Suppose that 0 ≤ 1 +p1 ≤ +d +d+1 ≤ 1 +p3 ≤ 1. By Theorem 5.4 (ii), to prove that Λ(p1, p2, p3) ≲ 1, +it will be enough to show that +1 +dp1 ++ 1 +p2 ++ d +p3 +≤ d. +However, this inequality clearly follows from the third inequality in (5.3) since d ≥ 2. +Case 3: Suppose that 0 ≤ +1 +p3 ≤ +d +d+1 ≤ +1 +p1 ≤ 1. By Theorem 5.4 (iii), it suffices to show that +d +p1 + 1 +p2 + +1 +dp3 ≤ d. However, this inequality can be easily obtained from the first inequality in (5.3). +Case 4: Suppose that +d +d+1 ≤ +1 +p1, 1 +p3 ≤ 1. By Theorem 5.4 (iv), to show that Λ(p1, p2, p3) ≲ 1, we +only need to prove that +d +p1 ++ 1 +p2 ++ d +p3 +≤ 2d − 1. +However, this inequality can be easily proven as follows: +d +p1 ++ 1 +p2 ++ d +p3 += +� d +p1 ++ 1 +p2 ++ 1 +p3 +� ++ d − 1 +p3 +≤ d + d − 1 +p3 +≤ 2d − 1, +where the first inequality follows from the first inequality in (5.3), and the last inequality follows +from a simple fact that 1 ≤ p3 ≤ ∞. +22 + +Remark 5.7. The reverse statement of Theorem 5.6 cannot be true. Indeed, we know by Corollary +5.5 that Λ +� +d+1 +d , d+1 +d−1, d+1 +d +� +≲ 1. However, ∆ +� +d+1 +d , d+1 +d−1, d+1 +d +� +cannot be bounded, which can be +easily shown by considering Proposition 4.2, namely, the necessary conditions for the boundedness +of ∆(p1, p2, p3). +We invoke Theorem 5.6 to deduce the following result. +Corollary 5.8. Conjecture 1.5 is true for the graph K3 and its subgraph P2 in Fd +q, d ≥ 2. +Proof. It is clear that P2 is a subgraph of K3 in Fd +q. Sine δ(K3) = 2, d ≥ 2, and δ(P2) = 1, we have +min{δ(K3), d} = 2 > δ(P2) = 1. Hence, all assumptions of Conjecture 1.5 are satisfied for K3 and +P2. Then the statement of the corollary follows immediately from Theorem 5.6. +6 +Mapping properties for the (C4 + t)-Operator +We investigate the mapping properties of the operator associated with the graph C4 + diagonal. +Throughout the remaining sections, we assume that t is a non-zero element in F∗ +q. Let fi, 1 ≤ i ≤ 4, +be non-negative real-valued functions on Fd +q. +The operator ♦t is associated with the graph C4 + diagonal t (Figure 1d), and we define ♦t(f1, f2, f3, f4) +as the quantity +1 +qd|St||Sd−2 +t +|2 +� +x1,x2,x3,x4∈Fdq +St(x1 −x2)St(x2 −x3)St(x3 −x4)St(x4 −x1)St(x1 −x3) +4 +� +i=1 +fi(xi). (6.1) +The operator ♦t is referred to as the (C4 + t)-Operator on Fd +q. Here, notice that we take the +quantity qd|St||Sd−2 +t +|2 as the normalizing factor N(G) in (1.4). +Applying a change of variables by letting x = x1, u = x1 − x2, v = x1 − x3, w = x1 − x4, we see +that +♦t(f1, f2, f3, f4) = 1 +qd +� +x∈Fdq +f1(x)T(f2, f3, f4)(x) =< f1, T(f2, f3, f4) >, +(6.2) +where the operator T(f2, f3, f4) is defined by +T(f2, f3, f4)(x) := +1 +|St||Sd−2 +t +|2 +� +u,v,w∈St +St(v − u)St(w − v)f2(x − u)f3(x − v)f4(x − w). +(6.3) +Definition 6.1. Let 1 ≤ p1, p2, p3, p4 ≤ ∞. We define ♦t(p1, p2, p3, p4) to be the smallest constant +such that the following estimate holds for all non-negative real-valued functions fi, 1 ≤ i ≤ 4, on +Fd +q : ♦t(f1, f2, f3, f4) ≤ ♦t(p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4. +We are asked to find 1 ≤ p1, p2, p3, p4 ≤ ∞ such that +♦t(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4 +(6.4) +23 + +holds for all non-negative real-valued functions fi, 1 ≤ i ≤ 4, on Fd +q. In other words, our main +problem is to determine all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ♦t(p1, p2, p3, p4) ≲ 1. +Lemma 6.2 (Necessary conditions for the boundedness of ♦t(p1, p2, p3, p4) ). Let ♦t be the (C4+t)- +Operator on Fd +q. If ♦t(p1, p2, p3, p4) ≲ 1, then we have +1 +p1 ++ 1 +p2 ++ d +p3 ++ 1 +p4 +≤ d, +d +p1 ++ 1 +p2 ++ 1 +p3 ++ 1 +p4 +≤ d, +and +1 +p1 ++ d +p2 ++ 1 +p3 ++ d +p4 +≤ 2d − 2. +Also, under this assumption when d = 2, it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3, 1/p4) +is contained in the convex hull of the points (0, 0, 1, 0), (0, 1, 0, 0), (0, 0, 0, 1), (1/2, 0, 1/2, 1/2), +(2/3, 2/3, 0, 0), (1, 0, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), (2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 0, 0), +(0, 0, 2/3, 2/3). +Proof. We prove the first conclusion that +1 +p1 ++ 1 +p2 ++ d +p3 ++ 1 +p4 +≤ d. +(6.5) +We notice from (6.2) and (6.3) that ♦t(f1, f2, f3, f4) becomes +1 +qd +� +x∈Fdq +f1(x) + + +1 +|St||Sd−2 +t +|2 +� +v∈St +f3(x − v) + + � +u,w∈St +St(v − u)St(w − v)f2(x − u)f4(x − w) + + + + . +Taking f1 = f2 = f4 = 1St, and f3 = δ0, we see that +||f1||p1||f2||p2|||f3||p3||f4||p4 ∼ q− 1 +p1 q− 1 +p2 q− d +p3 q− 1 +p4 +and +♦t(f1, f2, f3, f4) = 1 +qd +� +x∈St +1 +|St||Sd−2 +t +|2 + + +� +u∈St:||x−u||=t +1 + + + + +� +w∈St:||x−w||=t +1 + + ∼ q−d, +which yields the required necessary condition (6.5) , where the last similarity above follows from +Corollary 12.4 in Appendix. +The second conclusion follows by symmetry from the first conclusion. +To prove the third conclusion that is +1 +p1 + d +p2 + 1 +p3 + d +p4 ≤ 2d − 2, we test the inequality (6.4) with +f1 = f3 = 1St and f2 = f4 = δ0. +Then we have +||f1||p1||f2||p2||f3||p3||f4||p4 = +�|St| +qd +�1/p1 � 1 +qd +�1/p2 �|St| +qd +�1/p3 � 1 +qd +�1/p4 +∼ q− 1 +p1 − d +p2 − 1 +p3 − d +p4 . +24 + +On the other hand, taking f1 = f3 = 1St and f2 = f4 = δ0 in the definition (6.1), we see that +♦t(f1, f2, f3, f4) = +1 +qd|St||Sd−2 +t +|2 +� +x1∈St + + � +x3∈St +St(x1 − x3) + + ≲ +1 +qd|Sd−2 +t +| +∼ q−2d+2. +Hence, by (6.4) we must have the required third conclusion that +1 +p1 + d +p2 + 1 +p3 + d +p4 ≤ 2d − 2. +6.1 +Boundedness results for ♦t on Fd +q +Given a rhombus with a fixed diagonal(the graph C4 + diagonal), we will show that by removing +the vertex x2 or the vertex x4, +♦t(f1, f2, f3, f4) ≲ ||f2||∞∆(f1, f3, f4) +and +♦t(f1, f2, f3, f4) ≲ ||f4||∞∆(f1, f2, f3). +Hence, upper bounds of ♦t(p1, ∞, p3, p4) and ♦t(p1, p2, p3, ∞) can be controlled by upper bounds +of the ∆(p1, p3, p4) and ∆(p1, p2, p3), respectively. More precisely, we have the following relation. +Proposition 6.3. Suppose that ∆(p, s, r) ≲ 1 for 1 ≤ p, s, r ≤ ∞. Then we have +♦t(p, ∞, s, r) ≲ 1 +and +♦t(p, s, r, ∞) ≲ 1. +Proof. Since ∆(p, s, r) ≲ 1 for 1 ≤ p, s, r ≤ ∞, we see that for all non-negative functions f, g, h +on Fd +q, ∆(f, g, h) ≲ ||f||p||g||s||h||r. Thus, to complete the proof, it will be enough to establish the +following estimates: for all non-negative functions fi, i = 1, 2, 3, 4, +♦t(f1, f2, f3, f4) ≲ ||f2||∞∆(f1, f3, f4) +(6.6) +and +♦t(f1, f2, f3, f4) ≲ ||f4||∞∆(f1, f2, f3). +(6.7) +Since the proofs of both (6.6) and (6.7) are the same, we only provide the proof of the estimate +(6.7). Notice by the definition of ♦t(f1, f2, f3, f4) in (6.1) that ♦t(f1, f2, f3, f4) can be written as +the form +1 +qd|St||Sd−2 +t +| +� +x1,x2,x3∈Fd +q +:||x1−x2||=||x2−x3||=||x1−x3||=t +� 3 +� +i=1 +fi(xi) +�  + +1 +|Sd−2 +t +| +� +x4∈Fdq +St(x3 − x4)St(x4 − x1)f4(x4) + + . +For each x1, x3 ∈ Fd +q with ||x1 − x3|| = t, we define M(x1, x3) as the value in the above bracket. +Then, recalling the definition of ∆(f1, f2, f3) given in (4.1), we see that +♦t(f1, f2, f3, f4) ≤ + + + + +max +x1,x3∈Fd +q +:||x1−x3||=t +M(x1, x3) + + + + ∆(f1, f2, f3). +25 + +Hence, the estimate (6.7) follows immediately by proving the following claim: +M := +max +x1,x3∈Fd +q +:||x1−x3||=t +1 +|Sd−2 +t +| +� +x4∈Fdq +St(x3 − x4)St(x4 − x1) ≲ 1. +(6.8) +To prove this claim, we first apply a change of variables by letting x = x1, y = x1 − x3. Then it +follows that +M = +max +x∈Fdq,y∈St +1 +|Sd−2 +t +| +� +x4∈Fdq +St(x − y − x4)St(x4 − x). +Letting z = x − x4, we have +M = +max +x∈Fdq,y∈St +1 +|Sd−2 +t +| +� +z∈St:||z−y||=t +1 ∼ +1 +qd−2 max +y∈St +� +z∈St:||z−y||=t +1 +By Corollary 12.4 in Appendix, we conclude that M ≲ 1, as required. +In arbitrary dimensions d ≥ 2, we have the following consequences. +Theorem 6.4. Suppose that 1 ≤ a, b ≤ ∞ satisfy that +1 +a + d +b ≤ d +and +d +a + 1 +b ≤ d. +(6.9) +Namely, let (1/a, 1/b) be contained in the convex hull of points (0, 0), (0, 1), (d/(d + 1), d/(d + 1)), +(1, 0). Then we have ♦t(a, ∞, b, ∞) ≲ 1, ♦t(a, b, ∞, ∞) ≲ 1, ♦t(a, ∞, ∞, b) ≲ 1, ♦t(∞, ∞, a, b) ≲ +1, ♦t(∞, a, b, ∞) ≲ 1. +Proof. From Theorem 4.5, we know that the assumption (6.9) implies that ∆(a, b, ∞) ≲ 1, +∆(a, ∞, b) ≲ 1 and ∆(∞, a, b) ≲ 1. Hence, the statement of the theorem follows immediately +by combining these and Proposition 6.3. +6.2 +Sharp boundedness results up to endpoints for ♦t on F2 +q +In this subsection, we collect our boundedness results for the operator ♦t in two dimensions. +Theorem 6.5. Let ♦t be the (C4 + t)-Operator on F2 +q. Let 1 ≤ p1, p2, p3 ≤ ∞. +(i) Suppose that (p1, p2, p3) ̸= (2, 2, 2) satisfies the following equations: +2 +p1 ++ 1 +p2 ++ 1 +p3 +≤ 2, +1 +p1 ++ 2 +p2 ++ 1 +p3 +≤ 2, +1 +p1 ++ 1 +p2 ++ 2 +p3 +≤ 2. +Then we have ♦t(p1, ∞, p2, p3) ≲ 1 and ♦t(p1, p2, p3, ∞) ≲ 1. +(ii) In addition, we have ♦t(2, ∞, 2, 2) ⪅ 1 and ♦t(2, 2, 2, ∞) ⪅ 1, where ⪅ is used to denote that +the boundedness of ♦t holds for all indicator test functions. +26 + +Proof. Notice that Proposition 6.3 still holds after replacing ≲ by ⪅ . Hence, the statement of the +theorem is directly obtained by combining Proposition 6.3 and Theorem 4.11 (ii). +Theorem 6.5 guarantees the sharp boundedness for the operator ⋄t up to endpoints. Indeed, we +have the following result. +Theorem 6.6. Let ♦t be the (C4+t)-Operator on F2 +q. The necessary conditions for ♦t(p1, p2, p3, p4) ≲ +1 given in Lemma 6.2 are sufficient except for the two points (p1, p2, p3, p4) = (2, 2, 2, ∞), (2, ∞, 2, 2). +In addition, we have +♦t(2, ∞, 2, 2) ⪅ 1 +and +♦t(2, 2, 2, ∞) ⪅ 1. +(6.10) +Proof. The statement (6.10) was already proven in Theorem 6.5 (ii). Hence, using the interpolation +theorem and the second part of Lemma 6.2, the matter is reducing to proving ♦t(p1, p2, p3, p4) ≲ 1 +for the critical endpoints (1/p1, 1/p2, 1/p3, 1/p4) including all the following points: (0, 0, 0, 0), +(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (2/3, 2/3, 0, 0), (2/3, 0, 2/3, 0), (2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), +(0, 0, 2/3, 2/3). +In other words, the proof will be complete by proving the following estimates: ♦t(∞, ∞, ∞, ∞) ≲ +1, +♦t(1, ∞, ∞, ∞) ≲ 1, +♦t(∞, 1, ∞, ∞) ≲ 1, +♦t(∞, ∞, 1, ∞) ≲ 1, +♦t(∞, ∞, ∞, 1) ≲ 1, +♦t(3/2, 3/2, ∞, ∞) ≲ 1, ♦t(3/2, ∞, 3/2, ∞) ≲ 1, ♦t(3/2, ∞, ∞, 3/2) ≲ 1, ♦t(∞, 3/2, 3/2, ∞) ≲ +1, ♦t(∞, ∞, 3/2, 3/2) ≲ 1. +However, by a direct computation, these estimates follow immediately from Theorem 6.5 (i). +7 +Boundedness problem for the C4-Operator +Let t ∈ F∗ +q. Given non-negative real-valued functions fi, 1 ≤ i ≤ 4, on Fd +q, we define ✸(f1, f2, f3, f4) +to be the following value: +1 +qd|St|2|Sd−2 +t +| +� +x1,x2,x3,x4∈Fdq +St(x1 − x2)St(x2 − x3)St(x3 − x4)St(x4 − x1) +4 +� +i=1 +fi(xi), +(7.1) +where the quantity qd|St|2|Sd−2 +t +| stands for the normalizing factor N(G) in (1.4) when G = C4. +Since the operator ✸ is associated with the graph C4, it is named as the C4-Operator on Fd +q. +Definition 7.1. Let 1 ≤ p1, p2, p3, p4 ≤ ∞. We denote by ✸(p1, p2, p3, p4) the smallest constant +such that the following estimate holds for all non-negative real-valued functions fi, 1 ≤ i ≤ 4, on +Fd +q : +✸(f1, f2, f3, f4) ≤ ✸(p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4. +Main problem is to find all exponents 1 ≤ p1, p2, p3, p4 ≤ ∞ such that the inequality +✸(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4 +(7.2) +27 + +holds for all non-negative real-valued functions fi, 1 ≤ i ≤ 4, on Fd +q. In other words, our main +problem is to determine all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ✸(p1, p2, p3, p4) ≲ 1. +Lemma 7.2 (Necessary conditions for the boundedness of ✸(p1, p2, p3, p4) ). Suppose that (7.2) +holds, namely ✸(p1, p2, p3, p4) ≲ 1. Then we have +1 +p1 ++ 1 +p2 ++ 1 +p3 ++ d +p4 +≤ d + 1, +1 +p1 ++ 1 +p2 ++ d +p3 ++ 1 +p4 +≤ d + 1, +1 +p1 ++ d +p2 ++ 1 +p3 ++ 1 +p4 +≤ d + 1, +d +p1 ++ 1 +p2 ++ 1 +p3 ++ 1 +p4 +≤ d + 1, +d +p1 ++ 1 +p2 ++ d +p3 ++ 1 +p4 +≤ 2d − 2, +and +1 +p1 ++ d +p2 ++ 1 +p3 ++ d +p4 +≤ 2d − 2. +In particular, when d = 2, it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3, 1/p4) is con- +tained in the convex hull of the points (0, 0, 1, 0), (0, 0, 0, 1), (0, 1, 0, 0), (2/3, 0, 0, 2/3), (2/3, 2/3, 0, 0), +(1, 0, 0, 0), (0, 0, 0, 0), (0, 2/3, 2/3, 0), (0, 0, 2/3, 2/3). +Remark 7.3. When d = 2, 3, the first four inequalities in the conclusion are not necessary. We only +need the last two. +Proof. The 6 inequalities in the conclusion can be easily deduced by testing the inequality (7.2) +with the following specific functions, respectively: We leave the proofs to the readers. +1) f1 = f2 = f3 = 1St, and f4 = δ0. +2) f1 = f2 = f4 = 1St, and f3 = δ0. +3) f1 = f3 = f4 = 1St, and f2 = δ0. +4) f2 = f3 = f4 = 1St, and f1 = δ0. +5) f2 = f4 = 1St, and f1 = f3 = δ0. +6) f1 = f3 = 1St, and f2 = f4 = δ0. +7.1 +Boundedness results for the C4-Operator ✸ on Fd +q +In this subsection, we provide some exponents 1 ≤ pi ≤ ∞, 1 ≤ i ≤ 4, such that ✸(p1, p2, p3, p4) ≲ 1 +in the specific case when one of pi is ∞. However, our result will correspond to all dimensions d ≥ 2. +In general, it is very hard to deduce non-trivial boundedness results for the C4-Operator on Fd +q. +We begin by observing that an upper bound of ✸(f1, f2, f3, f4) can be controlled by estimating for +both the K2-Operator L and the P2-Operator Λ. +Proposition 7.4. For all non-negative functions fi, i = 1, 2, 3, 4, on Fd +q, d ≥ 2, we have +✸(f1, f2, f3, f4) ≲ + + + + + + + + + + + + + + + +� +1 +|Sd−2 +t +|L(f1f3, f2) + Λ(f1, f2, f3) +� +||f4||∞, +� +1 +|Sd−2 +t +|L(f2f4, f1) + Λ(f4, f1, f2) +� +||f3||∞, +� +1 +|Sd−2 +t +|L(f1f3, f4) + Λ(f3, f4, f1) +� +||f2||∞, +� +1 +|Sd−2 +t +|L(f2f4, f3) + Λ(f2, f3, f4) +� +||f1||∞. +28 + +Proof. We only provide the proof of the first inequality, +✸(f1, f2, f3, f4) ≲ +1 +|Sd−2 +t +| +L(f1f3, f2)||f4||∞ + Λ(f1, f2, f3)||f4||∞, +(7.3) +since other inequalities can be easily proven in the same way by replacing the role of f4 with +f3, f2, f1, respectively. By definition, the value of ✸(f1, f2, f3, f4) is equal to +1 +qd|St|2|Sd−2 +t +| +� +x1,x2,x3∈Fdq +St(x1−x2)St(x2−x3) +� 3 +� +i=1 +fi(xi) +�  + � +x4∈Fdq +St(x3 − x4)St(x4 − x1)f4(x4) + + . +For fixed x1, x3 ∈ Fd +q, the sum in the above bracket can be estimated as follows: +� +x4∈Fdq +St(x3 − x4)St(x4 − x1)f4(x4) ≲ +� +|St|||f4||∞ +if +x1 = x3, +qd−2||f4||∞ +if +x1 ̸= x3. +Notice that this estimates are easily obtained by invoking Corollary 12.4 in Appendix after using +a change of variables. +Let ✸(f1, f2, f3, f4) = ✸1+✸2, where ✸1 denotes the contribution to ✸(f1, f2, f3, f4) when x1 = x3, +and ✸2 does it when x1 ̸= x3. Then it follows that +✸1 ≲ +||f4||∞ +qd|St||Sd−2 +t +| +� +x1,x2∈Fdq +St(x1 − x2)(f1f3)(x1)f2(x2) = +1 +|Sd−2 +t +| +L(f1f3, f2)||f4||∞, +✸2 ≲ ||f4||∞ +qd|St|2 +� +x1,x2,x3∈Fdq:x1̸=x3 +St(x1 − x2)St(x2 − x3)f1(x1)f2(x2)f3(x3) ≲ Λ(f1, f2, f3)||f4||∞. +Hence, we obtain the required estimate (7.3). +In Proposition 7.4, we obtained four different kinds of the upper bounds of the ✸(f1, f2, f3, f4). +Using each of them, we are able to deduce exponents p1, p2, p3, p4 with ✸(p1, p2, p3, p4) ≲ 1, where +at least one of pj, j = 1, 2, 3, 4, takes ∞. +The following result can be proven by applying the first upper bound of ✸(f1, f2, f3, f4) in Propo- +sition 7.4 together with Theorem 3.5 and Theorem 5.4. +Theorem 7.5. Let ✸ be defined on functions on Fd +q, d ≥ 2. Let 1 ≤ p1, p2, p3 ≤ ∞. Then the +following statements are true. +(i) If +1 +p1 + d +p2 + 1 +p3 ≤ d and +d +p1 + 1 +p2 + d +p3 ≤ d, then ✸(p1, p2, p3, ∞) ≲ 1. +(ii) If +d +p1 + 1 +p2 + 1 +p4 ≤ d and +1 +p1 + d +p2 + d +p4 ≤ d, then ✸(p1, p2, ∞, p4) ≲ 1. +(iii) If +1 +p1 + 1 +p3 + d +p4 ≤ d and +d +p1 + d +p3 + 1 +p4 ≤ d, then ✸(p1, ∞, p3, p4) ≲ 1. +(iv) If +1 +p2 + d +p3 + 1 +p4 ≤ d and +d +p2 + 1 +p3 + d +p4 ≤ d, then ✸(∞, p2, p3, p4) ≲ 1. +29 + +Proof. We will only provide the proof of the first part of the theorem since the proofs of other parts +are the same in the sense that the proof of the first part uses the first upper bound of Proposition +7.4 and the proofs of other parts can also use their corresponding upper bounds of Proposition 7.4 +to complete the proofs. +Let us start proving the first part of the theorem. To complete the proof, we aim to show that for +all non-negative functions fi, i = 1, 2, 3, 4, on Fd +q, +✸(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||∞, +whenever the exponents 1 ≤ p1, p2, p3 ≤ ∞ satisfy the following conditions: +1 +p1 ++ d +p2 ++ 1 +p3 +≤ d +and +d +p1 ++ 1 +p2 ++ d +p3 +≤ d. +(7.4) +By the first part of Proposition 7.4, it follows that +✸(f1, f2, f3, f4) ≲ +� +1 +|Sd−2 +t +| +L(f1f3, f2) + Λ(f1, f2, f3) +� +||f4||∞. +Therefore, under the assumptions (7.4), our problem is reducing to establishing the following two +estimates: +L(f1f3, f2) ≲ ||f1||p1||f2||p2||f3||p3, +(7.5) +Λ(f1, f2, f3) ≲ ||f1||p1||f2||p2||f3||p3. +(7.6) +For 1 ≤ p1, p3 ≤ ∞, let 1/r = 1/p1 + 1/p3. Then the conditions (7.4) are the same as +1 +r + d +p2 +≤ d +and +d +r + 1 +p2 +≤ d. +So these conditions enable us to invoke Theorem 3.5 so that we obtain the estimate (7.5) as follows: +L(f1f3, f2) ≲ ||f1f3||r||f2||p2 ≤ ||f1|||p1||f2||p2||f3||p3, +where we used H¨older’s inequality in the last inequality. +It remains to prove the estimate (7.6) under the assumptions (7.4). To do this, we shall use Theorem +5.4, which gives sufficient conditions for Λ(p1, p2, p3) ≲ 1. We directly compare the conditions (7.4) +with the assumptions of Theorem 5.4. Then it is not hard to observe the following statements. +• (Case 1) In the case when 0 ≤ +1 +p1, 1 +p3 ≤ +d +d+1, the conditions (7.4) imply the hypothesis of the +first part of Theorem 5.4. +• (Case 2) In the case when 0 ≤ 1 +p1 ≤ +d +d+1 ≤ 1 +p3 ≤ 1, the conditions (7.4) imply the hypothesis +of the second part of Theorem 5.4. To see this, notice that if d/p1 + 1/p2 + d/p3 ≤ d, then +1/(dp1) + 1/p2 + d/p3 ≤ d. +30 + +• (Case 3) In the case when 0 ≤ 1 +p3 ≤ +d +d+1 ≤ 1 +p1 ≤ 1, the conditions (7.4) imply the hypothesis +of the third part of Theorem 5.4. +• (Case 4) In the case when +d +d+1 ≤ +1 +p1, 1 +p3 ≤ 1, the conditions (7.4) imply the hypothesis of the +fourth part of Theorem 5.4. +Hence, we conclude from Theorem 5.4 that Λ(p1, p2, p3) ≲ 1 under the assumptions (7.4), as +desired. +7.2 +Sharp boundedness results for the C4-Operator ✸ on F2 +q +Recall that Theorem 7.5 provides sufficient conditions for ⋄(p1, p2, p2, p4) ≲ 1 in any dimensions +d ≥ 2. In this section, we show that Theorem 7.5 is sharp in two dimensions. More precisely, using +Theorem 7.5 we will prove the following optimal result. +Theorem 7.6. Let ✸ be the C4-Operator on F2 +q. For 1 ≤ pi ≤ ∞, 1 ≤ i ≤ 4, we have +✸(p1, p2, p3, p4) ≲ 1 if and only if +2 +p1 ++ 1 +p2 ++ 2 +p3 ++ 1 +p4 +≤ 2, +and +1 +p1 ++ 2 +p2 ++ 1 +p3 ++ 2 +p4 +≤ 2. +Proof. The necessary conditions for ✸(p1, p2, p3, p4) ≲ 1 follow immediately from Lemma 7.2 for +d = 2 (see Remark 7.3). +Conversely, suppose that 1 ≤ p1, p2, p3, p4 ≤ ∞ satisfy the following two inequalities: +2 +p1 ++ 1 +p2 ++ 2 +p3 ++ 1 +p4 +≤ 2, +and +1 +p1 ++ 2 +p2 ++ 1 +p3 ++ 2 +p4 +≤ 2. +(7.7) +Then, as mentioned in Lemma 7.2, it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3, 1/p4) +is contained in the convex hull of the points (0, 0, 1, 0), (0, 0, 0, 1), (0, 1, 0, 0), (2/3, 0, 0, 2/3), +(2/3, 2/3, 0, 0), (1, 0, 0, 0), (0, 0, 0, 0), (0, 2/3, 2/3, 0), (0, 0, 2/3, 2/3). +By interpolating the above 9 critical points, to prove ✸(p1, p2, p3, p4) ≲ 1 for all pi, 1 ≤ i ≤ 4 satisfy- +ing the inequalities in (7.7), it will be enough to prove it for the 9 critical points (1/p1, 1/p2, 1/p3, 1/p4). +This can be easily proven by using Theorem 7.5. For example, for the point (1/p1, 1/p2, 1/p3, 1/p4) = +(2/3, 0, 0, 2/3), a direct computation shows that the assumptions in Theorem 7.5 (ii) are satisfied +and thus ✸(p1, p2, p3, p4) = ✸(3/2, ∞, ∞, 3/2) ≲ 1. For other critical points, we can easily prove +them in the same way so that we omit the detail proofs. +Notice that the graph C4 is a subgraph of the graph C4 + diagonal, and they are associated with +the operators ✸ and ♦t, respectively. +Hence, the following theorem shows that the answer to +Question 1.2 is negative when G is the C4 + diagonal, and G′ is the C4. However, this does not +mean that Conjecture 1.5 is not true since the C4 and the C4 + diagonal do not satisfy the main +hypothesis (1.6) of Conjecture 1.5. +Theorem 7.7. Let ♦t, ✸ be the (C4 + t)-Operator and the C4-Operator on F2 +q, respectively. Let +1 ≤ p1, p2, p3, p4 ≤ ∞. Then the following statements hold. +31 + +(i) If ✸(p1, p2, p3, p4) ≲ 1, then ♦t(p1, p2, p3, p4) ≲ 1. +(ii) Moreover, there exist exponents 1 ≤ a, b, c, d ≤ ∞ such that ♦t(a, b, c, d) ≲ 1 but ✸(a, b, c, d) +is not bounded. +Proof. First, let us prove the statement (ii) in the conclusion. To prove this, we choose (a, b, c, d) = +(3/2, ∞, 3/2, ∞). From Theorem 6.5 (i), we can easily note that ♦t(3/2, ∞, 3/2, ∞) ≲ 1. However, +it is impossible that ✸(3/2, ∞, 3/2, ∞) ≲ 1, which can be shown from Theorem 7.6. +Next, let us prove the first conclusion of the theorem. +Suppose that ✸(p1, p2, p3, p4) ≲ 1 for +1 ≤ pi ≤ ∞, 1 ≤ i ≤ 4. Then, as mentioned in the second conclusion of Lemma 7.2, the point +(1/p1, 1/p2, 1/p3, 1/p4) lies on the convex body with the critical endpoints: (0, 0, 1, 0), (0, 0, 0, 1), +(0, 1, 0, 0), (2/3, 0, 0, 2/3), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (0, 0, 0, 0), (0, 2/3, 2/3, 0), (0, 0, 2/3, 2/3). +Invoking the interpolation theorem, to prove the conclusion that ♦t(p1, p2, p3, p4) ≲ 1, it will be +enough to establish the boundedness only for those 9 critical points (1/p1, 1/p2, 1/p3, 1/p4). More +precisely, it remains to establish the following estimates: +♦t(∞, ∞, ∞, ∞) ≲ 1, ♦t(1, ∞, ∞, ∞) ≲ 1, ♦t(∞, 1, ∞, ∞) ≲ 1, ♦t(∞, ∞, 1, ∞) ≲ 1, +♦t(∞, ∞, ∞, 1) ≲ 1, ♦t(3/2, 3/2, ∞, ∞) ≲ 1, ♦t(3/2, ∞, ∞, 3/2) ≲ 1, ♦t(∞, 3/2, 3/2, ∞) ≲ 1, +♦t(∞, ∞, 3/2, 3/2) ≲ 1. +However, these estimates follow by applying Theorem 6.5 (i). +8 +Boundedness problem for the P3-Operator +For t ∈ F∗ +q and non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd +q, we define ⊓(f1, f2, f3, f4) +as the following value: +1 +qd|St|3 +� +x1,x2,x3,x4∈Fdq +St(x1 − x2)St(x2 − x3)St(x3 − x4) +4 +� +i=1 +fi(xi). +(8.1) +This operator ⊓ will be named the P3-Operator on Fd +q since it is related to the graph P3 with +vertices in Fd +q, d ≥ 2. Note that in the definition of ⊓(f1, f2, f3, f4), we take the normalizing fact +qd|St|3, which is corresponding to N(G) in (1.4) when G is the P3. +Definition 8.1. Let 1 ≤ p1, p2, p3, p4 ≤ ∞. We define ⊓(p1, p2, p3, p4) as the smallest constant +such that the following estimate holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on +Fd +q : +⊓(f1, f2, f3, f4) ≤ ⊓(p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4. +We want to determine 1 ≤ p1, p2, p3, p4 ≤ ∞ such that +⊓ (f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4 +(8.2) +32 + +holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd +q. In other words, our main +problem is to find all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ⊓(p1, p2, p3, p4) ≲ 1. +Lemma 8.2 (Necessary conditions for ⊓(p1, p2, p3, p4) ≲ 1). Suppose that ⊓(p1, p2, p3, p4) ≲ 1. +Then we have +1 +p1 + d +p2 + 1 +p3 ≤ d, +1 +p2 + d +p3 + 1 +p4 ≤ d, +d +p1 + 1 +p2 + 1 +p3 + 1 +p4 ≤ d+2, +1 +p1 + 1 +p2 + 1 +p3 + d +p4 ≤ d+2, +1 +p1 + d +p2 + 1 +p3 + d +p4 ≤ 2d − 1, +d +p1 + 1 +p2 + d +p3 + 1 +p4 ≤ 2d − 1, and +d +p1 + 1 +p2 + 1 +p3 + d +p4 ≤ 2d +In particular, when d = 2, by using Polymake [1, 6], it can be shown that (1/p1, 1/p2, 1/p3, 1/p4) +is contained in the convex hull of the points: (0, 1, 0, 1/2), (0, 1, 0, 0), (1/2, 0, 1/2, 1), (1/2, 0, 1, 0), +(1, 1/2, 0, 0), (1, 0, 0, 0), (1, 1/3, 1/3, 0), (1, 0, 1/2, 0), (1/2, 1/3, 5/6, 0), (1, 0, 0, 1), (1/2, 1/2, 1/2, 1/2), +(1, 1/2, 0, 1/2), (0, 5/6, 1/3, 1/2), (0, 0, 0, 1), (0, 1/2, 0, 1), (0, 2/3, 2/3, 0), (0, 1/3, 1/3, 1), (0, 0, 1/2, 1), +(0, 0, 1, 0), (0, 0, 0, 0). +Remark 8.3. When d = 2, the third, fourth, and seventh inequalities above are not necessary. +When d = 3, the third and fourth inequalities above are not necessary. +Proof. As in the proofs of Propositions 3.2, 4.2, and 5.2, the conclusions of the statement follow +by testing the inequality (8.2) with the following specific functions, respectively: +1) f1 = f3 = 1St, f2 = δ0, and f4 = 1Fdq. +2) f1 = 1Fdq, f2 = f4 = 1St, and f3 = δ0. +3) f2 = f3 = f4 = 1St, and f1 = δ0. +4) f1 = f2 = f3 = 1St, and f4 = δ0. +5) f1 = f3 = 1St, and f2 = f4 = δ0. +6) f2 = f4 = 1St, and f1 = f3 = δ0. +7) f2 = f3 = 1St, and f1 = f4 = δ0. +8.1 +Boundedness results for the P3-Operator ⊓ on Fd +q +We begin by observing that an upper bound of ⊓(f1, f2, f3, f4) can be controlled by the value +Λ(f1, f2, f3). +Proposition 8.4. Let 1 ≤ a, b, c ≤ ∞. If Λ(a, b, c) ≲ 1, then ⊓(a, b, c, ∞) ≲ 1 and ⊓(∞, a, b, c) ≲ 1. +Proof. For all non-negative functions fi, i = 1, 2, 3, 4, on Fd +q, our task is to prove the following +inequalities: +⊓ (f1, f2, f3, f4) ≲ +� +Λ(f1, f2, f3) ||f4||∞, +||f1||∞ Λ(f2, f3, f4). +(8.3) +We will only prove the first inequality, that is +⊓ (f1, f2, f3, f4) ≲ Λ(f1, f2, f3) ||f4||. +(8.4) +By symmetry, the second inequality can be easily proven in the same way. By definition in (8.1), +33 + +we can write ⊓(f1, f2, f3, f4) as +1 +qd|St|2 +� +x1,x2,x3∈Fdq +St(x1 − x2)St(x2 − x3) +� 3 +� +i=1 +fi(xi) +�  + 1 +|St| +� +x4∈Fdq +f4(x4)St(x3 − x4) + + . +Since the value in the above bracket is Af4(x3), which is clearly dominated by ||Af4||∞, the required +estimate (8.4) follows immediately from the definition of Λ(f1, f2, f3) in (5.1). +The following theorem can be deduced from Proposition 8.4 and Theorem 5.4. +Theorem 8.5. Consider the P3-Operator ⊓ on Fd +q. Suppose that the exponents 1 ≤ a, b, c ≤ ∞ +satisfy one of the following conditions: +(i) 0 ≤ 1 +a, 1 +c ≤ +d +d+1 and 1 +a + d +b + 1 +c ≤ d +(ii) 0 ≤ 1 +a ≤ +d +d+1 ≤ 1 +c ≤ 1, and +1 +da + 1 +b + d +c ≤ d +(iii) 0 ≤ 1 +c ≤ +d +d+1 ≤ 1 +a ≤ 1, and d +a + 1 +b + 1 +dc ≤ d +(iv) +d +d+1 ≤ 1 +a, 1 +c ≤ 1 and d +a + 1 +b + d +c ≤ 2d − 1. +Then we have ⊓(a, b, c, ∞) ≲ 1 and ⊓(∞, a, b, c) ≲ 1. +Proof. Using Theorem 5.4 with p1 = a, p2 = b, p3 = c, it is clear that Λ(a, b, c) ≲ 1 for all +exponents a, b, c in our assumption. Hence, the statement follows immediately from Proposition +8.4. +Now we prove that the value ⊓(f1, f2, f3, f4) can be expressed in terms of the averaging operator +over spheres. For functions f, g, h on Fd +q, let us denote +< f, g, h >:= ||fgh||1 = 1 +qd +� +x∈Fdq +f(x)g(x)h(x). +Proposition 8.6. Let fi, i = 1, 2, 3, 4, be non-negative real-valued functions on Fd +q. Then we have +⊓(f1, f2, f3, f4) =< Af1, f2, A(f3 · Af4) >=< A(f2 · Af1), f3, Af4 > . +Proof. By symmetry, to complete the proof, it suffices to prove the first equality, that is +⊓(f1, f2, f3, f4) =< Af1, f2, A(f3 · Af4) > . +Combining the definition in (8.1) and the definition of the spherical averaging operator A, it follows +34 + +that +⊓(f1, f2, f3, f4) = 1 +qd +� +x2∈Fdq +f2(x2)Af1(x2) + + 1 +|St| +� +x3∈Fdq +f3(x3)St(x2 − x3)Af4(x3) + + += 1 +qd +� +x2∈Fdq +f2(x2)Af1(x2)A(f3 · Af4)(x2). +This gives the required estimate. +Combining Proposition 8.6 and the averaging estimate over spheres, we are able to deduce sufficient +conditions for the boundedness of the P3-Operator ⊓ on Fd +q. +Theorem 8.7. Let 1 ≤ p1, p2, p3, p4 ≤ ∞ be exponents satisfying one of the following conditions: +(i) 0 ≤ +1 +p1, 1 +p4, +1 +p3 + +1 +dp4 ≤ +d +d+1, and +1 +dp1 + 1 +p2 + +1 +dp3 + +1 +d2p4 ≤ 1. +(ii) 0 ≤ +1 +p1, 1 +p4 ≤ +d +d+1 ≤ 1 +p3 + +1 +dp4 ≤ 1, and +1 +dp1 + 1 +p2 + d +p3 + 1 +p4 ≤ d. +(iii) 0 ≤ +1 +p1, +1 +p3 + d +p4 − d + 1 ≤ +d +d+1 ≤ +1 +p4 ≤ 1, and +1 +p1 + d +p2 + 1 +p3 + d +p4 ≤ 2d − 1. +(iv) 0 ≤ +1 +p1 ≤ +d +d+1 ≤ 1 +p4, +1 +p3 + d +p4 − d + 1 ≤ 1, and +1 +dp1 + 1 +p2 + d +p3 + d2 +p4 ≤ d2. +(v) 0 ≤ +1 +p4, +1 +p3 + +1 +dp4 ≤ +d +d+1 ≤ 1 +p1 ≤ 1, and +d +p1 + 1 +p2 + +1 +dp3 + +1 +d2p4 ≤ d. +(vi) 0 ≤ +1 +p4 ≤ +d +d+1 ≤ 1 +p1, +1 +p3 + +1 +dp4 ≤ 1, and +d +p1 + 1 +p2 + d +p3 + 1 +p4 ≤ 2d − 1. +(vii) 0 ≤ +1 +p3 + d +p4 − d + 1 ≤ +d +d+1 ≤ +1 +p1, 1 +p4 ≤ 1, and d2 +p1 + d +p2 + 1 +p3 + d +p4 ≤ d2 + d − 1. +(viii) +d +d+1 ≤ +1 +p1, 1 +p4, 1 +p3 + d +p4 − d + 1 ≤ 1, and +d +p1 + 1 +p2 + d +p3 + d2 +p4 ≤ d2 + d − 1. +Then we have ⊓(p1, p2, p3, p4) ≲ 1 and ⊓(p4, p3, p2, p1) ≲ 1. +Proof. By symmetry, it will be enough to prove the first part of conclusions, that is ⊓(p1, p2, p3, p4) ≲ +1. To complete the proof, we will first find the general conditions that guarantee this conclusion. +Next we will demonstrate that each of the hypotheses in the theorem satisfies the general condi- +tions. +To derive the first general condition, we assume that 1 ≤ r1, p2, r ≤ ∞ satisfy that +1 +r1 ++ 1 +p2 ++ 1 +r ≤ 1. +(8.5) +Then by Proposition 8.6 and H¨older’s inequality, +⊓(f1, f2, f3, f4) ≤ ||Af1||r1||f2||p2||A(f3 · Af4)||r, +where we also used the nesting property of norms associated with the normalizing counting mea- +sure. Assume that 1 ≤ p1, s ≤ ∞ satisfy the following averaging estimates over spheres: +A(p1 → r1) ≲ 1 +and +A(s → r) ≲ 1. +(8.6) +35 + +It follows that ⊓(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3 · Af4||s. Now we assume that 1 ≤ p3, t ≤ ∞ +satisfy that +1 +s = 1 +p3 ++ 1 +t . +(8.7) +Then, by the H¨older’s inequality, we see that +⊓(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||Af4||t. +Finally, if we assume that 1 ≤ p4 ≤ ∞ satisfies the following averaging estimate: +A(p4 → t) ≲ 1, +(8.8) +then we obtain that ⊓(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4. +In summary, we see that ⊓(p1, p2, p3, p4) ≲ 1 provided that the numbers 1 ≤ pi ≤ ∞, i = 1, 2, 3, 4, +satisfy all the conditions (8.5), (8.6), (8.7), (8.8). Thus, to finish the proof, we will show that each +of the 8 hypotheses in the theorem satisfies all these conditions. +Given 1 ≤ p1, p4 ≤ ∞, by Lemma 2.5 we can chose 1 ≤ r1, t ≤ ∞ such that the first averaging +estimate in (8.6) and the averaging estimate (8.8) hold respectively. More precisely, we can select +0 ≤ 1/r1, 1/t ≤ 1 as follows: +• If 0 ≤ +1 +p1 ≤ +d +d+1, then we take 1/r1 = 1/(dp1). +• If 0 ≤ +1 +p4 ≤ +d +d+1, then we take 1/t = 1/(dp4). +• If +d +d+1 ≤ 1 +p1 ≤ 1, then we choose 1/r1 = d/p1 − d + 1. +• If +d +d+1 ≤ 1 +p4 ≤ 1, then we choose 1/t = d/p4 − d + 1. +In the next step, we determine 1 ≤ r ≤ ∞ by using the condition (8.7) and the second averaging +estimate in (8.6). Since two kinds of t values can be chosen as above, the condition (8.7) becomes +1 +s = 1 +p3 ++ +1 +dp4 +or +1 +s = 1 +p3 ++ d +p4 +− d + 1. +Combining these s values with the second averaging estimate in (8.6), the application of Lemma +2.5 enables us to choose 1/r values as follows: +• If 0 ≤ 1 +s = 1 +p3 + +1 +dp4 ≤ +d +d+1, then we take 1 +r = +1 +dp3 + +1 +d2p4. +• If +d +d+1 ≤ 1 +s = 1 +p3 + +1 +dp4 ≤ 1, then we take 1 +r = d +p3 + 1 +p4 − d + 1. +• If 0 ≤ 1 +s = 1 +p3 + d +p4 − d + 1 ≤ +d +d+1, then we take 1 +r = +1 +dp3 + 1 +p4 − 1 + 1 +d. +• If +d +d+1 ≤ 1 +s = 1 +p3 + d +p4 − d + 1 ≤ 1, then we take 1 +r = d +p3 + d2 +p4 − d2 + 1. +Finally, use the condition (8.5) together with previously selected two values for r1 and four values +for r. Then we obtain the required remaining conditions. +36 + +Remark 8.8. Notice that Theorem 8.5 is a special case of Theorem 8.7. However, the proof of +Theorem 8.5 is much simpler than that of Theorem 8.5. +We do not know if the consequences from Theorems 8.5 and 8.7 imply the sharp boundedness +results for the P3-Operator ⊓ on Fd +q. However, they play an important role in proving the theorem +below, which states that the exponents for ⊓(p1, p2, p3, p4) ≲ 1 are less restricted than those for +♦t(p1, p2, p3, p4) ≲ 1. The precise statement is as follows. +Theorem 8.9. Let ♦t and ⊓ be the operators acting on the functions on F2 +q. If ♦t(p1, p2, p3, p4) ≲ 1 +for 1 ≤ p1, p2, p3, p4 ≤ ∞, then ⊓(p1, p2, p3, p4) ≲ 1. +Proof. Assume that ♦t(p1, p2, p3, p4) ≲ 1 for 1 ≤ p1, p2, p3, p4 ≤ ∞. Then, by Lemma 6.2, the +point (1/p1, 1/p2, 1/p3, 1/p4) is contained in the convex hull of the following points: (0, 0, 1, 0), +(0, 1, 0, 0), (0, 0, 0, 1), (1/2, 0, 1/2, 1/2), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), +(2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3). +To complete the proof, by the interpolation theorem, it suffices to show that for each of the above +critical points (1/p1, 1/p2, 1/p3, 1/p4), we have +⊓(p1, p2, p3, p4) ≲ 1. +To prove this, we will use Theorem 8.7 and Theorem 8.5. By Theorem 8.7 with the hypothesis (i), +one can notice that ⊓(3/2, ∞, ∞, 3/2) ≲ 1, which is corresponding to the point (1/p1, 1/p2, 1/p3, 1/p4) = +(2/3, 0, 0, 2/3). Similarly, Theorem 8.7 with the hypothesis (ii) can be used for the point (1/2, 0, 1/2, 1/2), +namely, ⊓(2, ∞, 2, 2) ≲ 1. +For any other points, we can invoke Theorem 8.5. More precisely, we can apply Theorem 8.5 +with the hypothesis (i) for the points (0, 1, 0, 0), (2/3, 2/3, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), +(0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3). The points (0, 0, 1, 0), (0, 0, 0, 1) can be obtained by Theo- +rem 8.5 with the hypothesis (ii). Finally, for the point (1, 0, 0, 0), we can prove that ⊓(1, ∞, ∞, ∞) ≲ +1 by using Theorem 8.5 with the hypothesis (iii). This completes the proof. +Remark 8.10. The reverse statement of Theorem 8.9 is not true in general. As a counterexample, +we can take p1 = 3/2, p2 = 3, p3 = 3/2, p4 = ∞. Indeed, the assumption (i) of Theorem 8.5 with +d = 2 implies that ⊓(3/2, 3, 3/2, ∞) ≲ 1. However, ♦t(3/2, 3, 3/2, ∞) cannot be bounded, which +follows from Lemma 6.2. +The following corollary is a consequence of Theorem 8.9. +Corollary 8.11. Conjecture 1.5 is valid for the graph C4 + diagonal and its subgraph P3 in F2 +q. +Proof. It is obvious that the P3 is a subgraph of C4 + diagonal in F2 +q. For d = 2, it is plain to +notice that min{δ(C4 + diagonal), d} = 2 > δ(P3) = 1. Thus, the graph C4 + diagonal and its +subgraph P3 satisfy all assumptions of Conjecture 1.5. Then the statement of the corollary follows +37 + +immediately from Theorem 8.9 since the operators ♦t and ⊓ are related to the C4 + diagonal and +its subgraph P3, respectively. +The following theorem provides a concrete example for a positive answer to Question 1.2 since the +operators ✸ and ⊓ are related to the graph C4 and its subgraph P3, respectively. Furthermore, +the graphs also satisfy Conjecture 1.5 (see Corollary 8.14 below). +Theorem 8.12. Let ✸ and ⊓ be the operators acting on the functions on F2 +q. If ✸(p1, p2, p3, p4) ≲ 1, +1 ≤ p1, p2, p3, p4 ≤ ∞, then ⊓(p1, p2, p3, p4) ≲ 1. +Proof. By Theorem 7.7 (i), if ✸(p1, p2, p3, p4) ≲ 1, then ♦t(p1, p2, p3, p4) ≲ 1. By Theorem 8.9, if +♦t(p1, p2, p3, p4) ≲ 1, then ⊓(p1, p2, p3, p4) ≲ 1. Hence, the statement follows. +Remark 8.13. The reverse statement of Theorem 8.12 cannot hold. As in Remark 8.10, if we can +take p1 = 3/2, p2 = 3, p3 = 3/2, p4 = ∞, then ⊓(3/2, 3, 3/2, ∞) ≲ 1. However, ✸(3/2, 3, 3/2, ∞) +cannot be bounded, which follows from Theorem 7.6. +Corollary 8.14. Conjecture 1.5 holds true for the graph C4 and its subgraph P3 on F2 +q. +Proof. The main hypothesis (1.6) of Conjecture 1.5 is satisfied for the the graph C4 and its subgraph +P3 on F2 +q : +min{δ(C4), 2} = 2 > 1 = δ(P3). +Since the operators ✸ and ⊓ are associated to the graph C4 and its subgraph P3, respectively, the +statement of the corollary follows from Theorem 8.12. +9 +Operators associated to the graph K3 + tail (a kite) +Given t ∈ F∗ +q and functions fi, i = 1, 2, 3, 4, on Fd +q, we define ✂(f1, f2, f3, f4) as the following value: +1 +qd|St|2|Sd−2 +t +| +� +x1,x2,x3,x4∈Fdq +St(x1 − x2)St(x2 − x3)St(x3 − x4)St(x3 − x1) +4 +� +i=1 +fi(xi). +Note that this operator ✂ is related to the graph K3 + tail (Figure 1g), and so the normalizing +factor N(G) in (1.4) can be taken as the quantity qd|St|2|Sd−2 +t +|. +Definition 9.1. For 1 ≤ p1, p2, p3, p4 ≤ ∞, we define ✂(p1, p2, p3, p4) as the smallest constant +such that the following estimate holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on +Fd +q : +✂(f1, f2, f3, f4) ≤ ✂(p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4. +Here, our main problem is to determine all exponents 1 ≤ p1, p2, p3, p4 ≤ ∞ such that +✂ (f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4 +(9.1) +38 + +holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd +q. In other words, we are asked +to determine all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ✂(p1, p2, p3, p4) ≲ 1. +Recall that when d = 2, we assume that 3 ∈ Fq is a square number. +Lemma 9.2 (Necessary conditions for the boundedness of ✂(p1, p2, p3, p4) ). Suppose that (9.1) +holds, namely ✂(p1, p2, p3, p4) ≲ 1. Then we have +1 +p1 ++ 1 +p2 ++ d +p3 ++ 1 +p4 +≤ d, +1 +p1 ++ d +p2 ++ 1 +p3 +≤ d, +d +p1 ++ 1 +p2 ++ 1 +p3 +≤ d, +1 +p1 ++ d +p2 ++ 1 +p3 ++ d +p4 +≤ 2d − 1, +and +d +p1 ++ 1 +p2 ++ 1 +p3 ++ d +p4 +≤ 2d − 1. +In particular, if d = 2, then it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3, 1/p4) +is contained in the convex hull of the points: (0, 0, 1, 0), (0, 1, 0, 0), (0, 1, 0, 1/2), (2/3, 0, 2/3, 0), +(1/2, 1/2, 1/2, 0), (5/6, 0, 1/3, 1/2), (1, 0, 0, 0), (1/3, 0, 1/3, 1), (5/8, 5/8, 1/8, 1/2), (1/2, 0, 0, 1), +(1/4, 1/4, 1/4, 1), (1/3, 1/3, 0, 1), (1, 0, 0, 1/2), (2/3, 2/3, 0, 1/2), (2/3, 2/3, 0, 0), (0, 1/2, 0, 1), +(0, 1/3, 1/3, 1), (0, 0, 0, 1), (0, 5/6, 1/3, 1/2), (0, 0, 1/2, 1), (0, 0, 0, 0), (0, 2/3, 2/3, 0). +Proof. To deduce the first inequality, we test (9.1) with f1 = f2 = f4 = 1St and f3 = δ0. To obtain +the second one, we test (9.1) with f1 = f3 = 1St, f2 = δ0, and f4 = 1Fdq. To get the third one, +we test (9.1) with f1 = δ0, f2 = f3 = 1St, and f4 = 1Fdq. To prove the fourth one, we test (9.1) +with f1 = f3 = 1St and f2 = f4 = δ0. Finally, to obtain the fifth inequality, we test (9.1) with +f1 = f4 = δ0 and f2 = f3 = 1St. +9.1 +Sufficient conditions for the boundedness of ✂ on Fd +q +When one of exponents p1, p2, p4 is ∞, the boundedness problem of ✂(p1, p2, p3, p4) can be reduced +to that for the K3-Operator ∆ or the P2-Operator Λ. +Proposition 9.3. Let 1 ≤ a, b, c ≤ ∞. +(i) If ∆(a, b, c) ≲ 1, then ✂(a, b, c, ∞) ≲ 1. +(ii) If Λ(a, b, c) ≲ 1, then ✂(∞, a, b, c) ≲ 1 and ✂(a, ∞, b, c) ≲ 1. +Proof. For all non-negative functions fi, i = 1, 2, 3, 4, on Fd +q. we aim to prove the following inequal- +ities: +✂ (f1, f2, f3, f4) ≲ + + + + + + + +∆(f1, f2, f3) ||f4||∞, +||f1||∞ Λ(f2, f3, f4), +||f2||∞ Λ(f1, f3, f4). +(9.2) +By the definition, ✂(f1, f2, f3, f4) can be expressed as +1 +qd|St||Sd−2 +t +| +� +x1,x2,x3∈Fdq +St(x1−x2)St(x2−x3)St(x3−x1) +� 3 +� +i=1 +fi(xi) +�  + 1 +|St| +� +x4∈Fdq +St(x3 − x4)f4(x4) + + . +39 + +The sum in the above bracket is clearly dominated by ||f4||∞ for all x3 ∈ Fd +q. Hence, recalling the +definition of ∆(f1, f2, f3) in (4.1), we get the first inequality in (9.2): +✂(f1, f2, f3, f4) ≤ ∆(f1, f2, f3)||f4||∞. +Now we prove the second and third inequalities in (9.2). We will only provide the proof of the +second inequality, that is +✂ (f1, f2, f3, f4) ≤ ||f1||∞ Λ(f2, f3, f4). +(9.3) +The third inequality can be similarly proved by switching the roles of variables x1, x2. We write +✂(f1, f2, f3, f4) as follows: +1 +qd|St|2 +� +x2,x3,x4∈Fdq +St(x2−x3)St(x3−x4) +� 4 +� +i=2 +fi(xi) +�  + +1 +|Sd−2 +t +| +� +x1∈Fdq +St(x1 − x2)St(x3 − x1)f1(x1) + + . +Recall the definition of Λ(f2, f3, f4) in (5.1). Then, to prove the inequality (9.3), it will be enough +to show that for all x2, x3 ∈ Fd +q with ||x2 −x3|| = t ̸= 0, the value in the above bracket is ≲ ||f1||∞. +Now by a simple change of variables, the value in the above bracket is the same as +1 +|Sd−2 +t +| +� +x1∈St +St((x3 − x2) − x1)f1(x1 + x2). +This is clearly dominated by +1 +|Sd−2 +t +| +� +x1∈St +St((x3 − x2) − x1)||f1||∞. +Since ||x3 − x2|| = t ̸= 0, applying Corollary 12.4 in Appendix gives us the desirable estimate. +We address sufficient conditions for the boundedness of ✂ on Fd +q. +The following result can be obtained from Proposition 9.3 (i). +Theorem 9.4. Let ✂ be defined on the functions on Fd +q, d ≥ 2. Suppose that 1 ≤ a, b ≤ ∞ satisfies +the following equations: +1 +a + d +b ≤ d +and +d +a + 1 +b ≤ d. +Then we have ✂(a, b, ∞, ∞) ≲ 1, ✂ (a, ∞, b, ∞) ≲ 1, ✂ (∞, a, b, ∞) ≲ 1. +Proof. The statement follows immediately by combining Proposition 9.3 (i) with Theorem 4.5. +Proposition 9.3 (ii) can be used to deduce the following result. +Theorem 9.5. Let ✂ be defined on the functions on Fd +q, d ≥ 2. Suppose that 1 ≤ a, b, c ≤ ∞ +satisfies one of the following conditions: +40 + +(i) 0 ≤ 1 +a, 1 +c ≤ +d +d+1 and 1 +a + d +b + 1 +c ≤ d, +(ii) 0 ≤ 1 +a ≤ +d +d+1 ≤ 1 +c ≤ 1, and +1 +da + 1 +b + d +c ≤ d, +(iii) 0 ≤ 1 +c ≤ +d +d+1 ≤ 1 +a ≤ 1, and d +a + 1 +b + 1 +dc ≤ d, +(iv) +d +d+1 ≤ 1 +a, 1 +c ≤ 1 and d +a + 1 +b + d +c ≤ 2d − 1. +Then we have ✂(∞, a, b, c) ≲ 1 and +✂ (a, ∞, b, c) ≲ 1. +Proof. From our assumptions on the numbers a, b, c, Theorem 5.4 implies that Λ(a, b, c) ≲ 1. +Hence, the statement follows by applying Proposition 9.3 (ii). +9.2 +Boundedness of ✂ in two dimensions +Theorems 9.4, 9.5 provide non-trivial results available in higher dimensions. In this section we +will show that further improvements can be made in two dimensions. Before we state and prove +the improvements, we collect the results in two dimensions, which can be direct consequences of +Theorems 9.4, 9.5. +To deduce the following theorem, we will apply Theorem 9.4 with d = 2. +Theorem 9.6. Let ✂ be defined on functions on F2 +q. Then we have ✂(p1, p2, p3, p4) ≲ 1 pro- +vided that (p1, p2, p3, p4) is one of the following points: (∞, ∞, ∞, ∞), (1, ∞, ∞, ∞), (∞, 1, ∞, ∞), +(∞, ∞, 1, ∞), (3/2, 3/2, ∞, ∞), (3/2, ∞, 3/2, ∞), (∞, 3/2, 3/2, ∞). +Proof. Using the first conclusion of Theorem 9.4 with d = 2, we see that ✂(p1, p2, p3, p4) ≲ 1 +whenever (p1, p2, p3, p4) takes the following points: (∞, ∞, ∞, ∞), (1, ∞, ∞, ∞), (∞, 1, ∞, ∞), +(3/2, 3/2, ∞, ∞). +Next, the second conclusion of Theorem 9.4 with d = 2 implies that ✂(p1, p2, p3, p4) ≲ 1 for the +points (p1, p2, p3, p4) = (∞, ∞, 1, ∞), (3/2, ∞, 3/2, ∞). Finally, it follows from the third conclusion +of Theorem 9.4 with d = 2 that ✂(p1, p2, p3, p4) ≲ 1 for (p1, p2, p3, p4) = (∞, 3/2, 3/2, ∞). Hence, +the proof is complete. +The following theorem will be proven by applying Theorem 9.5 with d = 2. +Theorem 9.7. Let ✂ be defined on the functions on F2 +q. Suppose that (p1, p2, p3, p4) is one of +the following points: (∞, ∞, ∞, 1), (2, ∞, 2, 2), (3/2, ∞, ∞, 3/2), (∞, ∞, 3/2, 3/2). Then we have +✂(p1, p2, p3, p4) ≲ 1. +Proof. We get that ✂(∞, ∞, ∞, 1) ≲ 1 by using the assumption (ii) and the first conclusion of The- +orem 9.5. Invoking the assumption (i) and the second conclusion of Theorem 9.5, one can directly +note that ✂(2, ∞, 2, 2) ≲ 1 and ✂(3/2, ∞, ∞, 3/2) ≲ 1. Finally, to prove that ✂(∞, ∞, 3/2, 3/2) ≲ +1, one can use the assumption (i) and the first conclusion of Theorem 9.5. +We now introduce the connection between ✂(f1, f2, f3, f4) and the bilinear averaging operator. +41 + +Proposition 9.8. Let B be the bilinear operator defined as in (4.2). Then, for any non-negative +real-valued functions fi, i = 1, 2, 3, 4, on F2 +q, we have +✂(f1, f2, f3, f4) = ||B(f1, f2) · f3 · Af4||1, +where A denotes the averaging operator over the circle in F2 +q. +Proof. In two dimensions, ✂(f1, f2, f3, f4) can be rewritten as the following form: +1 +q2|St|2 +� +x1,x2,x3,x4∈F2q +St(x1 − x2)St(x3 − x2)St(x3 − x4)St(x3 − x1) +4 +� +i=1 +fi(xi). +By the change of variables by putting y1 = x3 − x1, y2 = x3 − x2, y3 = x3, y4 = x3 − x4, the +value ✂(f1, f2, f3, f4) becomes +1 +q2|St|2 +� +y1,y2,y3,y4∈F2q +St(y2 − y1)St(y2)St(y4)St(y1)f1(y3 − y1)f2(y3 − y2)f3(y3)f4(y3 − y4). +This can be expressed as follows: +1 +q2 +� +y3∈F2q +f3(y3) + + 1 +|St| +� +y4∈St +f4(y3 − y4) + + + + 1 +|St| +� +y1,y2∈St:||y2−y1||=t +f1(y3 − y1)f2(y3 − y2) + + . +Recalling the definitions of the averaging operator in (2.1) and the bilinear averaging operator in +(4.2), it follows that +✂(f1, f2, f3, f4) = 1 +q2 +� +y3∈F2q +f3(y3)Af4(y3)B(f1, f2)(y3). +By the definition of the normalized norm || ||1, the statement follows. +For 1 ≤ p1, p2, p3, p4 ≤ ∞, recall that the notation ✂(p1, p2, p3, p4) ⪅ 1 is used if the following +estimate holds for all subsets E, F, G, H of F2 +q: +✂(E, F, G, H) ≲ ||E||p1||F||p2||G||p3||H||p4, +and this estimate is referred to as the restricted strong-type ✂(p1, p2, p3, p4) estimate. +The following theorem is our main result in two dimensions, which gives a new restricted strong- +type estimate for the boundedness on the operator ✂. +Theorem 9.9. Let ✂ be defined on functions on F2 +q. Let 1 ≤ p3, p4 ≤ ∞. Then the following +statements are valid for all subsets E, F of F2 +q and all non-negative functions f3, f4 on F2 +q. +42 + +(i) If 2 ≤ p3 ≤ ∞, 3/2 ≤ p4 ≤ ∞, and +1 +p3 + +1 +2p4 ≤ 1 +2, then we have +✂(E, F, f3, f4) ≲ ||E||2||F||2||f3||p3||f4||p4. +(ii) If 2 ≤ p3 ≤ ∞, 4/3 ≤ p4 ≤ 3/2, and +1 +p3 + 2 +p4 ≤ 3 +2, then we have +✂(E, F, f3, f4) ≲ ||E||2||F||2||f3||p3||f4||p4. +Proof. Let E, F be subsets of F2 +q and f, g be non-negative real-valued functions on F2 +q. By Propo- +sition 9.8 and H¨older’s inequality, it follows that for 2 ≤ p3 ≤ ∞, +✂(E, F, f3, f4) ≤ ||B(E, F)||2||f3||p3||Af4|| 2p3 +p3−2. +Here, we also notice that 2 ≤ +2p3 +p3−2 ≤ ∞. Since ||B(E, F)||2 ≲ ||E||2||F||2 by Lemma 4.10, we see +that +✂(E, F, f3, f4) ≤ ||E||2||F||2||f3||p3||Af4|| 2p3 +p3−2. +Hence, to complete the proof, it suffices to show that for all exponents p3, p4 satisfying the as- +sumptions of the theorem, we have +A +� +p4 → +2p3 +p3 − 2 +� +≲ 1. +(9.4) +To prove this, we first recall from Theorem 2.3 with d = 2 that A(p → r) ≲ 1 for any numbers +1 ≤ p, r ≤ ∞ such that (1/p, 1/r) lies on the convex hull of points (0, 0), (0, 1), (1, 1), and (2 +3, 1 +3). +Also invoke Lemma 2.5 to find the equations indicating the endpoint estimates for A(p → r) ≲ 1. +Using those averaging estimates with p = p4, r = +2p3 +p3−2, the inequality (9.4) can be obtained by a +direct computation, where we also use the fact that 2 ≤ r = +2p3 +p3−2 ≤ ∞. +The following corollary is a direct consequence of Theorem 9.9. +Corollary 9.10. Let ✂ be defined on functions on F2 +q. Then we have ✂(2, 2, 2, ∞) ⪅ 1. +Proof. The statement follows by a direct application of Theorem 9.9 (i). +The theorem below shows that the exponents for ♦t(p1, p2, p3, p4) ≲ 1 are more restricted than +those for ✂(p1, p2, p3, p4) ≲ 1 up to the endpoints. This also provides a positive answer to Question +1.2 since the graph K3 + tail is a subgraph of the graph C4 + diagonal. +Theorem 9.11. Let ♦t and ✂ be the operators acting on the functions on F2 +q. Suppose that +♦t(p1, p2, p3, p4) ≲ 1 for 1 ≤ p1, p2, p3, p4 ≤ ∞. Then we have ✂(p1, p2, p3, p4) ≲ 1 except for the +point (2, 2, 2, ∞). In addition, we have ✂(2, 2, 2, ∞) ⪅ 1. +Proof. Assume that ♦t(p1, p2, p3, p4) ≲ 1 for 1 ≤ p1, p2, p3, p4 ≤ ∞. Then, by Lemma 6.2, the +point (1/p1, 1/p2, 1/p3, 1/p4) is contained in the convex hull of the following points: (0, 0, 1, 0), +43 + +(0, 1, 0, 0), (0, 0, 0, 1), (1/2, 0, 1/2, 1/2), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), +(2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3). +By Theorems 9.6, 9.7, the strong type estimate ✂(p1, p2, p3, p4) ≲ 1 holds for all the above points +(1/p1, 1/p2, 1/p3, 1/p4) except for (1/2, 1/2, 1/2, 0). Moreover, we know from Corollary 9.10 that +✂(2, 2, 2, ∞) ⪅ 1. Hence, the statement follows by interpolating those points. +Remark 9.12. The reverse statement of Theorem 9.11 is not true. +To see this, observe from +Theorem 9.9 (ii) that ✂(2, 2, 6, 3/2) ⪅ 1. In addition, by Lemma 6.2, notice that ♦t(2, 2, 6, 3/2) +cannot be bounded. +Corollary 9.13. Conjecture 1.5 holds up to endpoints for the graph C4 + diagonal and its subgraph +K3 + tail in F2 +q. +Proof. The operators ♦t and ✂ are associated with the C4 + diagonal and its subgraph K3 + tail +in F2 +q, respectively. Hence, invoking Theorem 9.11, the proof is reduced to showing that the C4 + +diagonal and its subgraph K3 + tail satisfy the main hypothesis (1.6) of Conjecture 1.5. However, +it is clear that +min{δ(C4 + diagonal), 2} = 2 > 1 = δ(K3 + tail). +Thus, the proof is complete. +The following theorem shows that there exists an inclusive relation between boundedness exponents +for the operators corresponding to the graphs C4 and K3 + tail, although they are not subgraphs +of each other. +Theorem 9.14. Let ✸ and ✂ be defined on functions on F2 +q and let 1 ≤ p1, p2, p3, p4 ≤ ∞. Then +if ✸(p1, p2, p3, p4) ≲ 1, we have ✂(p1, p2, p3, p4) ≲ 1. +Proof. First, by Theorem 7.6, note that ✸(2, 2, 2, ∞) cannot be bounded. +Now suppose that +✸(p1, p2, p3, p4) ≲ 1. Then (p1, p2, p3, p4) ̸= (2, 2, 2, ∞). Using Theorem 7.7, we get ♦t(p1, p2, p3, p4) ≲ +1. Then the statement follows immediately from Theorem 9.11. +By combining Remark 9.12 and Theorem 7.7, it is clear that the reverse of Theorem 9.14 does +not hold. Notice that Theorem 9.14 provides an example to satisfy Conjecture 1.5 without the +hypothesis that G′ is a subgraph of the graph G. +10 +Boundedness problems for the Y -shaped graph +In this section, we study the boundedness of the operator for the Y -shaped graph in Figure 1h. +For t ∈ F∗ +q, the Y -shaped operator Y is defined by +Y (f1, f2, f3, f4) = +1 +qd|St|3 +� +x1,x2,x3,x4∈Fdq +St(x3 − x1)St(x3 − x2)St(x3 − x4) +4 +� +i=1 +fi(xi), +(10.1) +44 + +where functions fi, i = 1, 2, 3, 4, are defined on Fd +q. Note that this operator Y is related to the +Y -shaped graph, and so the normalizing factor N(G) in (1.4) can be taken as qd|St|3. +The operator norm of the Y -shaped operator is defined in a standard way. +Definition 10.1. For 1 ≤ p1, p2, p3, p4 ≤ ∞, the operator norm Y (p1, p2, p3, p4) is defined as the +smallest constant such that the following estimate holds for all non-negative real-valued functions +fi, i = 1, 2, 3, 4, on Fd +q : +Y (f1, f2, f3, f4) ≤ Y (p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4. +(10.2) +We aim to find all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ✂(p1, p2, p3, p4) ≲ 1. +Lemma 10.2 (Necessary conditions for the boundedness of Y (p1, p2, p3, p4)). Let 1 ≤ pi ≤ ∞, 1 ≤ +i ≤ 4. Suppose that Y (p1, p2, p3, p4) ≲ 1. Then all the following inequalities are satisfied: +1 +p1 + 1 +p2 + d +p3 + 1 +p4 ≤ d, +d +p1 + d +p2 + 1 +p3 + d +p4 ≤ 3d − 2, +d +p1 + d +p2 + 1 +p3 ≤ 2d − 1, +d +p1 + 1 +p3 + d +p4 ≤ 2d − 1, +d +p2 + 1 +p3 + d +p4 ≤ 2d − 1, +d +p1 + 1 +p3 ≤ d, +d +p2 + 1 +p3 ≤ d, +1 +p3 + d +p4 ≤ d. +In particular, if d = 2, then it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3, 1/p4) is +contained in the convex hull of the points: (0, 0, 1, 0), (1, 0, 0, 1/2), (1, 0, 1/2, 0), (1/2, 1, 0, 1/2), +(1, 1/2, 0, 1/2), (1/2, 1/2, 0, 1), (1, 0, 1/3, 1/3), (1/2, 5/6, 1/3, 0), (1, 1/2, 0, 0), (1/2, 0, 1/3, 5/6), +(1/2, 1, 0, 0), (1, 1/3, 1/3, 0), (1/2, 0, 0, 1), (1, 0, 0, 0), (0, 0, 0, 0), (0, 0, 0, 1), (0, 1, 0, 0), (0, 0, 2/3, 2/3), +(0, 5/6, 1/3, 1/2), (0, 1, 0, 1/2), (0, 1/2, 1/3, 5/6), (0, 1/2, 0, 1), (0, 2/3, 2/3, 0). +Proof. By a direct computation, the conclusions of the lemma easily follow by testing the inequality +(10.2) with the following specific functions, respectively: +1) f1 = f2 = f4 = 1St, f3 = δ0. +2) f1 = f2 = f4 = δ0, f3 = 1St. +3) f1 = f2 = δ0, f3 = 1St, f4 = 1Fdq. +4) f1 = f2 = δ0, f2 = 1Fdq, f3 = 1St. +5) f1 = 1Fdq, f2 = f4 = δ0, f3 = 1St. +6) f1 = δ0, f2 = f4 = 1Fdq, f3 = 1St. +7) f1 = f4 = 1Fdq, f2 = δ0, f3 = 1St. +8) f1 = f2 = 1Fdq, f3 = 1St, f4 = δ0. +10.1 +Sufficient conditions for the boundedness of Y on Fd +q +It is not hard to observe that the boundedness problem for the Y -shaped operator can be reduced +to the spherical averaging estimate. Indeed, the value Y (f1, f2, f3, f4) in (10.1) can be written by +Y (f1, f2, f3, f4) = 1 +qd +� +x3∈Fdq +f3(x3) +� +i=1,2,4 + + 1 +|St| +� +xi∈Fdq +St(x3 − xi)fi(xi) + + . +45 + +Invoking the definition of the averaging operator A = ASt in (2.2), we get +Y (f1, f2, f3, f4) = 1 +qd +� +x3∈Fdq +f3(x3)Af1(x3)Af2(x3)Af4(x4) = ||Af1 · Af2 · f3 · Af4||1. +By H˝older’s inequality and the nesting property of the norm || · ||p, we get +Y (f1, f2, f3, f4) ≤ ||Af1||r1||Af2||r2||f||p3||Af4||r4 +if +1 +r1 ++ 1 +r2 ++ 1 +p3 ++ 1 +r4 +≤ 1. +(10.3) +Proposition 10.3. Let 1 ≤ p1, p2, p3, p4, r1, r2, r4 ≤ ∞ be extended real numbers which satisfy the +following assumptions: +1 +r1 + 1 +r2 + 1 +p3 + 1 +r4 ≤ 1 and A(pi → ri) ≲ 1 for all i = 1, 2, 4. Then we have +Y (p1, p2, p3, p4) ≲ 1. +Proof. By combining the inequality (10.3) with our assumptions on the averaging estimates, it +follows that for all functions fi, i = 1, 2, 3, 4, on Fd +q, +Y (f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4. +This completes the proof. +The following theorem provides lots of sufficient conditions for the boundedness of the Y -shaped +operator. +Theorem 10.4. Let 1 ≤ p1, p2, p3, p4 ≤ ∞, and Y be the Y -shaped operator on Fd +q. Then +Y (p1, p2, p3, p4) ≲ 1 provided that one of the following conditions is satisfied: +(i) 0 ≤ +1 +p1, 1 +p2, 1 +p4 ≤ +d +d+1 and +1 +p1 + 1 +p2 + d +p3 + 1 +p4 ≤ d. +(ii) 0 ≤ +1 +p1, 1 +p2 ≤ +d +d+1 ≤ 1 +p4 and +1 +dp1 + +1 +dp2 + 1 +p3 + d +p4 ≤ d. +(iii) 0 ≤ +1 +p1, 1 +p4 ≤ +d +d+1 ≤ 1 +p2 ≤ 1 and +1 +dp1 + d +p2 + 1 +p3 + +1 +dp4 ≤ d. +(iv) 0 ≤ +1 +p2, 1 +p4 ≤ +d +d+1 ≤ 1 +p1 ≤ 1 and +d +p1 + +1 +dp2 + 1 +p3 + +1 +dp4 ≤ d. +(v) 0 ≤ +1 +p1 ≤ +d +d+1 ≤ 1 +p2, 1 +p4 ≤ 1 and +1 +dp1 + d +p2 + 1 +p3 + d +p4 ≤ 2d − 1. +(vi) 0 ≤ +1 +p2 ≤ +d +d+1 ≤ 1 +p1, 1 +p4 ≤ 1 and +d +p1 + +1 +dp2 + 1 +p3 + d +p4 ≤ 2d − 1. +(vii) 0 ≤ +1 +p4 ≤ +d +d+1 ≤ 1 +p1, 1 +p2 ≤ 1 and +d +p1 + d +p2 + 1 +p3 + +1 +dp4 ≤ 2d − 1. +(viii) +d +d+1 ≤ +1 +p1, 1 +p2, 1 +p4 ≤ 1 and +d +p1 + d +p2 + 1 +p3 + d +p4 ≤ 3d − 2. +Proof. The proof uses Proposition 10.3 and the sharp averaging estimates in Lemma 2.5. The +proof of this theorem is similar to that of Theorem 5.4. Therefore, we leave the detail of the proof +to readers. +Conjecture 1.5 is also supported by the following theorem. +46 + +Theorem 10.5. Let ♦t and Y be the operators acting on the functions on F2 +q. If ♦t(p1, p2, p3, p4) ≲ +1 with 1 ≤ p1, p2, p3, p4 ≤ ∞, then Y (p1, p2, p3, p4) ≲ 1. +Proof. Assume that ♦t(p1, p2, p3, p4) ≲ 1. Then, by Lemma 6.2, (1/p1, 1/p2, 1/p3, 1/p4) is contained +in the convex hull of the points (0, 0, 1, 0), (0, 1, 0, 0), (0, 0, 0, 1), (1/2, 0, 1/2, 1/2), (2/3, 2/3, 0, 0), +(1, 0, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), (2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3). +By interpolating those critical points, it suffices to check that each critical point above satisfies +one of the 8 hypotheses of Theorem 10.4 with d = 2. However, this can be easily shown by a direct +computation. For example, for the critical point (1/p1, 1/p2, 1/p3, 1/p4) = (1/2, 1/2, 1/2, 0), we +can invoke the hypothesis (i) of Theorem 10.4 with d = 2 and obtain that Y (2, 2, 2, ∞) ≲ 1. In a +same way, it can be easily proven for other critical points. +Remark 10.6. The reverse statement of Theorem 10.5 is not true. To find a counterexample, we +can take p1 = p3 = ∞, p2 = p4 = 3/2. Indeed, by the hypothesis (5) of Theorem 10.4 with d = 2, +we see that Y (∞, 3/2, ∞, 3/2) ≲ 1. However, ♦t(∞, 3/2, ∞, 3/2) is not bounded, which follows +from Lemma 6.2 with d = 2. +The following corollary proposes some possibility that the assumption of the subgraph in Conjec- +ture 1.5 can be dropped. +Corollary 10.7. Let ✸ and Y be the operators acting on the functions on F2 +q. If ✸(p1, p2, p3, p4) ≲ 1 +with 1 ≤ p1, p2, p3, p4 ≤ ∞, then Y (p1, p2, p3, p4) ≲ 1. +Proof. The statement of the corollary follows immediately by combining Theorem 7.7 and Theorem +10.5. +Combining all the results obtained so far, we get the following theorem: +Theorem 10.8. When d = 2 and n = 3, 4, Conjecture 1.5 is true, where we accept boundedness +results up to endpoints in the case when G is the C4 + diagonal and its subgraph G′ is the K3+tail. +Proof. By Corollary 5.8 for n = 3, and by Corollaries 8.11, 8.14, 9.13, 10.7 for n = 4, we have proven +that for d = 2 and n = 3, 4, there is the required inclusive boundedness relationship between any +two operators corresponding to arbitrary connected ordered graph G and its subgraph G′ except +for the following three cases: +(I) G = C4 + diagonal and G′ = C4. +(II) K3 + tail and G′ = Y -shape. +(III) G = K3 + tail and G′ = P3. +However, since δ(G) = δ(G′) for each case of (I), (II), (III), they do not satisfy the main hypothesis +(1.6) of Conjecture 1.5. Hence, they cannot be counterexamples contradicting Conjecture 1.5 and +so there is no counterexample against Conjecture 1.5, as required. +47 + +11 +Acknowledgements +A. Iosevich and P. Bhowmik were supported in part by the National Science Foundation grant no. +HDR TRIPODS–1934962 and the National Science Foundation grant DMS–2154232. D. Koh was +supported by Basic Science Research Programs through National Research Foundation of Korea +(NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07044469). T. Pham would like +to thank to the VIASM for the hospitality and for the excellent working conditions. +12 +Appendix +In this appendix, we introduce the number of intersection points of two spheres in Fd +q. Let η denote +the quadratic character of F∗ +q, namely, η(s) = 1 for a square number s in F∗ +q, and η(s) = −1 +otherwise. +Definition 12.1. Given a non-zero vector m in Fd +q, and t, b ∈ Fq, we define N(m, t, b) to be the +number of common solutions x ∈ Fd +q of the following equations: ||x|| = t, +m · x = b. +Notice that the value of N(m, t, b) is the number of all intersection points between the sphere St +and the plane {x ∈ Fd +q : m · x = b}. The explicit value of it is well known as follows. +Lemma 12.2. Let b, t ∈ Fq, and let m be a non-zero element in Fd +q, d ≥ 2. Then the following +statements hold: +(i) If ||m|| ̸= 0 and b2 − t||m|| = 0, then +N(m, t, b) = + + + +qd−2 +if d is even, +qd−2 + q +d−3 +2 (q − 1)η +� +(−1) +d−1 +2 ||m|| +� +if d is odd. +(ii) If ||m|| ̸= 0 and b2 − t||m|| ̸= 0, then +N(m, t, b) = + + + +qd−2 + q +d−2 +2 η +� +(−1) +d +2 (b2 − t||m||) +� +if d is even, +qd−2 − q +d−3 +2 η +� +(−1) +d−1 +2 ||m|| +� +if d is odd. +(iii) If ||m|| = 0 = b2 − t||m||, then +N(m, t, b) = + + + +qd−2 + ν(t)q +d−2 +2 η +� +(−1) +d +2 ) +� +if d is even, +qd−2 − q +d−1 +2 η +� +(−1) +d−1 +2 t +� +if d is odd, +where ν(t) = −1 if t ∈ F∗ +q and ν(0) = q − 1. +(iv) If ||m|| = 0 and b2 − t||m|| ̸= 0, then N(m, t, b) = qd−2. +Proof. See Exercises 6.31–6.34 in [16], or one can prove it by using the discrete Fourier analysis +with the explicit value of the Gauss sum. +48 + +By a direct application of Lemma 12.2, one can find the explicit number of the intersections of two +spheres over finite fields. Precisely we have the following result. +Theorem 12.3. Given a non-zero vector m ∈ Fd +q and t, j ∈ Fq, let +Θ(m, t, j) := {x ∈ St : ||x − m|| = j}|. +If m ∈ Sℓ, then |Θ(m, t, j)| = N +� +m, t, t+ℓ−j +2 +� +. +Proof. Since ||x − m|| = t + ℓ − 2m · x for x ∈ St, m ∈ Sℓ, it is clear that Θ(m, t, j) is the number +of common solutions x of the following equations: +||x|| = t, +m · x = t + ℓ − j +2 +. +Hence, by the definition of N, we obtain the required conclusion. +Corollary 12.4. Let t ∈ F∗ +q and ℓ ∈ Fq. Then, for every non-zero vector m ∈ Sℓ, we have +� +x∈St:||x−m||=t +1 ∼ qd−2 +excepting for the following three cases: +1) d = 2, ℓ ̸= 0, η(tℓ − ℓ2/4) = −1. +2) d = 2, ℓ = 0, η(−1) = 1. +3) d = 3, ℓ = 0, η(−t) = 1. +For each of those three cases, the value in the above sum takes zero. +On the other hand, if +d = 2, ℓ ̸= 0, and η(tℓ − ℓ2/4) = 1, the value in the above sum is exactly two. +Proof. It follows from Theorem 12.3 that for any ||m|| = ℓ, +� +x∈St:||x−m||=t +1 = N +� +m, t, ℓ +2 +� +, +and so the corollary is a direct consequence of Lemma 12.2 (i), (ii), (iii). +References +[1] B. Assarf, E. Gawrilow, K. Herr, M. Joswig, B. Lorenz, A. Paffenholz and T. Rehn, Computing +convex hulls and counting integer points with polymake, Math. Program. Comput. 9 (2017), +no.1, 1-38. +[2] M. Bennett, D. Hart, A. Iosevich, J. Pakianathan, and M. Rudnev, Group actions and geo- +metric combinatorics in Fd +q, Forum Math. 29 (2017), no.1, 91-110. +[3] A. Carbery, B. Stones, and J. Wright, Averages in vector spaces over finite fields, Math. Proc. +Cambridge Philos. Soc. 144 (2008), no.1, 13-27. +49 + +[4] J. Chapman, M. Erdo˜gan, D. Hart, A. Iosevich, and D. Koh, Pinned distance sets, Wolff’s +exponent in finite fields and sum-product estimates, Math.Z. 271, (2012), no.1-2, 63-93. +[5] X. Du, A. Iosevich, Y. Ou, H. Wang and R. 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Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. +51 + diff --git a/qNAyT4oBgHgl3EQfl_iN/content/tmp_files/load_file.txt b/qNAyT4oBgHgl3EQfl_iN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..960a6e823e6c1d55e95682e108084c09cc443681 --- /dev/null +++ b/qNAyT4oBgHgl3EQfl_iN/content/tmp_files/load_file.txt @@ -0,0 +1,2245 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf,len=2244 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='00463v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='CA] 1 Jan 2023 Multi-linear forms, graphs, and Lp-improving measures in Fd q Pablo Bhowmick Alex Iosevich Doowon Koh Thang Pham January 3, 2023 Contents 1 Introduction 2 2 The spherical averaging problem 8 3 Sharp mapping properties for the K2-Operator 10 4 Boundedness problem for the K3-Operator 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 Boundedness results for ∆ on Fd q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 Boundedness of ✂ in two dimensions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 41 10 Boundedness problems for the Y -shaped graph 44 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 Sufficient conditions for the boundedness of Y on Fd q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 45 11 Acknowledgements 48 12 Appendix 48 1 Abstract The purpose of this paper is to introduce and study the following graph theoretic paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let TKf(x) = � K(x, y)f(y)dµ(y), where f : X → R, X a set, finite or infinite, and K and µ denote a suitable kernel and a measure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Given a connected ordered graph G on n vertices, consider the multi-linear form ΛG(f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , fn) = � x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=',xn∈X � (i,j)∈E(G) K(xi, xj) n � l=1 fl(xl)dµ(xl), where E(G) is the edge set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Define ΛG(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) as the smallest constant C > 0 such that the inequality ΛG(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , fn) ≤ C n � i=1 ||fi||Lpi(X,µ) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) holds for all non-negative real-valued functions fi, 1 ≤ i ≤ n, on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The basic question is, how does the structure of G and the mapping properties of the operator TK influence the sharp exponents in (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this paper, this question is investigated mainly in the case X = Fd q, the d-dimensional vector space over the field with q elements, and K(xi, xj) is the indicator function of the sphere evaluated at xi − xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This provides a connection with the study of Lp-improving measures and distance set problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1 Introduction One of the fundamental objects in harmonic analysis is the operator of the form TKf(x) = � Rd K(x, y)f(y)dy, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) where K : Rd × Rd → R is a suitable kernel and f is a locally integrable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' See [21] and the references contained therein for a variety of manifestations of operators of this type and their bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The purpose of this paper is to study operators from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) in the context of vector spaces over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let Fq denote the finite field with q elements and Fd q the d-dimensional vector space over this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let K : Fd q × Fd q → C be a suitable kernel, and define TKf(x) = � y∈Fdq K(x, y)f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Operators of this type have been studied before ([3, 12, 14, 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In particular, the operator TK with K(x, y) = St(x − y), where St is the indicator function of the sphere St = {x ∈ Fd q : ||x|| = t}, ||x|| = x2 1 + x2 2 + · · · + x2 d comes up naturally in the study of the Erd˝os-Falconer distance problem 2 in vector spaces over finite fields, namely the question of how large a subset E ⊂ Fd q needs to be to ensure that if ∆(E) = {||x − y|| : x, y ∈ E}, then |∆(E)| ≥ q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Here and throughout, |S|, with S a finite set, denotes the number of elements in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' See, for example ([2, 4, 10, 11, 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If one is interested in studying more complicated geometric objects than distances, an interesting modification of the spherical averaging operator needs to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Indeed, let E ⊂ Fd q, and suppose that we want to know how many equilateral triangles of side-length 1 it determines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The quantity that counts such triangles is given by � x,y,z∈Fdq K(x, y)K(x, z)K(y, z)E(x)E(y)E(z), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) where K(x, y) = S1(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let us interpret the quantity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let us view x, y, z as vertices, and let us view the presence of K(x, y) as determining the edge connecting x and y, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this way, the quantity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) is associated with the graph K3, the complete graph on three vertices (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Another natural example is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let K(x, y) = S1(x−y), and consider the quantity that counts rhombi of side-length 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' � x,y,z,w∈Fdq K(x, y)K(y, z)K(z, w)K(w, x)E(x)E(y)E(z)E(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) Arguing as above, we associate this form with the graph C4, the cycle on four vertices (Figure 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In general, let K be a kernel function, and let G be a connected ordered graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Define ΛG(f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , fn) = 1 N(G) � x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=',xn∈Fdq � (i,j)∈E(G) K(xi, xj) n � l=1 fl(xl), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) where E(G) is the edge set of G and N(G) is the normalizing factor defined as the number of distinct embeddings of G in Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We note in passing that the paradigm we just introduced extends readily to the setting of hy- pergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If we replace our basic object, the linear operator TK, by an m-linear operator MK, the problem transforms to the setting where the edges dictated by the kernel K are replaced by hyperedges induced by the multi-linear kernel K(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , xm+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We shall address this formulation of the problem in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The norm ||f||p, 1 ≤ p < ∞, is defined to be associated with normalizing counting measure on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 3 More precisely, given a function f on Fd q, we define ||f||p := \uf8eb \uf8edq−d � x∈Fdq |f(x)|p \uf8f6 \uf8f8 1 p (1 ≤ p < ∞), and ||f||∞ := max x∈Fdq |f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ pi ≤ ∞, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , n, be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We define ΛG(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) as the smallest constant C > 0 such that the following inequality ΛG(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , fn) ≤ C n � i=1 ||fi||pi (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) holds for all non-negative real-valued functions fi, 1 ≤ i ≤ n, on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If there is a uniform constant C ≥ 1 independent of q such that ΛG(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) ≤ C, then we use the notation ΛG(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For the remainder of this paper, the kernel function K(x, y) is assumed to be St(x − y) with t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In addition, when the dimension d is two, we assume that the number 3 in Fq is a square number so that we can exclude the trivial case in which the shape of an equilateral triangle in F2 q does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The main purpose of this paper is to determine all numbers 1 ≤ pi ≤ ∞, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , n, such that ΛG(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This problem will be called the boundedness problem for the operator ΛG on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this paper we shall mainly confine ourselves to the graphs G which possess one of the following specific structures: K2 (the graph with two vertices and one edge), K3 (the cycle with three vertices and three edges), K3 + tail (a kite), P2 (the path of length 2), P3 (the path of length 3), C4 (the cycle with 4 vertices and 4 edges), C4 + diagonal, Y -shape (a space station).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is worth noting that the study of the distribution of these graphs in a given set has been received much attention from people in Discrete Geometry and Geometric Measure Theory during the last decade, see for example [5, 9, 7, 8, 13, 17, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When the graph G is the K2, the complete answer to the boundedness problem will be given in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To deduce the result, we will invoke the spherical averaging estimates over finite fields (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When the number of the vertices of the graph G is 3 or 4, we will obtain reasonably good bound- edness results in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In particular, in the case when the degree of each vertex is at least two (K3, C4+ a diagonal, and C4), we shall prove sharp results (up to the endpoints) for the operators on F2 q (see Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For three and higher dimensions, the boundedness problem is not simple and we will address partial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 4 (a) G = K2 (b) G = K3 (c) G = P2 (d) G = C4+ diagonal (e) G = C4 (f) G = P3 (g) G = K3+ tail (h) G = Y -shape It is very natural to ask whether or not one can prove a general theorem that addresses all connected graphs on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Unfortunately, such a result is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The main difficulties arise when the maximal degree is large or the edge set is dense, or if the graph contains a cycle or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' All of these issues will be illustrated in the proofs of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notation: Let 1 ≤ pi ≤ ∞ and fi, 1 ≤ i ≤ n, be non-negative real-valued functions on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When n = 2, we denote L(f1, f2) := ΛG(f1, f2) and L(p1, p2) := ΛG(p1, p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (Case of n = 3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If G is the graph K3, we denote ∆(f1, f2, f3) := ΛG(f1, f2, f3) and ∆(p1, p2, p3) := ΛG(p1, p2, p3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If G is the graph P2, we denote Λ(f1, f2, f3) := ΛG(f1, f2, f3) and Λ(p1, p2, p3) := ΛG(p1, p2, p3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (Case of n = 4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If G is the C4 + diagonal, we denote ♦t(f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and ♦t(p1, p2, p3, p4) := ΛG(p1, p2, p3, p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If G is the C4, we denote ✸(f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and ✸(p1, p2, p3, p4) := ΛG(p1, p2, p3, p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If G is the P3, we denote ⊓(f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and ⊓(p1, p2, p3, p4) := 5 ΛG(p1, p2, p3, p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If G is the K3 + tail (a kite), we denote ✂(f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and ✂(p1, p2, p3, p4) := ΛG(p1, p2, p3, p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If G is the Y -shape (a space station), we denote Y (f1, f2, f3, f4) := ΛG(f1, f2, f3, f4) and Y (p1, p2, p3, p4) := ΛG(p1, p2, p3, p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We denote ΛG(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) ⪅ 1 if the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) holds true for all characteristic functions on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By F∗ q, we mean the set of all non-zero elements in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For t ∈ F∗ q, we denote by Sn−1 t the sphere of radius t centered at the origin in Fn q : Sn−1 t := {x ∈ Fn q : ||x|| = t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Unless otherwise specified in this paper, d represents the general dimension of Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When n = d, we write St instead of Sd−1 t for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We identify the set St with its indicator function 1St, namely, St(x) = 1St(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We write δ0 for the indicator function of the set of the zero vector in Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For positive numbers A, B > 0, we write A ≲ B if A ≤ CB for some constant C > 0 independent of q, the size of the underlying finite field Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The notation A ∼ B means that A ≲ B and B ≲ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We also study the boundedness relation between the operators associated with a graph G and its subgraph G′ with n-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Throughout the paper, we always assume that the graph G and its subgraph G′ are connected ordered graphs with |G| = |G′| in Fd q, and two vertices x, y in G is connected if ||x − y|| = t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6, we will see that any exponents 1 ≤ p1, p2, p3 ≤ ∞ with ∆(p1, p2, p3) ≲ 1 satisfy that Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that P2 can be considered as a subgraph of K3, and the operators Λ and ∆ are related to the graphs P2 and K3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, in view of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6, one may have a question that, “Compared to a graph G, does the operator associated with its subgraph yield less restricted mapping exponents?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, one may pose the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that G′ is a subgraph of the graph G with n vertices in Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ pi ≤ ∞, 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If ΛG(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) ≲ 1, is it true that ΛG′(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) ≲ 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Somewhat surprisingly, the answer turns out to be no!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When G = K3 and G′ = P2, the answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 is positive as Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, it turns out that there exist a graph G and its subgraph G′ yielding a negative answer, although the answers are positive for the most graphs which we consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For example, the answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 is negative when G is the C4 + diagonal and G′ is the C4 (see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6 Since the general answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 is not always positive, we pose the following natural question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Find general properties of the graph G and its subgraph G′ which yield a positive answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The main goal of this paper is to address a conjecture on this problem and to confirm it in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To precisely state our conjecture on the problem, let us review the standard definition and notation for the minimal degree of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The Minimum Degree of a graph G, denoted by δ(G), is defined as the degree of the vertex with the least number of edges incident to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We propose the following conjecture which can be a solution of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let G′ be a subgraph of the graph G in Fd q, d ≥ 2, with n vertices, and let 1 ≤ pi ≤ ∞, 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In addition, assume that min {δ(G), d} > δ(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) Then if ΛG(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) ≲ 1, we have ΛG′(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , pn) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Note that the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is equivalent to the following: (i) δ(G) > δ(G′) and (ii) d > δ(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) We have some comments and further questions below, regarding the above conjecture and our main theorems which we will state and prove in the body of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Our results in this paper confirm Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, possibly up to endpoints, for all graphs G and their subgraphs G′ on n = 3, 4 vertices in F2 q (see Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Also note that when d = 2 and n = 3, 4, the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) is equivalent to the first condition (i) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) since δ(G) ≤ 2 (see the figures above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It will be shown that the conclusion of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 cannot be reversed at least for n = 3, 4 in two dimensions (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 for n = 3, and see Remarks 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='13, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='12, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6 for n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=') It is worth investigating whether the key hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 can be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The conclusion of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 does not hold in general if the > in the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) is replaced by ≥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To see this, consider G = C4+diagonal and G′ = C4 on 4 vertices in Fd q, with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is obvious that G′ is a subgraph of G and min{δ(G), d} = δ(G′) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 (ii) implies that the conclusion of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We are not sure what can we say about the conclusion of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 if the main hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is relaxed by the second one of the conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To be precise, when (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) is replaced by the second statement of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7), that is, d > δ(G′), we do not have 7 a definitive answer even for n = 4 in F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For instance, let G = Y -shape and G′ = K3+ a tail on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then it is clear that d = 2 > δ(G′) = 1 and 1 = δ(G) = δ(G′) and so this provides an example that does not satisfy the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 but satisfy the second statement of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Unfortunately, in this paper we have not found any inclusive bounedness relations between the operators corresponding to such graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In order to exclude this uncertain case, both conditions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) were taken as the hypothesis for Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, namely the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The rest of this paper is organized as follows: In Section 2, we recall known results on the spherical averaging operator, which functions as a fundamental tool to prove our theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Sections 3 to 10 are devoted to the presentation and proofs of our main results associated to the graphs mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The appendix contains some technical lemmas on the number of intersection points of two spheres in Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2 The spherical averaging problem In the finite field setting, Carbery, Stones, and Wiright [3] initially formulated and studied the averaging problem over the varieties defined by vector valued polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This problem for general varieties was studied by Chun-Yen Shen and the third listed author [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Here, we introduce the standard results on the averaging problem over the spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We adopt the notation in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let dx be the normalizing counting measure on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For each non-zero t, we endow the sphere St with the normalizing surface measure dσt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We recall that dσt(x) = qd |St|1St(x)dx so that we can identify the measure dσt with the function qd |St|1St on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The spherical averaging operator ASt is defined by AStf(x) = f ∗ dσt(x) = � St f(x − y)dσt(y) = 1 |St| � y∈St f(x − y), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) where f is a function on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By a change of variables, we also have AStf(x) = 1 |St| � y∈Fdq St(x − y)f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) For 1 ≤ p, r ≤ ∞, we define ASt(p → r) to be the smallest number such that the averaging estimate ||f ∗ dσt||Lr(Fdq,dx) ≤ ASt(p → r)||f||Lp(Fdq,dx) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) holds for all test functions f on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 8 Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 (Spherical averaging problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Determine all exponents 1 ≤ p, r ≤ ∞ such that ASt(p → r) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From now on, we simply write A for the spherical averaging operator ASt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By testing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) with f = δ0 and by using the duality of the averaging operator, it is not hard to notice that the necessary conditions for the boundedness of A(p → r) are as follows: (1/p, 1/r) is contained in the convex hull of points (0, 0), (0, 1), (1, 1), and ( d d+1, 1 d+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Using the Fourier decay estimate on St and its cardinality, it can be shown that these necessary conditions are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For the reader’s convenience, we give a detail proof although the argument is standard, as is well known in the literature such as [3, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p, r ≤ ∞ be numbers such that (1/p, 1/r) lies on the convex hull of points (0, 0), (0, 1), (1, 1), and ( d d+1, 1 d+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have A(p → r) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since both dσt and dx have total mass 1, it follows from Young’s inequality for convolution functions that if 1 ≤ r ≤ p ≤ ∞, then ∥f ∗ dσt∥Lr(Fdq,dx) ≤ ∥f∥Lp(Fdq,dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) We notice that these results do not hold for the Euclidean Averaging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By the interpolation and the duality, we only need to establish the following critical estimate: A �d + 1 d → d + 1 � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is well known that for non-zero t, |(dσt)∨(m)| := ����� 1 |St| � x∈St χ(m · x) ����� ≲ q− (d−1) 2 for all m ̸= (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , 0), where χ denotes a non-trivial additive character of Fq (see the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since |St| ∼ qd−1, we complete the proof by combining this Fourier decay estimate with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let dσ be the normalized surface measure on an variety S in Fd q with |S| ∼ qd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If |(dσ)∨(m)| ≲ q− k 2 for all m ∈ Fd q \\ (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' , 0) and for some k > 0, then we have A �k + 2 k + 1 → k + 2 � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 Define K = (dσ)∨ − δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We want to prove that for every function f on 9 (Fd q, dx), ∥f ∗ dσ∥Lk+2(Fdq,dx) ≲ ∥f∥ L k+2 k+1 (Fdq,dx), where dx is the normalized counting measure on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since dσ = �K + �δ0 = �K + 1 and ∥f ∗ 1∥Lk+2(Fdq,dx) ≲ ∥f∥ L k+2 k+1 (Fdq,dx), it suffices to show that for every f on (Fd q, dx), ∥f ∗ �K∥Lk+2(Fdq,dx) ≲ ∥f∥ L k+2 k+1 (Fdq,dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) Notice that this inequality can be obtained by interpolating the following two estimates: ∥f ∗ �K∥L2(Fdq,dx) ≲ q− k 2 ∥f∥L2(Fdq,dx), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) and ∥f ∗ �K∥L∞(Fdq,dx) ≲ q∥f∥L1(Fdq,dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) It remains to prove the above two estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) follows from the Plancherel theorem, the size assumption of |(dσ)∨|, and the definition of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' On the other hand, the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) follows from Young’s inequality and the observation that ∥ �K∥L∞(Fdq,dx) ≲ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The boundary points of the convex hull play an important role in the application of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, we will apply the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p, r ≤ ∞ and let A denote the averaging operator over the sphere St, t ̸= 0, in Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (i) If 1 ≤ 1 p ≤ d d+1 and 1 r = 1 dp, then A(p → r) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) If d d+1 ≤ 1 p ≤ 1 and 1 r = d p − d + 1, then A(p → r) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let L1 and L2 denote the line segments connecting the two points (0, 0), (d/(d+1), 1/(d+1)) and the two points (d/(d + 1), 1/(d + 1)), (1, 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 implies that for any point (1/p, 1/r) lying on either L1 or L2, we have A(p → r) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the first part of the lemma follows since L1 = {(1/p, 1/r) : 0 ≤ 1/p ≤ d/(d + 1), 1/r = 1/(dp)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Similarly, since L2 = {(1/p, 1/r) : d/(d + 1) ≤ 1/p ≤ 1, 1/r = d/p − d + 1}, the second part of the lemma is obtained, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 3 Sharp mapping properties for the K2-Operator In this section, we provide the sharp mapping properties of the operator associated to the graph K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To this end, as described below, we relate the problem to the spherical averaging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' As usual, the inner product of the non-negative real-valued functions f, g on Fd q is defined as < f, g >:= ||fg||1 = 1 qd � x∈Fdq f(x)g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 10 Let t ∈ F∗ q and let f1, f2 be non-negative real-valued functions on Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the operator L associated with the graph K2 is defined as L(f1, f2) = 1 qd|St| � x1,x2∈Fdq St(x1 − x2)f1(x1)f2(x2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) and it is called the K2-Operator on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Here, the quantity qd|St| represents the normalizing factor N(G) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) when the graph G is K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By a change of variables, we can write L(f1, f2) = 1 qd � x1∈Fdq f1(x1) \uf8eb \uf8ed 1 |St| � x2∈Fdq f2(x1 − x2)St(x2) \uf8f6 \uf8f8 =< f1, Af2 >, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) where A denotes the averaging operator related to the sphere St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Likewise, we also obtain that L(f1, f2) =< Af1, f2 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In a usual way, we define the operator norm of the K2-Operator L as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We define L(p1, p2) as the smallest constant such that the following estimate holds for all functions f1, f2 on Fd q : L(f1, f2) ≤ L(p1, p2)||f1||p1||f2||p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The main goal of this section is to address all numbers 1 ≤ p1, p2 ≤ ∞ satisfying L(p1, p2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We begin with the necessary conditions for the boundedness of the K2-Operator L on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that L(p1, p2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have 1 p1 + d p2 ≤ d and d p1 + 1 p2 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By symmetry, it is clear that L(p1, p2) ≲ 1 ⇐⇒ L(p2, p1) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, it suffices to prove the first listed conclusion that 1 p1 + d p2 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) and our assumption that L(p1, p2) ≲ 1, we must have L(f1, f2) := 1 qd � x1∈Fdq f1(x1) \uf8eb \uf8ed 1 |St| � x2∈Fdq f2(x1 − x2)St(x2) \uf8f6 \uf8f8 ≲ ||f1||p1||f2||p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We test this inequality with f1 = 1St and f2 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then L(f1, f2) = 1 qd � x1∈St 1 |St| = q−d, and ||f1||p1||f2||p2 ∼ (q−d|St|)1/p1(q−d)1/p2 ∼ q− 1 p1 − d p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 11 By a direct comparison, we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For 1 ≤ p1, p2 ≤ ∞, one can note that 1 p1 + d p2 ≤ d and d p1 + 1 p2 ≤ d if and only if (1/p1, 1/p2) ∈ [0, 1] × [0, 1] lies on the convex hull of points (0, 0), (0, 1), ( d d+1 , d d+1), (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let us move to the sufficient conditions on the exponents 1 ≤ p1, p2 ≤ ∞ such that L(p1, p2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We will invoke the following lemma which demonstrates that the boundedness of the K2-Operator L can be directly determined by the spherical averaging estimates over the finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For 1 ≤ p ≤ ∞, we denote by p′ the H¨older conjugate of p, namely, 1/p + 1/p′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that A(p2 → p′ 1) ≲ 1 with 1 ≤ p1, p2 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have L(p1, p2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since L(f1, f2) =< f1, Af2 >, it follows by H¨older’s inequality that L(f1, f2) ≤ ||f1||p1||Af||p′ 1 ≲ ||f1||p1||f2||p2, where the averaging assumption was used in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We now show that the necessary conditions are in fact sufficient conditions for L(p1, p2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (Sharp boundedness result for the K2-Operator on Fd q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have L(p1, p2) ≲ 1 if and only if 1 p1 + d p2 ≤ d, d p1 + 1 p2 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, it will be enough to prove that L(p1, p2) ≲ 1 for all 1 ≤ p1, p2 ≤ ∞ satisfying 1 p1 + d p2 ≤ d, d p1 + 1 p2 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By the interpolation theorem and the nesting property of the norm, it suffices to establish the estimates on the critical end-points (1/p1, 1/p2) ∈ [0, 1] × [0, 1], which are (0, 1), (1, 0), and (d/(d + 1), d/(d + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In other words, it remains to prove the following estimates: L(∞, 1) ≲ 1, L(1, ∞) ≲ 1, L �d + 1 d , d + 1 d � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4, matters are reduced to establishing the following averaging estimates: A(1 → 1) ≲ 1, A(∞ → ∞) ≲ 1, A �d + 1 d → d + 1 � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, these averaging estimates are clearly valid by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following result is a special case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, but it is very useful in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 12 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For any dimensions d ≥ 2, we have L � d+1 d , d+1 d � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that if p1 = p2 = d+1 d , then it satisfies that 1 p1 + d p2 ≤ d and d p1 + 1 p2 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the statement follows immediately from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 4 Boundedness problem for the K3-Operator Let t ∈ F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The operator ∆ related to the graph K3 can be defined as ∆(f1, f2, f3) = 1 qd 1 |St||Sd−2 t | � x1,x2,x3∈Fdq St(x1 − x2)St(x2 − x3)St(x3 − x1) 3 � i=1 fi(xi), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) where each fi, i = 1, 2, 3, is a non-negative real-valued function on Fd q, and the quantity qd|St||Sd−2 t | stands for the normalizing factor N(G) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) when G = K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We name the operator ∆ as the K3-Operator on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We define ∆(p1, p2, p3) as the best constant such that the following estimate holds for all non-negative real-valued functions fi, i = 1, 2, 3, on Fd q : ∆(f1, f2, f3) ≤ ∆(p1, p2, p3)||f1||p1||f2||p2||f3||p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The purpose of this section is to find the numbers 1 ≤ p1, p2, p3 ≤ ∞ such that ∆(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When the dimension d is two, we will settle this problem up to the endpoint estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To this end, we relate our problem to the estimate of the Bilinear Averaging Operator (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2)) for which we establish the sharp bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' On the other hand, as we shall see, in three and higher dimensions d ≥ 3, it is not easy to deduce the sharp results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, when one of the exponents p1, p2, p3 is ∞, we will able to obtain the optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This will be done by applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, the boundedness result for the K2-Operator L on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We begin by deducing necessary conditions for our problem in Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Recall that for d = 2, we pose an additional restriction that 3 ∈ Fq is a square number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 (Necessary conditions for the boundedness of ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that ∆(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have d p1 + 1 p2 + 1 p3 ≤ d, 1 p1 + d p2 + 1 p3 ≤ d, 1 p1 + 1 p2 + d p3 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In particular, when d = 2, it can be shown by Polymake1[1, 6] that (1/p1, 1/p2, 1/p3) is contained in the convex hull of the points: (0, 0, 1), (0, 1, 0), (2/3, 2/3, 0), (1/2, 1/2, 1/2), (2/3, 0, 2/3), (1, 0, 0), (0, 0, 0), (0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1Polymake is software for the algorithmic treatment of convex polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We only prove the first inequality in the conclusion since we can establish other inequalities by symmetric property of ∆(f1, f2, f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We will use the simple fact that x ∈ St if and only if −x ∈ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1), taking f1 = δ0, f2 = 1St, and f3 = 1St, we see that ||f1||p1||f2||p2|||f3||p3 ∼ q− d p1 q− 1 p2 q− 1 p3 , ∆(f1, f2, f3) = 1 qd 1 |St||Sd−2 t | \uf8eb \uf8ed � x2,x3∈St:||x2−x3||=t 1 \uf8f6 \uf8f8 ∼ q−d, where the last similarity above follows from Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 in Appendix with our assumption that 3 ∈ Fq is a square number for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By the direct comparison of these estimates, we obtain the required necessary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In order to prove that the necessary conditions in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 are sufficient con- ditions for d = 2, we only need to establish the following critical endpoint estimates: ∆(2, 2, 2) ≲ 1, ∆(∞, ∞, ∞) ≲ 1, ∆(1, ∞, ∞) ≲ 1, ∆(∞, 1, ∞) ≲ 1, ∆(∞, ∞, 1) ≲ 1, ∆ �3 2, 3 2, ∞ � ≲ 1, ∆ � 3 2, ∞, 3 2 � ≲ 1, ∆ � ∞, 3 2, 3 2 � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In fact, this claim follows by interpolating the critical points given in the second part of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 Boundedness results for ∆ on Fd q The graph K2 can be obtained by removing any one of three vertices in the graph K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Therefore, the boundedness of L(p1, p2) can determine the boundedness of ∆(p1, p2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ a, b ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If L(a, b) ≲ 1, then ∆(a, b, ∞) ≲ 1, ∆(a, ∞, b) ≲ 1, and ∆(∞, a, b) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By symmetry, to complete the proof, it suffices to prove that ∆(a, b, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Recall from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) that ∆(f1, f2, f3) = 1 qd 1 |St||Sd−2 t | � x1,x2,x3∈Fdq St(x1 − x2)St(x2 − x3)St(x3 − x1) 3 � i=1 fi(xi), and L(f1, f2) = 1 qd|St| � x1,x2∈Fdq St(x1 − x2)f1(x1)f2(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since fi, i = 1, 2, 3, are non-negative real-number functions on Fd q, we have ∆(f1, f2, f3) ≤ 1 |Sd−2 t | \uf8eb \uf8ed max x1,x2∈Fdq:||x1−x2||=t � x3∈Fdq:||x2−x3||=t=||x3−x1|| f3(x3) \uf8f6 \uf8f8 L(f1, f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 14 Since |Sd−2 t | ∼ qd−2 and L(f1, f2) ≲ ||f1||a||f2||b by our assumption, it suffices to prove that the maximum value in the above parenthesis is ≲ qd−2||f3||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let us denote by I the maximum above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By a change of variables, x = x1, y = x1 − x2, we see that I ≤ \uf8eb \uf8ed max x∈Fdq,y∈St � x3∈Fdq:||x−y−x3||=t=||x3−x|| 1 \uf8f6 \uf8f8 ||f3||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By another change of variables by putting z = x − x3, we get I ≤ \uf8eb \uf8ed max x∈Fdq,y∈St � z∈St:||z−y||=t 1 \uf8f6 \uf8f8 ||f3||∞ = \uf8eb \uf8edmax y∈St � z∈St:||z−y||=t 1 \uf8f6 \uf8f8 ||f3||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now applying Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 in Appendix, we conclude that I ≲ qd−2||f3||∞ as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When one of p1, p2, p3 is ∞, we are able to obtain sharp boundedness results for ∆(p1, p2, p3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ a, b ≤ ∞ satisfy that 1 a+ d b ≤ d and d a+ 1 b ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ∆(a, b, ∞) ≲ 1, ∆(a, ∞, b) ≲ 1 and ∆(∞, a, b) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The statement follows immediately by combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is not hard to see from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 that the above theorem cannot be improved in the case when one of the exponents p1, p2, p3 is ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In particular, we have the following critical endpoint estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We have ∆ � d+1 d , d+1 d , ∞ � ≲ 1, ∆ � d+1 d , ∞, d+1 d � ≲ 1, and ∆ � ∞, d+1 d , d+1 d � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Put a = b = d+1 d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we see that 1 a + d b = d = d a + 1 b, Hence, the statement of the corollary follows directly from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 also implies the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We have ∆(∞, ∞, ∞) ≲ 1, ∆(1, ∞, ∞) ≲ 1, ∆(∞, 1, ∞) ≲ 1, ∆(∞, ∞, 1) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By taking a = b = ∞ in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, we obtain the estimate that ∆(∞, ∞, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now, by symmetry, it will be enough to show that ∆(1, ∞, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, this is easily shown by taking a = 1, b = ∞ in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Thus, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7, and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3, we see that to completely solve the problem on the boundedness of ∆(p1, p2, p3) for d = 2, we only need to establish the following critical endpoint estimate ∆ �d + 2 d , d + 2 d , d + 2 d � = ∆(2, 2, 2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 Sharp restricted strong-type estimates in two dimensions Although Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is valid for all dimensions d ≥ 2, it is not sharp, compared to the necessary conditions given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this subsection, we will deduce the sharp boundedness results up to the endpoints for ∆(p1, p2, p3) in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To this end, we need the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ∆ be the K3-Operator on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, for all subsets E, F, H of F2 q, the following estimate holds: ∆(E, F, H) ≲ ||E||2||F||2||H||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For 1 ≤ p1, p2, p3 ≤ ∞, we say that the restricted strong-type ∆(p1, p2, p3) estimate holds if the estimate ∆(E, F, H) ≲ ||E||p1||F||p2||H||p3 is valid for all subsets E, F, H of F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this case, we write ∆(p1, p2, p3) ⪅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The proof proceeds with some reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When d = 2, by a change of variables by letting x = x3, y = x3 − x1, z = x3 − x2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) becomes ∆(f1, f2, f3) = 1 q2 � x∈F2q f3(x) \uf8ee \uf8f0 1 |St| � y,z∈F2q St(z − y)St(z)St(y)f1(x − y)f2(x − z) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We define B(f1, f2)(x) as the value in the bracket above, namely, B(f1, f2)(x) := 1 |St| � y,z∈St:||z−y||=t f1(x − y)f2(x − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) We refer to this operator B as “the bilinear averaging operator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is clear that ∆(f1, f2, f3) = 1 q2 � x∈F2q B(f1, f2)(x)f3(x) =< B(f1, f2), f3 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By H¨older’s inequality, we have ∆(f1, f2, f3) ≤ ||B(f1, f2)||2||f3||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Thus, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9 follows immediately from the reduction lemma below Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let B(f1, f2) be the bilinear averaging operator defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, for all subsets E, F of F2 q, we have ||B(E, F)||2 ≲ ||E||2||F||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We begin by representing the bilinear averaging operator B(f1, f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2), note that B(f1, f2)(x) = 1 |St| � y∈St f1(x − y) \uf8eb \uf8ed � z∈St:||z−y||=t f2(x − z) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 16 For each y ∈ St, let Θ(y) := {z ∈ St : ||z − y|| = t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' With this notation, the bilinear averaging operator is written as B(f1, f2)(x) = 1 |St| � y∈St f1(x − y) \uf8eb \uf8ed � z∈Θ(y) f2(x − z) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let η denote the quadratic character of F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Recall that η(s) = 1 for a square number s in F∗ q, and η(s) = −1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice from Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 in Appendix that Θ(y) is the empty set for all y ∈ St if d = 2 and η(3) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this case, the problem is trivial since B(f1, f2)(x) = 0 for all x ∈ F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Therefore, when d = 2, we always assume that η(3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that |Θ(y)| = 2 for all y ∈ St, which follows from the last statement of Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, for each y ∈ St, we can write Θ(y) = {θy, θ−1y}, where θy denotes the rotation of y by “60 degrees,” and ||y − θy|| = t = ||y − θ−1y||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From these observations, the bilinear averaging operator B(f1, f2) can be represented as follows: B(f1, f2)(x) = Bθ(f1, f2)(x) + Bθ−1(f1, f2)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) Here, we define Bθ(f1, f2)(x) := 1 |St| � y∈St f1(x − y)f2(x − θy), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) and Bθ−1(f1, f2) := 1 |St| � y∈St f1(x − y)f2(x − θ−1y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In order to complete the proof of the lemma, it suffices to establish the following two estimates: for all subsets E, F of Fd q, ||Bθ(E, F)||2 ≲ ||E||2||F||2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) and ||Bθ−1(E, F)||2 ≲ ||E||2||F||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) We will only provide the proof of the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) since the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now we start proving the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since ||E||2 2 = q−2|E| and ||F||2 2 = q−2|F|, it is enough to prove that ||Bθ(E, F)||2 2 ≲ q−4|E||F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) Without loss of generality, we may assume that |E| ≤ |F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By the definition, it follows that ||Bθ(E, F)||2 2 = q−2|St|−2 � x∈F2q � y,y′∈St E(x − y)E(x − y′)F(x − θy)F(x − θy′) = I + II, 17 where the first term I is the value corresponding to the case where y = y′, whereas the second term II is corresponding to the case where y ̸= y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We have I = q−2|St|−2 � y∈St � x∈F2q E(x − y)F(x − θy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Applying a change of variables by replacing x with x + y, we see that I = q−2|St|−2 � y∈St � x∈F2q E(x)F(x + y − θy) = q−2|St|−2 � x∈E \uf8eb \uf8ed� y∈St F(x + y − θy) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Observe that y − θy ̸= y′ − θy′ for all y, y′ in St with y ̸= y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we see that the value in the parentheses above is bounded above by |St ∩ F| ≤ |F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Therefore, we obtain the desired estimate: I ≤ q−2|St|−2|E||F| ∼ q−4|E||F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Next, it remains to show that II ≲ q−4|E||F|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since we have assumed that |E| ≤ |F|, it suffices to show that II ≲ q−4|E|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By the definition of II, it follows that II = q−2|St|−2 � x∈F2q � y,y′∈St:y̸=y′ E(x − y)E(x − y′)F(x − θy)F(x − θy′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is obvious that II ≤ q−2|St|−2 � y,y′∈St:y̸=y′ � x∈F2q E(x − y)E(x − y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We use a change of variables by replacing x with x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have II ≤ q−2|St|−2 � x∈F2q E(x) \uf8eb \uf8ed � y,y′∈St:y̸=y′ E(x + y − y′) \uf8f6 \uf8f8 = q−2|St|−2 � x∈F2q E(x) \uf8eb \uf8ed � 0̸=u∈F2q E(x + u)W(u) \uf8f6 \uf8f8 , where W(u) denotes the number of pairs (y, y′) ∈ St × St such that u = y − y′ and y ̸= y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is not hard to see that for any non-zero vector u ∈ F2 q, we have W(u) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' So we obtain that II ≲ q−2|St|−2 � x∈F2q E(x) \uf8eb \uf8ed � 0̸=u∈F2q E(x + u) \uf8f6 \uf8f8 ≲ q−4|E|2, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In two dimensions, we are able to obtain the optimal boundedness of ∆(p1, p2, p3) except for one endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Indeed, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 18 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3 ≤ ∞ and let ∆ be the K3-Operator on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (i) If ∆(p1, p2, p3) ≲ 1, then 2 p1 + 1 p2 + 1 p3 ≤ 2, 1 p1 + 2 p2 + 1 p3 ≤ 2, 1 p1 + 1 p2 + 2 p3 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8) (ii) Conversely, if (p1, p2, p3) satisfies all three inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8), then ∆(p1, p2, p3) ≲ 1 for (p1, p2, p3) ̸= (2, 2, 2), and we have ∆(2, 2, 2) ⪅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The first part of the theorem is the special case of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now we prove the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' As stated in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, one can notice by using Polymake [1, 6] that all the points � 1 p1, 1 p2, 1 p3 � ∈ [0, 1]3 satisfying all three inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8) are contained in the convex hull of the critical points (1/2, 1/2, 1/2), (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), �2 3, 2 3, 0 � , �2 3, 0, 2 3 � , � 0, 2 3, 2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9 with d = 2 that the restricted strong-type estimate for the operator ∆ holds for the point (1/p1, 1/p2, 1/p3) = (1/2, 1/2, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In addition, notice from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 with d = 2 that ∆(p1, p2, p3) ≲ 1 for the above critical points (1/p1, 1/p2, 1/p3) except for (1/2, 1/2, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the statement of the second part follows immediately by invoking the interpolation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 5 Boundedness results for the P2-Operator For t ∈ F∗ q and functions fi, i = 1, 2, 3, on Fd q, the operator Λ associated with the graph P2, called the P2-Operator on Fd q, is defined as Λ(f1, f2, f3) = 1 qd|St|2 � x1,x2,x3∈Fdq St(x1 − x2)St(x2 − x3)f1(x1)f2(x2)f3(x3), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) where the quantity qd|St|2 stands for the normalizing factor N(G) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) when G = P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Note that this can be written as Λ(f1, f2, f3) =< f2, Af1 · Af3 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We define Λ(p1, p2, p3) as the smallest constant such that the following estimate holds for all non-negative real-valued functions f1, f2, f3 on Fd q : Λ(f1, f2, f3) ≤ Λ(p1, p2, p3)||f1||p1||f2||p2||f3||p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this section, we study the problem determining all numbers 1 ≤ p1, p2, p3 ≤ ∞ satisfying Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Compared to the K3-Operator ∆, this problem is much hard to find the optimal answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Based on the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) with the averaging estimates in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, we are able to 19 address partial results on this problem (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 (Necessary conditions for the boundedness of Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have 1 p1 + d p2 + 1 p3 ≤ d, d p1 + 1 p2 ≤ d, 1 p2 + d p3 ≤ d, d p1 + 1 p2 + d p3 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Also, under this assumption when d = 2, it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3) is contained in the convex hull of points: (0, 1, 0), (1/2, 0, 1), (1, 0, 1/2),(1, 0, 0), (5/6, 1/3, 1/2), (1/2, 1/3, 5/6), (2/3, 2/3, 0), (0, 2/3, 2/3), (0, 0, 0), (0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, for all functions fi, i = 1, 2, 3, on Fd q, we have Λ(f1, f2, f3) = 1 qd|St|2 � x1,x2,x3∈Fdq St(x1 − x2)St(x2 − x3)f1(x1)f2(x2)f3(x3) ≲ ||f1||p1||f2||p2||f3||p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We test the above inequality with f2 = δ0, f1 = f3 = 1St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is plain to note that ||f1||p1||f2||p2||f3||p3 = (q−d|St|) 1 p1 q− d p2 (q−d|St|) 1 p3 ∼ q− 1 p1 − d p2 − 1 p3 , and Λ(f1, f2, f3) = q−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Therefore, we obtain that q−d ≲ q− 1 p1 − d p2 − 1 p3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This implies the first inequality in the conclusion that 1 p1 + d p2 + 1 p3 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To obtain the second inequality in the conclusion, we choose f1 = δ0, f2 = 1St, and f3 = 1Fdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then it is easy to check that ||f1||p1||f2||p2||f3||p3 ∼ q− d p1 − 1 p2 , and Λ(f1, f2, f3) = 1 qd|St|2 � x2∈St � x3∈Fdq St(x2 − x3) = q−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Comparing these estimates gives the second inequality in the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The third inequality in the conclusion can be easily obtained by switching the roles of f1 and f3 in the proof of the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To deduce the last inequality in the conclusion, we take f1 = f3 = δ0 and f2 = 1St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then ||f1||p1||f2||p2||f3||p3 ∼ q− d p1 − 1 p2 − d p3 , and Λ(f1, f2, f3) = 1 qd|St| ∼ q−2d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From these, we have the required result that d p1 + 1 p2 + d p3 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 20 By symmetry, it is not hard to note that Λ(p1, p2, p3) ≲ 1 ⇐⇒ Λ(p3, p2, p1) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In the following lemma, we prove that the boundedness question for the P2-Operator Λ is closely related to the spherical averaging problem over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that 1 r1 + 1 p2 + 1 r3 = 1, A(p1 → r1) ≲ 1, and A(p3 → r3) ≲ 1 for some 1 ≤ p1, p2, p3, r1, r3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since Λ(f1, f2, f3) =< f2, Af1 · Af3 >, we obtain by H¨older’s inequality with the first assumption that Λ(f1, f2, f3) ≤ ||Af1||r1||f2||p2||Af3||r3 ≲ ||f1||p1||f2||p2||f3||p3, where the averaging assumption was used for the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, Λ(p1, p2, p3) ≲ 1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now we state and prove our boundedness results of Λ(p1, p2, p3) on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, for the P2-Operator Λ on Fd q, the following four statements hold: (i) If 0 ≤ 1 p1, 1 p3 ≤ d d+1 and 1 p1 + d p2 + 1 p3 ≤ d, then Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) If 0 ≤ 1 p1 ≤ d d+1 ≤ 1 p3 ≤ 1 and 1 dp1 + 1 p2 + d p3 ≤ d, then Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iii) If 0 ≤ 1 p3 ≤ d d+1 ≤ 1 p1 ≤ 1 and d p1 + 1 p2 + 1 dp3 ≤ d, then Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iv) If d d+1 ≤ 1 p1, 1 p3 ≤ 1 and d p1 + 1 p2 + d p3 ≤ 2d − 1, then Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (i) By the nesting property of the norm, it suffices to prove it in the case when 0 ≤ 1 p1, 1 p3 ≤ d d+1 and 1 p1 + d p2 + 1 p3 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This equation can be rewritten as 1 dp1 + 1 p2 + 1 dp3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since 0 ≤ 1 p1, 1 p3 ≤ d d+1, we see from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (i) that letting 1 r1 = 1 dp1 , 1 r3 = 1 dp3, we have A(p1 → r1) ≲ 1 and A(p3 → r3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since 1 r1 + 1 p2 + 1 r3 = 1, applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 gives the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) As in the proof of the first part of the theorem, it will be enough to prove Λ(p1, p2, p3) ≲ 1 in the case when 0 ≤ 1 p1 ≤ d d+1 ≤ 1 p3 ≤ 1 and 1 dp1 + 1 p2 + d p3 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 r1 = 1 dp1 and 1 r3 = d p3 −d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 it follows that A(p1 → r1) ≲ 1 and A(p3 → r3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Also notice that 1 r1 + 1 p2 + 1 r3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 (ii) follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iii) Switching the roles of p1, p2, the proof is exactly the same as that of the second part of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iv) As before, it suffices to prove the case when d d+1 ≤ 1 p1, 1 p3 ≤ 1 and d p1 + 1 p2 + d p3 = 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Put 1 rk = d pk − d + 1 for k = 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we see from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (ii) that A(pk → rk) ≲ 1 for k = 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that 1 r1 + 1 p2 + 1 r3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Therefore, using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 we finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 21 As a special case of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For any dimensions d ≥ 2, we have Λ � d+1 d , d+1 d−1, d+1 d � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This clearly follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 by taking p1 = p3 = d+1 d , and p2 = d+1 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' While we do not know whether Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 is optimal or not, the result will play a crucial role in proving the following theorem which implies that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is true for the graph K3 and its subgraph P2 in all dimensions d ≥ 2 (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ∆ and Λ be the operators associated with K3 and P2, respectively, on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then if ∆(p1, p2, p3) ≲ 1 for 1 ≤ p1, p2, p3 ≤ ∞, we have Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that ∆(p1, p2, p3) ≲ 1 for 1 ≤ p1, p2, p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, the exponents p1, p2, p3 satisfy the following three inequalities: d p1 + 1 p2 + 1 p3 ≤ d, 1 p1 + d p2 + 1 p3 ≤ d, 1 p1 + 1 p2 + d p3 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) To complete the proof, it remains to show that Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We will prove this by considering the four cases depending on the sizes of p1 and p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Case 1: Suppose that 0 ≤ 1 p1, 1 p3 ≤ d d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) clearly implies that 1 p1 + d p2 + 1 p3 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Thus, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 (i), we obtain the required conclusion that Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Case 2: Suppose that 0 ≤ 1 p1 ≤ d d+1 ≤ 1 p3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 (ii), to prove that Λ(p1, p2, p3) ≲ 1, it will be enough to show that 1 dp1 + 1 p2 + d p3 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, this inequality clearly follows from the third inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) since d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Case 3: Suppose that 0 ≤ 1 p3 ≤ d d+1 ≤ 1 p1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 (iii), it suffices to show that d p1 + 1 p2 + 1 dp3 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, this inequality can be easily obtained from the first inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Case 4: Suppose that d d+1 ≤ 1 p1, 1 p3 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 (iv), to show that Λ(p1, p2, p3) ≲ 1, we only need to prove that d p1 + 1 p2 + d p3 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, this inequality can be easily proven as follows: d p1 + 1 p2 + d p3 = � d p1 + 1 p2 + 1 p3 � + d − 1 p3 ≤ d + d − 1 p3 ≤ 2d − 1, where the first inequality follows from the first inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3), and the last inequality follows from a simple fact that 1 ≤ p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 22 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The reverse statement of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6 cannot be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Indeed, we know by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 that Λ � d+1 d , d+1 d−1, d+1 d � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, ∆ � d+1 d , d+1 d−1, d+1 d � cannot be bounded, which can be easily shown by considering Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, namely, the necessary conditions for the boundedness of ∆(p1, p2, p3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We invoke Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6 to deduce the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is true for the graph K3 and its subgraph P2 in Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is clear that P2 is a subgraph of K3 in Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Sine δ(K3) = 2, d ≥ 2, and δ(P2) = 1, we have min{δ(K3), d} = 2 > δ(P2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, all assumptions of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 are satisfied for K3 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the statement of the corollary follows immediately from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6 Mapping properties for the (C4 + t)-Operator We investigate the mapping properties of the operator associated with the graph C4 + diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Throughout the remaining sections, we assume that t is a non-zero element in F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let fi, 1 ≤ i ≤ 4, be non-negative real-valued functions on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The operator ♦t is associated with the graph C4 + diagonal t (Figure 1d), and we define ♦t(f1, f2, f3, f4) as the quantity 1 qd|St||Sd−2 t |2 � x1,x2,x3,x4∈Fdq St(x1 −x2)St(x2 −x3)St(x3 −x4)St(x4 −x1)St(x1 −x3) 4 � i=1 fi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) The operator ♦t is referred to as the (C4 + t)-Operator on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Here, notice that we take the quantity qd|St||Sd−2 t |2 as the normalizing factor N(G) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Applying a change of variables by letting x = x1, u = x1 − x2, v = x1 − x3, w = x1 − x4, we see that ♦t(f1, f2, f3, f4) = 1 qd � x∈Fdq f1(x)T(f2, f3, f4)(x) =< f1, T(f2, f3, f4) >, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) where the operator T(f2, f3, f4) is defined by T(f2, f3, f4)(x) := 1 |St||Sd−2 t |2 � u,v,w∈St St(v − u)St(w − v)f2(x − u)f3(x − v)f4(x − w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We define ♦t(p1, p2, p3, p4) to be the smallest constant such that the following estimate holds for all non-negative real-valued functions fi, 1 ≤ i ≤ 4, on Fd q : ♦t(f1, f2, f3, f4) ≤ ♦t(p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We are asked to find 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ♦t(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) 23 holds for all non-negative real-valued functions fi, 1 ≤ i ≤ 4, on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In other words, our main problem is to determine all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ♦t(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 (Necessary conditions for the boundedness of ♦t(p1, p2, p3, p4) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ♦t be the (C4+t)- Operator on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If ♦t(p1, p2, p3, p4) ≲ 1, then we have 1 p1 + 1 p2 + d p3 + 1 p4 ≤ d, d p1 + 1 p2 + 1 p3 + 1 p4 ≤ d, and 1 p1 + d p2 + 1 p3 + d p4 ≤ 2d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Also, under this assumption when d = 2, it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3, 1/p4) is contained in the convex hull of the points (0, 0, 1, 0), (0, 1, 0, 0), (0, 0, 0, 1), (1/2, 0, 1/2, 1/2), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), (2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We prove the first conclusion that 1 p1 + 1 p2 + d p3 + 1 p4 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) We notice from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) that ♦t(f1, f2, f3, f4) becomes 1 qd � x∈Fdq f1(x) \uf8ee \uf8f0 1 |St||Sd−2 t |2 � v∈St f3(x − v) \uf8eb \uf8ed � u,w∈St St(v − u)St(w − v)f2(x − u)f4(x − w) \uf8f6 \uf8f8 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Taking f1 = f2 = f4 = 1St, and f3 = δ0, we see that ||f1||p1||f2||p2|||f3||p3||f4||p4 ∼ q− 1 p1 q− 1 p2 q− d p3 q− 1 p4 and ♦t(f1, f2, f3, f4) = 1 qd � x∈St 1 |St||Sd−2 t |2 \uf8eb \uf8ed � u∈St:||x−u||=t 1 \uf8f6 \uf8f8 \uf8eb \uf8ed � w∈St:||x−w||=t 1 \uf8f6 \uf8f8 ∼ q−d, which yields the required necessary condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) , where the last similarity above follows from Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The second conclusion follows by symmetry from the first conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To prove the third conclusion that is 1 p1 + d p2 + 1 p3 + d p4 ≤ 2d − 2, we test the inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) with f1 = f3 = 1St and f2 = f4 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ||f1||p1||f2||p2||f3||p3||f4||p4 = �|St| qd �1/p1 � 1 qd �1/p2 �|St| qd �1/p3 � 1 qd �1/p4 ∼ q− 1 p1 − d p2 − 1 p3 − d p4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 24 On the other hand, taking f1 = f3 = 1St and f2 = f4 = δ0 in the definition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1), we see that ♦t(f1, f2, f3, f4) = 1 qd|St||Sd−2 t |2 � x1∈St \uf8eb \uf8ed � x3∈St St(x1 − x3) \uf8f6 \uf8f8 ≲ 1 qd|Sd−2 t | ∼ q−2d+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) we must have the required third conclusion that 1 p1 + d p2 + 1 p3 + d p4 ≤ 2d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 Boundedness results for ♦t on Fd q Given a rhombus with a fixed diagonal(the graph C4 + diagonal), we will show that by removing the vertex x2 or the vertex x4, ♦t(f1, f2, f3, f4) ≲ ||f2||∞∆(f1, f3, f4) and ♦t(f1, f2, f3, f4) ≲ ||f4||∞∆(f1, f2, f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, upper bounds of ♦t(p1, ∞, p3, p4) and ♦t(p1, p2, p3, ∞) can be controlled by upper bounds of the ∆(p1, p3, p4) and ∆(p1, p2, p3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, we have the following relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that ∆(p, s, r) ≲ 1 for 1 ≤ p, s, r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ♦t(p, ∞, s, r) ≲ 1 and ♦t(p, s, r, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since ∆(p, s, r) ≲ 1 for 1 ≤ p, s, r ≤ ∞, we see that for all non-negative functions f, g, h on Fd q, ∆(f, g, h) ≲ ||f||p||g||s||h||r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Thus, to complete the proof, it will be enough to establish the following estimates: for all non-negative functions fi, i = 1, 2, 3, 4, ♦t(f1, f2, f3, f4) ≲ ||f2||∞∆(f1, f3, f4) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) and ♦t(f1, f2, f3, f4) ≲ ||f4||∞∆(f1, f2, f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) Since the proofs of both (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) are the same, we only provide the proof of the estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice by the definition of ♦t(f1, f2, f3, f4) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) that ♦t(f1, f2, f3, f4) can be written as the form 1 qd|St||Sd−2 t | � x1,x2,x3∈Fd q :||x1−x2||=||x2−x3||=||x1−x3||=t � 3 � i=1 fi(xi) � \uf8ee \uf8f0 1 |Sd−2 t | � x4∈Fdq St(x3 − x4)St(x4 − x1)f4(x4) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For each x1, x3 ∈ Fd q with ||x1 − x3|| = t, we define M(x1, x3) as the value in the above bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, recalling the definition of ∆(f1, f2, f3) given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1), we see that ♦t(f1, f2, f3, f4) ≤ \uf8eb \uf8ec \uf8ec \uf8ed max x1,x3∈Fd q :||x1−x3||=t M(x1, x3) \uf8f6 \uf8f7 \uf8f7 \uf8f8 ∆(f1, f2, f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 25 Hence, the estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) follows immediately by proving the following claim: M := max x1,x3∈Fd q :||x1−x3||=t 1 |Sd−2 t | � x4∈Fdq St(x3 − x4)St(x4 − x1) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8) To prove this claim, we first apply a change of variables by letting x = x1, y = x1 − x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then it follows that M = max x∈Fdq,y∈St 1 |Sd−2 t | � x4∈Fdq St(x − y − x4)St(x4 − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Letting z = x − x4, we have M = max x∈Fdq,y∈St 1 |Sd−2 t | � z∈St:||z−y||=t 1 ∼ 1 qd−2 max y∈St � z∈St:||z−y||=t 1 By Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 in Appendix, we conclude that M ≲ 1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In arbitrary dimensions d ≥ 2, we have the following consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that 1 ≤ a, b ≤ ∞ satisfy that 1 a + d b ≤ d and d a + 1 b ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9) Namely, let (1/a, 1/b) be contained in the convex hull of points (0, 0), (0, 1), (d/(d + 1), d/(d + 1)), (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ♦t(a, ∞, b, ∞) ≲ 1, ♦t(a, b, ∞, ∞) ≲ 1, ♦t(a, ∞, ∞, b) ≲ 1, ♦t(∞, ∞, a, b) ≲ 1, ♦t(∞, a, b, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, we know that the assumption (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9) implies that ∆(a, b, ∞) ≲ 1, ∆(a, ∞, b) ≲ 1 and ∆(∞, a, b) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the statement of the theorem follows immediately by combining these and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 Sharp boundedness results up to endpoints for ♦t on F2 q In this subsection, we collect our boundedness results for the operator ♦t in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ♦t be the (C4 + t)-Operator on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (i) Suppose that (p1, p2, p3) ̸= (2, 2, 2) satisfies the following equations: 2 p1 + 1 p2 + 1 p3 ≤ 2, 1 p1 + 2 p2 + 1 p3 ≤ 2, 1 p1 + 1 p2 + 2 p3 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ♦t(p1, ∞, p2, p3) ≲ 1 and ♦t(p1, p2, p3, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) In addition, we have ♦t(2, ∞, 2, 2) ⪅ 1 and ♦t(2, 2, 2, ∞) ⪅ 1, where ⪅ is used to denote that the boundedness of ♦t holds for all indicator test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 26 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 still holds after replacing ≲ by ⪅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the statement of the theorem is directly obtained by combining Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 guarantees the sharp boundedness for the operator ⋄t up to endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Indeed, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ♦t be the (C4+t)-Operator on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The necessary conditions for ♦t(p1, p2, p3, p4) ≲ 1 given in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 are sufficient except for the two points (p1, p2, p3, p4) = (2, 2, 2, ∞), (2, ∞, 2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In addition, we have ♦t(2, ∞, 2, 2) ⪅ 1 and ♦t(2, 2, 2, ∞) ⪅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The statement (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10) was already proven in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, using the interpolation theorem and the second part of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, the matter is reducing to proving ♦t(p1, p2, p3, p4) ≲ 1 for the critical endpoints (1/p1, 1/p2, 1/p3, 1/p4) including all the following points: (0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (2/3, 2/3, 0, 0), (2/3, 0, 2/3, 0), (2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In other words, the proof will be complete by proving the following estimates: ♦t(∞, ∞, ∞, ∞) ≲ 1, ♦t(1, ∞, ∞, ∞) ≲ 1, ♦t(∞, 1, ∞, ∞) ≲ 1, ♦t(∞, ∞, 1, ∞) ≲ 1, ♦t(∞, ∞, ∞, 1) ≲ 1, ♦t(3/2, 3/2, ∞, ∞) ≲ 1, ♦t(3/2, ∞, 3/2, ∞) ≲ 1, ♦t(3/2, ∞, ∞, 3/2) ≲ 1, ♦t(∞, 3/2, 3/2, ∞) ≲ 1, ♦t(∞, ∞, 3/2, 3/2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, by a direct computation, these estimates follow immediately from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 7 Boundedness problem for the C4-Operator Let t ∈ F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Given non-negative real-valued functions fi, 1 ≤ i ≤ 4, on Fd q, we define ✸(f1, f2, f3, f4) to be the following value: 1 qd|St|2|Sd−2 t | � x1,x2,x3,x4∈Fdq St(x1 − x2)St(x2 − x3)St(x3 − x4)St(x4 − x1) 4 � i=1 fi(xi), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) where the quantity qd|St|2|Sd−2 t | stands for the normalizing factor N(G) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) when G = C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since the operator ✸ is associated with the graph C4, it is named as the C4-Operator on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We denote by ✸(p1, p2, p3, p4) the smallest constant such that the following estimate holds for all non-negative real-valued functions fi, 1 ≤ i ≤ 4, on Fd q : ✸(f1, f2, f3, f4) ≤ ✸(p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Main problem is to find all exponents 1 ≤ p1, p2, p3, p4 ≤ ∞ such that the inequality ✸(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) 27 holds for all non-negative real-valued functions fi, 1 ≤ i ≤ 4, on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In other words, our main problem is to determine all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ✸(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 (Necessary conditions for the boundedness of ✸(p1, p2, p3, p4) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) holds, namely ✸(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have 1 p1 + 1 p2 + 1 p3 + d p4 ≤ d + 1, 1 p1 + 1 p2 + d p3 + 1 p4 ≤ d + 1, 1 p1 + d p2 + 1 p3 + 1 p4 ≤ d + 1, d p1 + 1 p2 + 1 p3 + 1 p4 ≤ d + 1, d p1 + 1 p2 + d p3 + 1 p4 ≤ 2d − 2, and 1 p1 + d p2 + 1 p3 + d p4 ≤ 2d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In particular, when d = 2, it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3, 1/p4) is con- tained in the convex hull of the points (0, 0, 1, 0), (0, 0, 0, 1), (0, 1, 0, 0), (2/3, 0, 0, 2/3), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (0, 0, 0, 0), (0, 2/3, 2/3, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When d = 2, 3, the first four inequalities in the conclusion are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We only need the last two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The 6 inequalities in the conclusion can be easily deduced by testing the inequality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) with the following specific functions, respectively: We leave the proofs to the readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1) f1 = f2 = f3 = 1St, and f4 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2) f1 = f2 = f4 = 1St, and f3 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 3) f1 = f3 = f4 = 1St, and f2 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 4) f2 = f3 = f4 = 1St, and f1 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 5) f2 = f4 = 1St, and f1 = f3 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6) f1 = f3 = 1St, and f2 = f4 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 Boundedness results for the C4-Operator ✸ on Fd q In this subsection, we provide some exponents 1 ≤ pi ≤ ∞, 1 ≤ i ≤ 4, such that ✸(p1, p2, p3, p4) ≲ 1 in the specific case when one of pi is ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, our result will correspond to all dimensions d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In general, it is very hard to deduce non-trivial boundedness results for the C4-Operator on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We begin by observing that an upper bound of ✸(f1, f2, f3, f4) can be controlled by estimating for both the K2-Operator L and the P2-Operator Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For all non-negative functions fi, i = 1, 2, 3, 4, on Fd q, d ≥ 2, we have ✸(f1, f2, f3, f4) ≲ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � 1 |Sd−2 t |L(f1f3, f2) + Λ(f1, f2, f3) � ||f4||∞, � 1 |Sd−2 t |L(f2f4, f1) + Λ(f4, f1, f2) � ||f3||∞, � 1 |Sd−2 t |L(f1f3, f4) + Λ(f3, f4, f1) � ||f2||∞, � 1 |Sd−2 t |L(f2f4, f3) + Λ(f2, f3, f4) � ||f1||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 28 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We only provide the proof of the first inequality, ✸(f1, f2, f3, f4) ≲ 1 |Sd−2 t | L(f1f3, f2)||f4||∞ + Λ(f1, f2, f3)||f4||∞, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) since other inequalities can be easily proven in the same way by replacing the role of f4 with f3, f2, f1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By definition, the value of ✸(f1, f2, f3, f4) is equal to 1 qd|St|2|Sd−2 t | � x1,x2,x3∈Fdq St(x1−x2)St(x2−x3) � 3 � i=1 fi(xi) � \uf8eb \uf8ed � x4∈Fdq St(x3 − x4)St(x4 − x1)f4(x4) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For fixed x1, x3 ∈ Fd q, the sum in the above bracket can be estimated as follows: � x4∈Fdq St(x3 − x4)St(x4 − x1)f4(x4) ≲ � |St|||f4||∞ if x1 = x3, qd−2||f4||∞ if x1 ̸= x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that this estimates are easily obtained by invoking Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 in Appendix after using a change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✸(f1, f2, f3, f4) = ✸1+✸2, where ✸1 denotes the contribution to ✸(f1, f2, f3, f4) when x1 = x3, and ✸2 does it when x1 ̸= x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then it follows that ✸1 ≲ ||f4||∞ qd|St||Sd−2 t | � x1,x2∈Fdq St(x1 − x2)(f1f3)(x1)f2(x2) = 1 |Sd−2 t | L(f1f3, f2)||f4||∞, ✸2 ≲ ||f4||∞ qd|St|2 � x1,x2,x3∈Fdq:x1̸=x3 St(x1 − x2)St(x2 − x3)f1(x1)f2(x2)f3(x3) ≲ Λ(f1, f2, f3)||f4||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, we obtain the required estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4, we obtained four different kinds of the upper bounds of the ✸(f1, f2, f3, f4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Using each of them, we are able to deduce exponents p1, p2, p3, p4 with ✸(p1, p2, p3, p4) ≲ 1, where at least one of pj, j = 1, 2, 3, 4, takes ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following result can be proven by applying the first upper bound of ✸(f1, f2, f3, f4) in Propo- sition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 together with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✸ be defined on functions on Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the following statements are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (i) If 1 p1 + d p2 + 1 p3 ≤ d and d p1 + 1 p2 + d p3 ≤ d, then ✸(p1, p2, p3, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) If d p1 + 1 p2 + 1 p4 ≤ d and 1 p1 + d p2 + d p4 ≤ d, then ✸(p1, p2, ∞, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iii) If 1 p1 + 1 p3 + d p4 ≤ d and d p1 + d p3 + 1 p4 ≤ d, then ✸(p1, ∞, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iv) If 1 p2 + d p3 + 1 p4 ≤ d and d p2 + 1 p3 + d p4 ≤ d, then ✸(∞, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We will only provide the proof of the first part of the theorem since the proofs of other parts are the same in the sense that the proof of the first part uses the first upper bound of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 and the proofs of other parts can also use their corresponding upper bounds of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 to complete the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let us start proving the first part of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To complete the proof, we aim to show that for all non-negative functions fi, i = 1, 2, 3, 4, on Fd q, ✸(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||∞, whenever the exponents 1 ≤ p1, p2, p3 ≤ ∞ satisfy the following conditions: 1 p1 + d p2 + 1 p3 ≤ d and d p1 + 1 p2 + d p3 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) By the first part of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4, it follows that ✸(f1, f2, f3, f4) ≲ � 1 |Sd−2 t | L(f1f3, f2) + Λ(f1, f2, f3) � ||f4||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Therefore, under the assumptions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4), our problem is reducing to establishing the following two estimates: L(f1f3, f2) ≲ ||f1||p1||f2||p2||f3||p3, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) Λ(f1, f2, f3) ≲ ||f1||p1||f2||p2||f3||p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) For 1 ≤ p1, p3 ≤ ∞, let 1/r = 1/p1 + 1/p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the conditions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) are the same as 1 r + d p2 ≤ d and d r + 1 p2 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' So these conditions enable us to invoke Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 so that we obtain the estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) as follows: L(f1f3, f2) ≲ ||f1f3||r||f2||p2 ≤ ||f1|||p1||f2||p2||f3||p3, where we used H¨older’s inequality in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It remains to prove the estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) under the assumptions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To do this, we shall use Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4, which gives sufficient conditions for Λ(p1, p2, p3) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We directly compare the conditions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) with the assumptions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then it is not hard to observe the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (Case 1) In the case when 0 ≤ 1 p1, 1 p3 ≤ d d+1, the conditions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) imply the hypothesis of the first part of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (Case 2) In the case when 0 ≤ 1 p1 ≤ d d+1 ≤ 1 p3 ≤ 1, the conditions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) imply the hypothesis of the second part of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To see this, notice that if d/p1 + 1/p2 + d/p3 ≤ d, then 1/(dp1) + 1/p2 + d/p3 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 30 (Case 3) In the case when 0 ≤ 1 p3 ≤ d d+1 ≤ 1 p1 ≤ 1, the conditions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) imply the hypothesis of the third part of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (Case 4) In the case when d d+1 ≤ 1 p1, 1 p3 ≤ 1, the conditions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) imply the hypothesis of the fourth part of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, we conclude from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 that Λ(p1, p2, p3) ≲ 1 under the assumptions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 Sharp boundedness results for the C4-Operator ✸ on F2 q Recall that Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 provides sufficient conditions for ⋄(p1, p2, p2, p4) ≲ 1 in any dimensions d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this section, we show that Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is sharp in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, using Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 we will prove the following optimal result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✸ be the C4-Operator on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For 1 ≤ pi ≤ ∞, 1 ≤ i ≤ 4, we have ✸(p1, p2, p3, p4) ≲ 1 if and only if 2 p1 + 1 p2 + 2 p3 + 1 p4 ≤ 2, and 1 p1 + 2 p2 + 1 p3 + 2 p4 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The necessary conditions for ✸(p1, p2, p3, p4) ≲ 1 follow immediately from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 for d = 2 (see Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Conversely, suppose that 1 ≤ p1, p2, p3, p4 ≤ ∞ satisfy the following two inequalities: 2 p1 + 1 p2 + 2 p3 + 1 p4 ≤ 2, and 1 p1 + 2 p2 + 1 p3 + 2 p4 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) Then, as mentioned in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, it can be shown by Polymake [1, 6] that (1/p1, 1/p2, 1/p3, 1/p4) is contained in the convex hull of the points (0, 0, 1, 0), (0, 0, 0, 1), (0, 1, 0, 0), (2/3, 0, 0, 2/3), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (0, 0, 0, 0), (0, 2/3, 2/3, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By interpolating the above 9 critical points, to prove ✸(p1, p2, p3, p4) ≲ 1 for all pi, 1 ≤ i ≤ 4 satisfy- ing the inequalities in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7), it will be enough to prove it for the 9 critical points (1/p1, 1/p2, 1/p3, 1/p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This can be easily proven by using Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For example, for the point (1/p1, 1/p2, 1/p3, 1/p4) = (2/3, 0, 0, 2/3), a direct computation shows that the assumptions in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (ii) are satisfied and thus ✸(p1, p2, p3, p4) = ✸(3/2, ∞, ∞, 3/2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For other critical points, we can easily prove them in the same way so that we omit the detail proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that the graph C4 is a subgraph of the graph C4 + diagonal, and they are associated with the operators ✸ and ♦t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the following theorem shows that the answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 is negative when G is the C4 + diagonal, and G′ is the C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, this does not mean that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is not true since the C4 and the C4 + diagonal do not satisfy the main hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ♦t, ✸ be the (C4 + t)-Operator and the C4-Operator on F2 q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 31 (i) If ✸(p1, p2, p3, p4) ≲ 1, then ♦t(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) Moreover, there exist exponents 1 ≤ a, b, c, d ≤ ∞ such that ♦t(a, b, c, d) ≲ 1 but ✸(a, b, c, d) is not bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' First, let us prove the statement (ii) in the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To prove this, we choose (a, b, c, d) = (3/2, ∞, 3/2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (i), we can easily note that ♦t(3/2, ∞, 3/2, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, it is impossible that ✸(3/2, ∞, 3/2, ∞) ≲ 1, which can be shown from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Next, let us prove the first conclusion of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that ✸(p1, p2, p3, p4) ≲ 1 for 1 ≤ pi ≤ ∞, 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, as mentioned in the second conclusion of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, the point (1/p1, 1/p2, 1/p3, 1/p4) lies on the convex body with the critical endpoints: (0, 0, 1, 0), (0, 0, 0, 1), (0, 1, 0, 0), (2/3, 0, 0, 2/3), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (0, 0, 0, 0), (0, 2/3, 2/3, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Invoking the interpolation theorem, to prove the conclusion that ♦t(p1, p2, p3, p4) ≲ 1, it will be enough to establish the boundedness only for those 9 critical points (1/p1, 1/p2, 1/p3, 1/p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, it remains to establish the following estimates: ♦t(∞, ∞, ∞, ∞) ≲ 1, ♦t(1, ∞, ∞, ∞) ≲ 1, ♦t(∞, 1, ∞, ∞) ≲ 1, ♦t(∞, ∞, 1, ∞) ≲ 1, ♦t(∞, ∞, ∞, 1) ≲ 1, ♦t(3/2, 3/2, ∞, ∞) ≲ 1, ♦t(3/2, ∞, ∞, 3/2) ≲ 1, ♦t(∞, 3/2, 3/2, ∞) ≲ 1, ♦t(∞, ∞, 3/2, 3/2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, these estimates follow by applying Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 8 Boundedness problem for the P3-Operator For t ∈ F∗ q and non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd q, we define ⊓(f1, f2, f3, f4) as the following value: 1 qd|St|3 � x1,x2,x3,x4∈Fdq St(x1 − x2)St(x2 − x3)St(x3 − x4) 4 � i=1 fi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) This operator ⊓ will be named the P3-Operator on Fd q since it is related to the graph P3 with vertices in Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Note that in the definition of ⊓(f1, f2, f3, f4), we take the normalizing fact qd|St|3, which is corresponding to N(G) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) when G is the P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We define ⊓(p1, p2, p3, p4) as the smallest constant such that the following estimate holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd q : ⊓(f1, f2, f3, f4) ≤ ⊓(p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We want to determine 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ⊓ (f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) 32 holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In other words, our main problem is to find all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ⊓(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 (Necessary conditions for ⊓(p1, p2, p3, p4) ≲ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that ⊓(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have 1 p1 + d p2 + 1 p3 ≤ d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1 p2 + d p3 + 1 p4 ≤ d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' d p1 + 1 p2 + 1 p3 + 1 p4 ≤ d+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1 p1 + 1 p2 + 1 p3 + d p4 ≤ d+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1 p1 + d p2 + 1 p3 + d p4 ≤ 2d − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' d p1 + 1 p2 + d p3 + 1 p4 ≤ 2d − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' and d p1 + 1 p2 + 1 p3 + d p4 ≤ 2d In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' when d = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' by using Polymake [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' it can be shown that (1/p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p4) is contained in the convex hull of the points: (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1),' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When d = 2, the third, fourth, and seventh inequalities above are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When d = 3, the third and fourth inequalities above are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' As in the proofs of Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, the conclusions of the statement follow by testing the inequality (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) with the following specific functions, respectively: 1) f1 = f3 = 1St, f2 = δ0, and f4 = 1Fdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2) f1 = 1Fdq, f2 = f4 = 1St, and f3 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 3) f2 = f3 = f4 = 1St, and f1 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 4) f1 = f2 = f3 = 1St, and f4 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 5) f1 = f3 = 1St, and f2 = f4 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6) f2 = f4 = 1St, and f1 = f3 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 7) f2 = f3 = 1St, and f1 = f4 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 Boundedness results for the P3-Operator ⊓ on Fd q We begin by observing that an upper bound of ⊓(f1, f2, f3, f4) can be controlled by the value Λ(f1, f2, f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ a, b, c ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If Λ(a, b, c) ≲ 1, then ⊓(a, b, c, ∞) ≲ 1 and ⊓(∞, a, b, c) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For all non-negative functions fi, i = 1, 2, 3, 4, on Fd q, our task is to prove the following inequalities: ⊓ (f1, f2, f3, f4) ≲ � Λ(f1, f2, f3) ||f4||∞, ||f1||∞ Λ(f2, f3, f4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) We will only prove the first inequality, that is ⊓ (f1, f2, f3, f4) ≲ Λ(f1, f2, f3) ||f4||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) By symmetry, the second inequality can be easily proven in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By definition in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1), 33 we can write ⊓(f1, f2, f3, f4) as 1 qd|St|2 � x1,x2,x3∈Fdq St(x1 − x2)St(x2 − x3) � 3 � i=1 fi(xi) � \uf8eb \uf8ed 1 |St| � x4∈Fdq f4(x4)St(x3 − x4) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since the value in the above bracket is Af4(x3), which is clearly dominated by ||Af4||∞, the required estimate (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) follows immediately from the definition of Λ(f1, f2, f3) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following theorem can be deduced from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Consider the P3-Operator ⊓ on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that the exponents 1 ≤ a, b, c ≤ ∞ satisfy one of the following conditions: (i) 0 ≤ 1 a, 1 c ≤ d d+1 and 1 a + d b + 1 c ≤ d (ii) 0 ≤ 1 a ≤ d d+1 ≤ 1 c ≤ 1, and 1 da + 1 b + d c ≤ d (iii) 0 ≤ 1 c ≤ d d+1 ≤ 1 a ≤ 1, and d a + 1 b + 1 dc ≤ d (iv) d d+1 ≤ 1 a, 1 c ≤ 1 and d a + 1 b + d c ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ⊓(a, b, c, ∞) ≲ 1 and ⊓(∞, a, b, c) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 with p1 = a, p2 = b, p3 = c, it is clear that Λ(a, b, c) ≲ 1 for all exponents a, b, c in our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the statement follows immediately from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now we prove that the value ⊓(f1, f2, f3, f4) can be expressed in terms of the averaging operator over spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For functions f, g, h on Fd q, let us denote < f, g, h >:= ||fgh||1 = 1 qd � x∈Fdq f(x)g(x)h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let fi, i = 1, 2, 3, 4, be non-negative real-valued functions on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ⊓(f1, f2, f3, f4) =< Af1, f2, A(f3 · Af4) >=< A(f2 · Af1), f3, Af4 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By symmetry, to complete the proof, it suffices to prove the first equality, that is ⊓(f1, f2, f3, f4) =< Af1, f2, A(f3 · Af4) > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Combining the definition in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) and the definition of the spherical averaging operator A, it follows 34 that ⊓(f1, f2, f3, f4) = 1 qd � x2∈Fdq f2(x2)Af1(x2) \uf8ee \uf8f0 1 |St| � x3∈Fdq f3(x3)St(x2 − x3)Af4(x3) \uf8f9 \uf8fb = 1 qd � x2∈Fdq f2(x2)Af1(x2)A(f3 · Af4)(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This gives the required estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Combining Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6 and the averaging estimate over spheres, we are able to deduce sufficient conditions for the boundedness of the P3-Operator ⊓ on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3, p4 ≤ ∞ be exponents satisfying one of the following conditions: (i) 0 ≤ 1 p1, 1 p4, 1 p3 + 1 dp4 ≤ d d+1, and 1 dp1 + 1 p2 + 1 dp3 + 1 d2p4 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) 0 ≤ 1 p1, 1 p4 ≤ d d+1 ≤ 1 p3 + 1 dp4 ≤ 1, and 1 dp1 + 1 p2 + d p3 + 1 p4 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iii) 0 ≤ 1 p1, 1 p3 + d p4 − d + 1 ≤ d d+1 ≤ 1 p4 ≤ 1, and 1 p1 + d p2 + 1 p3 + d p4 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iv) 0 ≤ 1 p1 ≤ d d+1 ≤ 1 p4, 1 p3 + d p4 − d + 1 ≤ 1, and 1 dp1 + 1 p2 + d p3 + d2 p4 ≤ d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (v) 0 ≤ 1 p4, 1 p3 + 1 dp4 ≤ d d+1 ≤ 1 p1 ≤ 1, and d p1 + 1 p2 + 1 dp3 + 1 d2p4 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (vi) 0 ≤ 1 p4 ≤ d d+1 ≤ 1 p1, 1 p3 + 1 dp4 ≤ 1, and d p1 + 1 p2 + d p3 + 1 p4 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (vii) 0 ≤ 1 p3 + d p4 − d + 1 ≤ d d+1 ≤ 1 p1, 1 p4 ≤ 1, and d2 p1 + d p2 + 1 p3 + d p4 ≤ d2 + d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (viii) d d+1 ≤ 1 p1, 1 p4, 1 p3 + d p4 − d + 1 ≤ 1, and d p1 + 1 p2 + d p3 + d2 p4 ≤ d2 + d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ⊓(p1, p2, p3, p4) ≲ 1 and ⊓(p4, p3, p2, p1) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By symmetry, it will be enough to prove the first part of conclusions, that is ⊓(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To complete the proof, we will first find the general conditions that guarantee this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Next we will demonstrate that each of the hypotheses in the theorem satisfies the general condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To derive the first general condition, we assume that 1 ≤ r1, p2, r ≤ ∞ satisfy that 1 r1 + 1 p2 + 1 r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) Then by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6 and H¨older’s inequality, ⊓(f1, f2, f3, f4) ≤ ||Af1||r1||f2||p2||A(f3 · Af4)||r, where we also used the nesting property of norms associated with the normalizing counting mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Assume that 1 ≤ p1, s ≤ ∞ satisfy the following averaging estimates over spheres: A(p1 → r1) ≲ 1 and A(s → r) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) 35 It follows that ⊓(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3 · Af4||s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now we assume that 1 ≤ p3, t ≤ ∞ satisfy that 1 s = 1 p3 + 1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) Then, by the H¨older’s inequality, we see that ⊓(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||Af4||t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Finally, if we assume that 1 ≤ p4 ≤ ∞ satisfies the following averaging estimate: A(p4 → t) ≲ 1, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8) then we obtain that ⊓(f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In summary, we see that ⊓(p1, p2, p3, p4) ≲ 1 provided that the numbers 1 ≤ pi ≤ ∞, i = 1, 2, 3, 4, satisfy all the conditions (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Thus, to finish the proof, we will show that each of the 8 hypotheses in the theorem satisfies all these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Given 1 ≤ p1, p4 ≤ ∞, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 we can chose 1 ≤ r1, t ≤ ∞ such that the first averaging estimate in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) and the averaging estimate (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8) hold respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, we can select 0 ≤ 1/r1, 1/t ≤ 1 as follows: If 0 ≤ 1 p1 ≤ d d+1, then we take 1/r1 = 1/(dp1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If 0 ≤ 1 p4 ≤ d d+1, then we take 1/t = 1/(dp4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If d d+1 ≤ 1 p1 ≤ 1, then we choose 1/r1 = d/p1 − d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If d d+1 ≤ 1 p4 ≤ 1, then we choose 1/t = d/p4 − d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In the next step, we determine 1 ≤ r ≤ ∞ by using the condition (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) and the second averaging estimate in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since two kinds of t values can be chosen as above, the condition (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7) becomes 1 s = 1 p3 + 1 dp4 or 1 s = 1 p3 + d p4 − d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Combining these s values with the second averaging estimate in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6), the application of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 enables us to choose 1/r values as follows: If 0 ≤ 1 s = 1 p3 + 1 dp4 ≤ d d+1, then we take 1 r = 1 dp3 + 1 d2p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If d d+1 ≤ 1 s = 1 p3 + 1 dp4 ≤ 1, then we take 1 r = d p3 + 1 p4 − d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If 0 ≤ 1 s = 1 p3 + d p4 − d + 1 ≤ d d+1, then we take 1 r = 1 dp3 + 1 p4 − 1 + 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If d d+1 ≤ 1 s = 1 p3 + d p4 − d + 1 ≤ 1, then we take 1 r = d p3 + d2 p4 − d2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Finally, use the condition (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5) together with previously selected two values for r1 and four values for r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we obtain the required remaining conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 36 Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is a special case of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is much simpler than that of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We do not know if the consequences from Theorems 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 imply the sharp boundedness results for the P3-Operator ⊓ on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, they play an important role in proving the theorem below, which states that the exponents for ⊓(p1, p2, p3, p4) ≲ 1 are less restricted than those for ♦t(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The precise statement is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ♦t and ⊓ be the operators acting on the functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If ♦t(p1, p2, p3, p4) ≲ 1 for 1 ≤ p1, p2, p3, p4 ≤ ∞, then ⊓(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Assume that ♦t(p1, p2, p3, p4) ≲ 1 for 1 ≤ p1, p2, p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, the point (1/p1, 1/p2, 1/p3, 1/p4) is contained in the convex hull of the following points: (0, 0, 1, 0), (0, 1, 0, 0), (0, 0, 0, 1), (1/2, 0, 1/2, 1/2), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), (2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To complete the proof, by the interpolation theorem, it suffices to show that for each of the above critical points (1/p1, 1/p2, 1/p3, 1/p4), we have ⊓(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To prove this, we will use Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 and Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 with the hypothesis (i), one can notice that ⊓(3/2, ∞, ∞, 3/2) ≲ 1, which is corresponding to the point (1/p1, 1/p2, 1/p3, 1/p4) = (2/3, 0, 0, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Similarly, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 with the hypothesis (ii) can be used for the point (1/2, 0, 1/2, 1/2), namely, ⊓(2, ∞, 2, 2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For any other points, we can invoke Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' More precisely, we can apply Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 with the hypothesis (i) for the points (0, 1, 0, 0), (2/3, 2/3, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), (0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The points (0, 0, 1, 0), (0, 0, 0, 1) can be obtained by Theo- rem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 with the hypothesis (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Finally, for the point (1, 0, 0, 0), we can prove that ⊓(1, ∞, ∞, ∞) ≲ 1 by using Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 with the hypothesis (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The reverse statement of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9 is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' As a counterexample, we can take p1 = 3/2, p2 = 3, p3 = 3/2, p4 = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Indeed, the assumption (i) of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 with d = 2 implies that ⊓(3/2, 3, 3/2, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, ♦t(3/2, 3, 3/2, ∞) cannot be bounded, which follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following corollary is a consequence of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is valid for the graph C4 + diagonal and its subgraph P3 in F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It is obvious that the P3 is a subgraph of C4 + diagonal in F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For d = 2, it is plain to notice that min{δ(C4 + diagonal), d} = 2 > δ(P3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Thus, the graph C4 + diagonal and its subgraph P3 satisfy all assumptions of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the statement of the corollary follows 37 immediately from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9 since the operators ♦t and ⊓ are related to the C4 + diagonal and its subgraph P3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following theorem provides a concrete example for a positive answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 since the operators ✸ and ⊓ are related to the graph C4 and its subgraph P3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Furthermore, the graphs also satisfy Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 (see Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='14 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✸ and ⊓ be the operators acting on the functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If ✸(p1, p2, p3, p4) ≲ 1, 1 ≤ p1, p2, p3, p4 ≤ ∞, then ⊓(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 (i), if ✸(p1, p2, p3, p4) ≲ 1, then ♦t(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9, if ♦t(p1, p2, p3, p4) ≲ 1, then ⊓(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The reverse statement of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='12 cannot hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' As in Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10, if we can take p1 = 3/2, p2 = 3, p3 = 3/2, p4 = ∞, then ⊓(3/2, 3, 3/2, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, ✸(3/2, 3, 3/2, ∞) cannot be bounded, which follows from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 holds true for the graph C4 and its subgraph P3 on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The main hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is satisfied for the the graph C4 and its subgraph P3 on F2 q : min{δ(C4), 2} = 2 > 1 = δ(P3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since the operators ✸ and ⊓ are associated to the graph C4 and its subgraph P3, respectively, the statement of the corollary follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 9 Operators associated to the graph K3 + tail (a kite) Given t ∈ F∗ q and functions fi, i = 1, 2, 3, 4, on Fd q, we define ✂(f1, f2, f3, f4) as the following value: 1 qd|St|2|Sd−2 t | � x1,x2,x3,x4∈Fdq St(x1 − x2)St(x2 − x3)St(x3 − x4)St(x3 − x1) 4 � i=1 fi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Note that this operator ✂ is related to the graph K3 + tail (Figure 1g), and so the normalizing factor N(G) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) can be taken as the quantity qd|St|2|Sd−2 t |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For 1 ≤ p1, p2, p3, p4 ≤ ∞, we define ✂(p1, p2, p3, p4) as the smallest constant such that the following estimate holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd q : ✂(f1, f2, f3, f4) ≤ ✂(p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Here, our main problem is to determine all exponents 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ✂ (f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) 38 holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In other words, we are asked to determine all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ✂(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Recall that when d = 2, we assume that 3 ∈ Fq is a square number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 (Necessary conditions for the boundedness of ✂(p1, p2, p3, p4) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) holds, namely ✂(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have 1 p1 + 1 p2 + d p3 + 1 p4 ≤ d, 1 p1 + d p2 + 1 p3 ≤ d, d p1 + 1 p2 + 1 p3 ≤ d, 1 p1 + d p2 + 1 p3 + d p4 ≤ 2d − 1, and d p1 + 1 p2 + 1 p3 + d p4 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' if d = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' then it can be shown by Polymake [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6] that (1/p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p4) is contained in the convex hull of the points: (0,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To deduce the first inequality, we test (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) with f1 = f2 = f4 = 1St and f3 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To obtain the second one, we test (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) with f1 = f3 = 1St, f2 = δ0, and f4 = 1Fdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To get the third one, we test (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) with f1 = δ0, f2 = f3 = 1St, and f4 = 1Fdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To prove the fourth one, we test (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) with f1 = f3 = 1St and f2 = f4 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Finally, to obtain the fifth inequality, we test (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) with f1 = f4 = δ0 and f2 = f3 = 1St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 Sufficient conditions for the boundedness of ✂ on Fd q When one of exponents p1, p2, p4 is ∞, the boundedness problem of ✂(p1, p2, p3, p4) can be reduced to that for the K3-Operator ∆ or the P2-Operator Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ a, b, c ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (i) If ∆(a, b, c) ≲ 1, then ✂(a, b, c, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) If Λ(a, b, c) ≲ 1, then ✂(∞, a, b, c) ≲ 1 and ✂(a, ∞, b, c) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For all non-negative functions fi, i = 1, 2, 3, 4, on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' we aim to prove the following inequal- ities: ✂ (f1, f2, f3, f4) ≲ \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∆(f1, f2, f3) ||f4||∞, ||f1||∞ Λ(f2, f3, f4), ||f2||∞ Λ(f1, f3, f4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) By the definition, ✂(f1, f2, f3, f4) can be expressed as 1 qd|St||Sd−2 t | � x1,x2,x3∈Fdq St(x1−x2)St(x2−x3)St(x3−x1) � 3 � i=1 fi(xi) � \uf8eb \uf8ed 1 |St| � x4∈Fdq St(x3 − x4)f4(x4) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 39 The sum in the above bracket is clearly dominated by ||f4||∞ for all x3 ∈ Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, recalling the definition of ∆(f1, f2, f3) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1), we get the first inequality in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2): ✂(f1, f2, f3, f4) ≤ ∆(f1, f2, f3)||f4||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now we prove the second and third inequalities in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We will only provide the proof of the second inequality, that is ✂ (f1, f2, f3, f4) ≤ ||f1||∞ Λ(f2, f3, f4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) The third inequality can be similarly proved by switching the roles of variables x1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We write ✂(f1, f2, f3, f4) as follows: 1 qd|St|2 � x2,x3,x4∈Fdq St(x2−x3)St(x3−x4) � 4 � i=2 fi(xi) � \uf8eb \uf8ed 1 |Sd−2 t | � x1∈Fdq St(x1 − x2)St(x3 − x1)f1(x1) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Recall the definition of Λ(f2, f3, f4) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, to prove the inequality (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3), it will be enough to show that for all x2, x3 ∈ Fd q with ||x2 −x3|| = t ̸= 0, the value in the above bracket is ≲ ||f1||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now by a simple change of variables, the value in the above bracket is the same as 1 |Sd−2 t | � x1∈St St((x3 − x2) − x1)f1(x1 + x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This is clearly dominated by 1 |Sd−2 t | � x1∈St St((x3 − x2) − x1)||f1||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since ||x3 − x2|| = t ̸= 0, applying Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 in Appendix gives us the desirable estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We address sufficient conditions for the boundedness of ✂ on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following result can be obtained from Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✂ be defined on the functions on Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that 1 ≤ a, b ≤ ∞ satisfies the following equations: 1 a + d b ≤ d and d a + 1 b ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ✂(a, b, ∞, ∞) ≲ 1, ✂ (a, ∞, b, ∞) ≲ 1, ✂ (∞, a, b, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The statement follows immediately by combining Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 (i) with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 (ii) can be used to deduce the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✂ be defined on the functions on Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that 1 ≤ a, b, c ≤ ∞ satisfies one of the following conditions: 40 (i) 0 ≤ 1 a, 1 c ≤ d d+1 and 1 a + d b + 1 c ≤ d, (ii) 0 ≤ 1 a ≤ d d+1 ≤ 1 c ≤ 1, and 1 da + 1 b + d c ≤ d, (iii) 0 ≤ 1 c ≤ d d+1 ≤ 1 a ≤ 1, and d a + 1 b + 1 dc ≤ d, (iv) d d+1 ≤ 1 a, 1 c ≤ 1 and d a + 1 b + d c ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ✂(∞, a, b, c) ≲ 1 and ✂ (a, ∞, b, c) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' From our assumptions on the numbers a, b, c, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 implies that Λ(a, b, c) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the statement follows by applying Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 Boundedness of ✂ in two dimensions Theorems 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 provide non-trivial results available in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In this section we will show that further improvements can be made in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Before we state and prove the improvements, we collect the results in two dimensions, which can be direct consequences of Theorems 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To deduce the following theorem, we will apply Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✂ be defined on functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ✂(p1, p2, p3, p4) ≲ 1 pro- vided that (p1, p2, p3, p4) is one of the following points: (∞, ∞, ∞, ∞), (1, ∞, ∞, ∞), (∞, 1, ∞, ∞), (∞, ∞, 1, ∞), (3/2, 3/2, ∞, ∞), (3/2, ∞, 3/2, ∞), (∞, 3/2, 3/2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Using the first conclusion of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 with d = 2, we see that ✂(p1, p2, p3, p4) ≲ 1 whenever (p1, p2, p3, p4) takes the following points: (∞, ∞, ∞, ∞), (1, ∞, ∞, ∞), (∞, 1, ∞, ∞), (3/2, 3/2, ∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Next, the second conclusion of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 with d = 2 implies that ✂(p1, p2, p3, p4) ≲ 1 for the points (p1, p2, p3, p4) = (∞, ∞, 1, ∞), (3/2, ∞, 3/2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Finally, it follows from the third conclusion of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 with d = 2 that ✂(p1, p2, p3, p4) ≲ 1 for (p1, p2, p3, p4) = (∞, 3/2, 3/2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following theorem will be proven by applying Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✂ be defined on the functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that (p1, p2, p3, p4) is one of the following points: (∞, ∞, ∞, 1), (2, ∞, 2, 2), (3/2, ∞, ∞, 3/2), (∞, ∞, 3/2, 3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ✂(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We get that ✂(∞, ∞, ∞, 1) ≲ 1 by using the assumption (ii) and the first conclusion of The- orem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Invoking the assumption (i) and the second conclusion of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, one can directly note that ✂(2, ∞, 2, 2) ≲ 1 and ✂(3/2, ∞, ∞, 3/2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Finally, to prove that ✂(∞, ∞, 3/2, 3/2) ≲ 1, one can use the assumption (i) and the first conclusion of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' We now introduce the connection between ✂(f1, f2, f3, f4) and the bilinear averaging operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 41 Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let B be the bilinear operator defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, for any non-negative real-valued functions fi, i = 1, 2, 3, 4, on F2 q, we have ✂(f1, f2, f3, f4) = ||B(f1, f2) · f3 · Af4||1, where A denotes the averaging operator over the circle in F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In two dimensions, ✂(f1, f2, f3, f4) can be rewritten as the following form: 1 q2|St|2 � x1,x2,x3,x4∈F2q St(x1 − x2)St(x3 − x2)St(x3 − x4)St(x3 − x1) 4 � i=1 fi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By the change of variables by putting y1 = x3 − x1, y2 = x3 − x2, y3 = x3, y4 = x3 − x4, the value ✂(f1, f2, f3, f4) becomes 1 q2|St|2 � y1,y2,y3,y4∈F2q St(y2 − y1)St(y2)St(y4)St(y1)f1(y3 − y1)f2(y3 − y2)f3(y3)f4(y3 − y4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This can be expressed as follows: 1 q2 � y3∈F2q f3(y3) \uf8eb \uf8ed 1 |St| � y4∈St f4(y3 − y4) \uf8f6 \uf8f8 \uf8eb \uf8ed 1 |St| � y1,y2∈St:||y2−y1||=t f1(y3 − y1)f2(y3 − y2) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Recalling the definitions of the averaging operator in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) and the bilinear averaging operator in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2), it follows that ✂(f1, f2, f3, f4) = 1 q2 � y3∈F2q f3(y3)Af4(y3)B(f1, f2)(y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By the definition of the normalized norm || ||1, the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For 1 ≤ p1, p2, p3, p4 ≤ ∞, recall that the notation ✂(p1, p2, p3, p4) ⪅ 1 is used if the following estimate holds for all subsets E, F, G, H of F2 q: ✂(E, F, G, H) ≲ ||E||p1||F||p2||G||p3||H||p4, and this estimate is referred to as the restricted strong-type ✂(p1, p2, p3, p4) estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following theorem is our main result in two dimensions, which gives a new restricted strong- type estimate for the boundedness on the operator ✂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✂ be defined on functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the following statements are valid for all subsets E, F of F2 q and all non-negative functions f3, f4 on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 42 (i) If 2 ≤ p3 ≤ ∞, 3/2 ≤ p4 ≤ ∞, and 1 p3 + 1 2p4 ≤ 1 2, then we have ✂(E, F, f3, f4) ≲ ||E||2||F||2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) If 2 ≤ p3 ≤ ∞, 4/3 ≤ p4 ≤ 3/2, and 1 p3 + 2 p4 ≤ 3 2, then we have ✂(E, F, f3, f4) ≲ ||E||2||F||2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let E, F be subsets of F2 q and f, g be non-negative real-valued functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Propo- sition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8 and H¨older’s inequality, it follows that for 2 ≤ p3 ≤ ∞, ✂(E, F, f3, f4) ≤ ||B(E, F)||2||f3||p3||Af4|| 2p3 p3−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Here, we also notice that 2 ≤ 2p3 p3−2 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since ||B(E, F)||2 ≲ ||E||2||F||2 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10, we see that ✂(E, F, f3, f4) ≤ ||E||2||F||2||f3||p3||Af4|| 2p3 p3−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, to complete the proof, it suffices to show that for all exponents p3, p4 satisfying the as- sumptions of the theorem, we have A � p4 → 2p3 p3 − 2 � ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) To prove this, we first recall from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 with d = 2 that A(p → r) ≲ 1 for any numbers 1 ≤ p, r ≤ ∞ such that (1/p, 1/r) lies on the convex hull of points (0, 0), (0, 1), (1, 1), and (2 3, 1 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Also invoke Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 to find the equations indicating the endpoint estimates for A(p → r) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Using those averaging estimates with p = p4, r = 2p3 p3−2, the inequality (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) can be obtained by a direct computation, where we also use the fact that 2 ≤ r = 2p3 p3−2 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following corollary is a direct consequence of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✂ be defined on functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ✂(2, 2, 2, ∞) ⪅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The statement follows by a direct application of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The theorem below shows that the exponents for ♦t(p1, p2, p3, p4) ≲ 1 are more restricted than those for ✂(p1, p2, p3, p4) ≲ 1 up to the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This also provides a positive answer to Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 since the graph K3 + tail is a subgraph of the graph C4 + diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ♦t and ✂ be the operators acting on the functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that ♦t(p1, p2, p3, p4) ≲ 1 for 1 ≤ p1, p2, p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have ✂(p1, p2, p3, p4) ≲ 1 except for the point (2, 2, 2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In addition, we have ✂(2, 2, 2, ∞) ⪅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Assume that ♦t(p1, p2, p3, p4) ≲ 1 for 1 ≤ p1, p2, p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, the point (1/p1, 1/p2, 1/p3, 1/p4) is contained in the convex hull of the following points: (0, 0, 1, 0), 43 (0, 1, 0, 0), (0, 0, 0, 1), (1/2, 0, 1/2, 1/2), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), (2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Theorems 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7, the strong type estimate ✂(p1, p2, p3, p4) ≲ 1 holds for all the above points (1/p1, 1/p2, 1/p3, 1/p4) except for (1/2, 1/2, 1/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Moreover, we know from Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='10 that ✂(2, 2, 2, ∞) ⪅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, the statement follows by interpolating those points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The reverse statement of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11 is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To see this, observe from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='9 (ii) that ✂(2, 2, 6, 3/2) ⪅ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In addition, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, notice that ♦t(2, 2, 6, 3/2) cannot be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 holds up to endpoints for the graph C4 + diagonal and its subgraph K3 + tail in F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The operators ♦t and ✂ are associated with the C4 + diagonal and its subgraph K3 + tail in F2 q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, invoking Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11, the proof is reduced to showing that the C4 + diagonal and its subgraph K3 + tail satisfy the main hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, it is clear that min{δ(C4 + diagonal), 2} = 2 > 1 = δ(K3 + tail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Thus, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following theorem shows that there exists an inclusive relation between boundedness exponents for the operators corresponding to the graphs C4 and K3 + tail, although they are not subgraphs of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✸ and ✂ be defined on functions on F2 q and let 1 ≤ p1, p2, p3, p4 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then if ✸(p1, p2, p3, p4) ≲ 1, we have ✂(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' First, by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6, note that ✸(2, 2, 2, ∞) cannot be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Now suppose that ✸(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then (p1, p2, p3, p4) ̸= (2, 2, 2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Using Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7, we get ♦t(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the statement follows immediately from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By combining Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='12 and Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7, it is clear that the reverse of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='14 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='14 provides an example to satisfy Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 without the hypothesis that G′ is a subgraph of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 10 Boundedness problems for the Y -shaped graph In this section, we study the boundedness of the operator for the Y -shaped graph in Figure 1h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For t ∈ F∗ q, the Y -shaped operator Y is defined by Y (f1, f2, f3, f4) = 1 qd|St|3 � x1,x2,x3,x4∈Fdq St(x3 − x1)St(x3 − x2)St(x3 − x4) 4 � i=1 fi(xi), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) 44 where functions fi, i = 1, 2, 3, 4, are defined on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Note that this operator Y is related to the Y -shaped graph, and so the normalizing factor N(G) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4) can be taken as qd|St|3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The operator norm of the Y -shaped operator is defined in a standard way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For 1 ≤ p1, p2, p3, p4 ≤ ∞, the operator norm Y (p1, p2, p3, p4) is defined as the smallest constant such that the following estimate holds for all non-negative real-valued functions fi, i = 1, 2, 3, 4, on Fd q : Y (f1, f2, f3, f4) ≤ Y (p1, p2, p3, p4)||f1||p1||f2||p2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) We aim to find all numbers 1 ≤ p1, p2, p3, p4 ≤ ∞ such that ✂(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 (Necessary conditions for the boundedness of Y (p1, p2, p3, p4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ pi ≤ ∞, 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Suppose that Y (p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then all the following inequalities are satisfied: 1 p1 + 1 p2 + d p3 + 1 p4 ≤ d, d p1 + d p2 + 1 p3 + d p4 ≤ 3d − 2, d p1 + d p2 + 1 p3 ≤ 2d − 1, d p1 + 1 p3 + d p4 ≤ 2d − 1, d p2 + 1 p3 + d p4 ≤ 2d − 1, d p1 + 1 p3 ≤ d, d p2 + 1 p3 ≤ d, 1 p3 + d p4 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' if d = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' then it can be shown by Polymake [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6] that (1/p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1/p4) is contained in the convex hull of the points: (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (1,' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By a direct computation, the conclusions of the lemma easily follow by testing the inequality (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2) with the following specific functions, respectively: 1) f1 = f2 = f4 = 1St, f3 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2) f1 = f2 = f4 = δ0, f3 = 1St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 3) f1 = f2 = δ0, f3 = 1St, f4 = 1Fdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 4) f1 = f2 = δ0, f2 = 1Fdq, f3 = 1St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 5) f1 = 1Fdq, f2 = f4 = δ0, f3 = 1St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 6) f1 = δ0, f2 = f4 = 1Fdq, f3 = 1St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 7) f1 = f4 = 1Fdq, f2 = δ0, f3 = 1St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 8) f1 = f2 = 1Fdq, f3 = 1St, f4 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1 Sufficient conditions for the boundedness of Y on Fd q It is not hard to observe that the boundedness problem for the Y -shaped operator can be reduced to the spherical averaging estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Indeed, the value Y (f1, f2, f3, f4) in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1) can be written by Y (f1, f2, f3, f4) = 1 qd � x3∈Fdq f3(x3) � i=1,2,4 \uf8eb \uf8ed 1 |St| � xi∈Fdq St(x3 − xi)fi(xi) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 45 Invoking the definition of the averaging operator A = ASt in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2), we get Y (f1, f2, f3, f4) = 1 qd � x3∈Fdq f3(x3)Af1(x3)Af2(x3)Af4(x4) = ||Af1 · Af2 · f3 · Af4||1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By H˝older’s inequality and the nesting property of the norm || · ||p, we get Y (f1, f2, f3, f4) ≤ ||Af1||r1||Af2||r2||f||p3||Af4||r4 if 1 r1 + 1 r2 + 1 p3 + 1 r4 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3, p4, r1, r2, r4 ≤ ∞ be extended real numbers which satisfy the following assumptions: 1 r1 + 1 r2 + 1 p3 + 1 r4 ≤ 1 and A(pi → ri) ≲ 1 for all i = 1, 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then we have Y (p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By combining the inequality (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3) with our assumptions on the averaging estimates, it follows that for all functions fi, i = 1, 2, 3, 4, on Fd q, Y (f1, f2, f3, f4) ≲ ||f1||p1||f2||p2||f3||p3||f4||p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following theorem provides lots of sufficient conditions for the boundedness of the Y -shaped operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let 1 ≤ p1, p2, p3, p4 ≤ ∞, and Y be the Y -shaped operator on Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then Y (p1, p2, p3, p4) ≲ 1 provided that one of the following conditions is satisfied: (i) 0 ≤ 1 p1, 1 p2, 1 p4 ≤ d d+1 and 1 p1 + 1 p2 + d p3 + 1 p4 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) 0 ≤ 1 p1, 1 p2 ≤ d d+1 ≤ 1 p4 and 1 dp1 + 1 dp2 + 1 p3 + d p4 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iii) 0 ≤ 1 p1, 1 p4 ≤ d d+1 ≤ 1 p2 ≤ 1 and 1 dp1 + d p2 + 1 p3 + 1 dp4 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iv) 0 ≤ 1 p2, 1 p4 ≤ d d+1 ≤ 1 p1 ≤ 1 and d p1 + 1 dp2 + 1 p3 + 1 dp4 ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (v) 0 ≤ 1 p1 ≤ d d+1 ≤ 1 p2, 1 p4 ≤ 1 and 1 dp1 + d p2 + 1 p3 + d p4 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (vi) 0 ≤ 1 p2 ≤ d d+1 ≤ 1 p1, 1 p4 ≤ 1 and d p1 + 1 dp2 + 1 p3 + d p4 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (vii) 0 ≤ 1 p4 ≤ d d+1 ≤ 1 p1, 1 p2 ≤ 1 and d p1 + d p2 + 1 p3 + 1 dp4 ≤ 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (viii) d d+1 ≤ 1 p1, 1 p2, 1 p4 ≤ 1 and d p1 + d p2 + 1 p3 + d p4 ≤ 3d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The proof uses Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 and the sharp averaging estimates in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The proof of this theorem is similar to that of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Therefore, we leave the detail of the proof to readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is also supported by the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 46 Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ♦t and Y be the operators acting on the functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If ♦t(p1, p2, p3, p4) ≲ 1 with 1 ≤ p1, p2, p3, p4 ≤ ∞, then Y (p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Assume that ♦t(p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, (1/p1, 1/p2, 1/p3, 1/p4) is contained in the convex hull of the points (0, 0, 1, 0), (0, 1, 0, 0), (0, 0, 0, 1), (1/2, 0, 1/2, 1/2), (2/3, 2/3, 0, 0), (1, 0, 0, 0), (2/3, 0, 2/3, 0), (1/2, 1/2, 1/2, 0), (2/3, 0, 0, 2/3), (0, 2/3, 2/3, 0), (0, 0, 0, 0), (0, 0, 2/3, 2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By interpolating those critical points, it suffices to check that each critical point above satisfies one of the 8 hypotheses of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, this can be easily shown by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For example, for the critical point (1/p1, 1/p2, 1/p3, 1/p4) = (1/2, 1/2, 1/2, 0), we can invoke the hypothesis (i) of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 with d = 2 and obtain that Y (2, 2, 2, ∞) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' In a same way, it can be easily proven for other critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The reverse statement of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' To find a counterexample, we can take p1 = p3 = ∞, p2 = p4 = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Indeed, by the hypothesis (5) of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4 with d = 2, we see that Y (∞, 3/2, ∞, 3/2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, ♦t(∞, 3/2, ∞, 3/2) is not bounded, which follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2 with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The following corollary proposes some possibility that the assumption of the subgraph in Conjec- ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let ✸ and Y be the operators acting on the functions on F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If ✸(p1, p2, p3, p4) ≲ 1 with 1 ≤ p1, p2, p3, p4 ≤ ∞, then Y (p1, p2, p3, p4) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The statement of the corollary follows immediately by combining Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 and Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Combining all the results obtained so far, we get the following theorem: Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' When d = 2 and n = 3, 4, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 is true, where we accept boundedness results up to endpoints in the case when G is the C4 + diagonal and its subgraph G′ is the K3+tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='8 for n = 3, and by Corollaries 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='11, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='14, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='13, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='7 for n = 4, we have proven that for d = 2 and n = 3, 4, there is the required inclusive boundedness relationship between any two operators corresponding to arbitrary connected ordered graph G and its subgraph G′ except for the following three cases: (I) G = C4 + diagonal and G′ = C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (II) K3 + tail and G′ = Y -shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (III) G = K3 + tail and G′ = P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' However, since δ(G) = δ(G′) for each case of (I), (II), (III), they do not satisfy the main hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='6) of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, they cannot be counterexamples contradicting Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5 and so there is no counterexample against Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='5, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 47 11 Acknowledgements A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Iosevich and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Bhowmik were supported in part by the National Science Foundation grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' HDR TRIPODS–1934962 and the National Science Foundation grant DMS–2154232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Koh was supported by Basic Science Research Programs through National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07044469).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Pham would like to thank to the VIASM for the hospitality and for the excellent working conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 12 Appendix In this appendix, we introduce the number of intersection points of two spheres in Fd q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let η denote the quadratic character of F∗ q, namely, η(s) = 1 for a square number s in F∗ q, and η(s) = −1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Given a non-zero vector m in Fd q, and t, b ∈ Fq, we define N(m, t, b) to be the number of common solutions x ∈ Fd q of the following equations: ||x|| = t, m · x = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Notice that the value of N(m, t, b) is the number of all intersection points between the sphere St and the plane {x ∈ Fd q : m · x = b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' The explicit value of it is well known as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let b, t ∈ Fq, and let m be a non-zero element in Fd q, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then the following statements hold: (i) If ||m|| ̸= 0 and b2 − t||m|| = 0, then N(m, t, b) = \uf8f1 \uf8f2 \uf8f3 qd−2 if d is even, qd−2 + q d−3 2 (q − 1)η � (−1) d−1 2 ||m|| � if d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (ii) If ||m|| ̸= 0 and b2 − t||m|| ̸= 0, then N(m, t, b) = \uf8f1 \uf8f2 \uf8f3 qd−2 + q d−2 2 η � (−1) d 2 (b2 − t||m||) � if d is even, qd−2 − q d−3 2 η � (−1) d−1 2 ||m|| � if d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iii) If ||m|| = 0 = b2 − t||m||, then N(m, t, b) = \uf8f1 \uf8f2 \uf8f3 qd−2 + ν(t)q d−2 2 η � (−1) d 2 ) � if d is even, qd−2 − q d−1 2 η � (−1) d−1 2 t � if d is odd, where ν(t) = −1 if t ∈ F∗ q and ν(0) = q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' (iv) If ||m|| = 0 and b2 − t||m|| ̸= 0, then N(m, t, b) = qd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' See Exercises 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='31–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='34 in [16], or one can prove it by using the discrete Fourier analysis with the explicit value of the Gauss sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 48 By a direct application of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='2, one can find the explicit number of the intersections of two spheres over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Precisely we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Given a non-zero vector m ∈ Fd q and t, j ∈ Fq, let Θ(m, t, j) := {x ∈ St : ||x − m|| = j}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' If m ∈ Sℓ, then |Θ(m, t, j)| = N � m, t, t+ℓ−j 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Since ||x − m|| = t + ℓ − 2m · x for x ∈ St, m ∈ Sℓ, it is clear that Θ(m, t, j) is the number of common solutions x of the following equations: ||x|| = t, m · x = t + ℓ − j 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Hence, by the definition of N, we obtain the required conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Let t ∈ F∗ q and ℓ ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Then, for every non-zero vector m ∈ Sℓ, we have � x∈St:||x−m||=t 1 ∼ qd−2 excepting for the following three cases: 1) d = 2, ℓ ̸= 0, η(tℓ − ℓ2/4) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 2) d = 2, ℓ = 0, η(−1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 3) d = 3, ℓ = 0, η(−t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' For each of those three cases, the value in the above sum takes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' On the other hand, if d = 2, ℓ ̸= 0, and η(tℓ − ℓ2/4) = 1, the value in the above sum is exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' It follows from Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content='3 that for any ||m|| = ℓ, � x∈St:||x−m||=t 1 = N � m, t, ℓ 2 � , and so the corollary is a direct consequence of Lemma 12.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Princeton Mathematical Series, 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Monographs in Harmonic Analysis, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' Princeton University Press, Princeton, NJ, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' xiv+695 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} +page_content=' 51' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAyT4oBgHgl3EQfl_iN/content/2301.00463v1.pdf'} diff --git a/qNFRT4oBgHgl3EQfeDf8/content/tmp_files/2301.13570v1.pdf.txt b/qNFRT4oBgHgl3EQfeDf8/content/tmp_files/2301.13570v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..034b08183277f18e351ffa579002ba46be91c51e --- /dev/null +++ b/qNFRT4oBgHgl3EQfeDf8/content/tmp_files/2301.13570v1.pdf.txt @@ -0,0 +1,2758 @@ +arXiv:2301.13570v1 [math.GR] 31 Jan 2023 +Conjugacy for certain automorphisms of the one-sided +shift via transducers +Collin Bleak and Feyishayo Olukoya +Abstract +We address the following open problem, implicit in the 1990 article Automorphisms +of one-sided subshifts of finite type of Boyle, Franks and Kitchens (BFK): +Does there exists an element ψ in the group of automorphisms of the one- +sided shift Aut({0, 1, . . . , n − 1}N, σn) so that all points of {0, 1, . . . , n − 1}N +have orbits of length n under ψ and ψ is not conjugate to a permutation? +Here, by a permutation we mean an automorphism of one-sided shift dynamical system +induced by a permutation of the symbol set {0, 1, . . . , n − 1}. +We resolve this question by showing that any ψ with properties as above must be +conjugate to a permutation. +Our techniques naturally extend those of BFK using the strongly synchronizing +automata technology developed here and in several articles of the authors and collab- +orators (although, this article has been written to be largely self-contained). +Contents +1 +Introduction +2 +2 +Preliminaries +5 +2.1 +The natural numbers and some of its subsets . . . . . . . . . . . . . . . . . . +5 +2.2 +Words and infinite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.3 +Automata and transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.4 +Increasing alphabet size and the dual automaton . . . . . . . . . . . . . . . . +9 +2.5 +Synchronizing automata and bisynchronizing transducers . . . . . . . . . . . +9 +2.6 +De Bruijn graphs and folded automata . . . . . . . . . . . . . . . . . . . . . +11 +2.7 +Automorphisms of digraphs underlying de Bruijn graphs and Hn . . . . . . . +12 +2.8 +Synchronizing sequences and collapse chains . . . . . . . . . . . . . . . . . . +13 +3 +Minimal actions of finite order elements of Hn +15 +3.1 +Duals and Splits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +3.2 +Notational inconvenience. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +3.3 +Finite order elements of Hn +. . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +1 + +3.3.1 +Building A (A∨ +k) from A . . . . . . . . . . . . . . . . . . . . . . . . . +17 +3.3.2 +Duals, automata, and automorphisms . . . . . . . . . . . . . . . . . . +20 +4 +Water for the witch – shrinking conjugacy class representatives +23 +4.1 +Relabellings and automata sequences . . . . . . . . . . . . . . . . . . . . . . +23 +4.1.1 +Constructing discriminant permutations disc(s, t, Q) . . . . . . . . . . +25 +4.1.2 +Discriminant permutations and amalgamation sequences +. . . . . . . +27 +4.2 +Relabellings along orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.3 +Shadow states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.4 +Relabelling through shadows . . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +5 +Conjugate to an n-cycle +45 +5.1 +An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +1 +Introduction +Let n be a positive integer and set Xn := {0, 1, . . . , n − 1}. We will use Xn to represent our +standard alphabet of size n and we will denote by σn the usual shift map on XN +n . The group +Aut(XN +n , σn) of homeomorphisms of XN +n which commute with the shift map is called the group +of automorphisms of the shift dynamical system. This is a well-studied group in symbolic +dynamics, with the special property (first given by Hedlund in [10]) that if φ ∈ Aut(XN +n , σn) +has (x0x1x2 . . .)φ = y0y1y2 . . . then there is an integer k so that for all indices i, the value yi +is determined by the finite word xixi+1 . . . xi+k. +The paper [6] characterises all of the finite subgroups of the group Aut(XN +n , σn), shows +that this group contains non-abelian free groups whenever n ≥ 3, and investigates other +algebraic structures of the group. The papers [7, 5] develop a conjugacy invariant for the +group Aut(XN +n , σn), arising from the action of the group on periodic words, which for an +automorphism φ we will denote as Sp(φ) (this invariant consists of a tuple: the well-known +gyration and sign functions, together with first return data: bundled data associated to the +permutation representation on prime words of length k). +This article resolves the following open problem, implicit in [6], which Mike Boyle sug- +gested to us for its own sake, and, as a test of our approach. +Let Σn represent the group of permutations of the set Xn. By a mild abuse of language, +we say φ ∈ Aut(XN +n , σn) is a permutation if there is a fixed permutation α ∈ Σn so that +if (x0x1x2 . . .)φ = y0y1y2 . . . then we have yi = (xi)α for all i. We say a permutation is a +rotation if the permutation from Σn is an n-cycle. We can now state the problem: +Does there exist an automorphism ψ ∈ Aut(XN +n , σn) of order n so that all points +of XN +n travel on orbits of size n, where ψ is not conjugate to a rotation? +In [6] Boyle, Franks and Kitchens show that if n is prime then any such ψ is in fact conjugate +to a rotation. We show that the Boyle, Franks, and Kitchens result holds for general n. +2 + +We have written this article so that it is essentially self-contained for general researchers +working with automorphisms of the shift. In particular, we gather definitions and key con- +structions from [15] and [4] here to simplify the presentation without insisting the reader +peruse those articles to follow our discussion. We use the highlighted technology to enhance +the key method in the article [6]. The paper [4] shows how to represent any automorphism +φ of the one-sided shift by a particularly nice family of transducers (finite state machines +that transform inputs sequentially) while [15] investigates the order problem for that same +family of transducers. A key idea of [4] is that any such transducer T representing φ can +be thought of as a triple (D, R, φ∗), where D and R are strongly synchronizing automata +(edge-labelled directed graphs with the particularly nice property of having a synchronizing +sequence) with D representing the domain and R representing the range, and where φ∗ is +an isomorphism of the underlying digraphs Γ(D) and Γ(R) of D and R determined by the +action of φ on periodic words. In the case of a finite order element, the domain and range +automata can also be chosen to be identical. +In the article [6] the central method for studying finite subgroups of Aut(XN +n , σn) is firstly +to find an action of the group on the underlying digraph of an automaton (now understood +to be a strongly synchronizing automaton). Once the first step is accomplished, the group +is decomposed as a composition series where each composition factor is isomorphic to a +subgroup of the symmetric group Σn on n-points. +This is accomplished by pushing the +action down along what is called an “amalgamation sequence” (see Section 4.1.2 here) of +the digraph until one has an action by automorphisms on a particularly nice digraph. The +construction typically requires passing through the automorphism groups of various one-sided +shifts of finite type via topological conjugations induced by the amalgamations. +Our first step simplifies this process. In particular we show that we can always find an +action of a finite subgroup of Aut(XN +n , σn) on the underlying digraph of a strongly synchro- +nizing automaton whose amalgamation and synchronizing sequences cohere (Section 4.1.2), +thus we can push down along the synchronizing sequence of that automaton without needing +to possibly change alphabet. This is already enough, when n is prime, to show that every +element of order p in Aut(XN +n , σn) is conjugate in Aut(XN +n , σn) to a rotation. +However, to answer the open problem above, we need to go beyond this. Suppose φ ∈ +Aut(XN +n , σn) has order n and with the condition (⋆) that all points of XN +n travel on orbits of +size n. It turns out that (⋆) is equivalent to the condition that for any transducer (A, A, φ∗) +representing φ, the action of φ∗ on Γ(A) has the property that for every (based) circuit C of +Γ(A) the orbit length of C under this action is n. (We are using based circuits here to avoid a +circuit returning to itself with some non-trivial rotation as counting as completing the orbit.) +When n is a prime p, it is not hard to see that the action of φ∗ on the underlying digraph is +limited in orbit lengths for edges and vertices to 1 and p. When n is not prime, orbit lengths +of edges and vertices can be any divisor of n even though all circuits have orbit length n. +This last issue creates problems when trying to implement the approach successfully carried +out by Boyle et al for n prime. +We overcome this issue for such a φ with representative transducer (A, A, φ∗) with several +technical lemmas. These aim to show that the automaton A can be “fluffed up” by adding +3 + +shadow states (Section 4.3) to create a new strongly synchronizing automaton B with an +induced and more informative action ψ∗ on Γ(B) so that (B, B, ψ∗) still represents φ. By +‘more informative’ we mean that the correct addition of shadow states results in states and +edges originally on orbits of length < n having resulting orbits of length n. This new action +makes it possible to find a conjugate action of φ on a strongly synchronizing automaton of +strictly smaller size than A (where the conjugacy occurs entirely with Aut(XN +n , σn)). +Our approach can now be summarised as follows. First we conjugate to get a (conjugate) +action of φ on a strongly synchronizing automaton whose synchronizing sequence coheres +with the amalgamation sequence of its underlying digraph. Then we have a series of “fluffing +up” moves followed by reductions via conjugation. Eventually, these processes result in a +conjugate action given by a transducer over a single state automaton with n labelled loops, +where each edge is on an orbit of length n; our original element φ must then be conjugate +to a rotation. +The example in Section 5.1 might prove helpful to the reader as an illustration of our +approach and of the difficulties discussed above. +The property of being a strongly synchronizing automaton is equivalent to that of being +a folded de Bruijn graph. Crucial to the approach we have sketched out is the process: given +a finite order element φ ∈ Aut(XN +n , σn), find the minimal folded de Bruijn graph Γ so that φ +acts faithfully on Γ by automorphisms. The following is essentially a result from [4] stated +in our context (see Lemma 3.4 and Theorem 3.5, below). +Theorem 1.1. Let n ≥ 2 be an integer and suppose φ ∈ Aut(XN +n , σn) is a finite order element. +There is an effective process for determining Γφ, the minimal folded de Bruijn graph on an +n letter alphabet, so that φ induces a natural automorphism of Γφ. +Finally, we can state the theorem which answers the question of Boyle. +Theorem 1.2. Let φ ∈ Aut(XN +n , σn) be an element of finite order. The following are equiv- +alent: +• φ is conjugate to a rotation; +• every element of XN +n is on an orbit of length n under the action of A; and +• for any folded de Bruijn graph Γφ admitting a faithful action by φ via an automorphism +φ∗, every (based) circuit of Γφ is on an orbit of length n under φ∗. +It is unclear at the moment how much our approach depends on the condition that “all +circuits are on orbits of length n”. In work in progress we aim to extend our current ideas +towards resolving the conjugacy problem for finite order elements of Aut(XN +n , σn). +Acknowledgements +The authors are grateful for partial support from EPSRC research grant EP/R032866/1. The +second author is additionally grateful for support from Leverhulme Trust Research Project +Grant RPG-2017-159 and LMS ECF grant ECF-1920-105. Finally, we are also grateful to +Mike Boyle for conversations on the question we address here. +4 + +2 +Preliminaries +2.1 +The natural numbers and some of its subsets +We use the notation N for the set {0, 1, 2, . . .}; for j ∈ N we write Nj for the set {i ∈ N : 1 ≥ j} +of all natural numbers which are bigger than or equal to j. +2.2 +Words and infinite sequences +In this subsection we set up necessary notation for words and sequences. +Firstly, we employ all of the usual notation around finitary words over the alphabet X. +Namely, for a base set X, and natural n, Xn is the set of ordered n-tuples with coordinates +from X. We call these the words of length n (over alphabet X). By convention, we set +X0 := {ε} and we refer to ε as the empty word or empty string, proclaiming this to be +the same object, independent of the (non-empty) set X used as our alphabet. +We set +X∗ := ∪n∈NXn, the words of finite length over X (this is the Kleene-star operator). We also +set X+ := X∗\{ε}, the non-trivial/non-empty finite length words over X. If w ∈ X∗ we set +|w| = n where w ∈ Xn, and we call |w| the length of w. If X has a linear order <, then we +give X∗ the induced dictionary order. If u ∈ Xn then we implicitly set values ui ∈ X for +0 ≤ i < n so that u = (u0, ui, . . . , un−1). In this context, from here forward we will simply +write u = u0u1 . . . un−1. For u ∈ Xn and i ≤ |u|, we write u[1,i] for the prefix u1 . . . ui of u. +Finally, if u, v ∈ X∗, so that u = u0u1 . . . ur−1 and v = v0v1 . . . vs−1 then uv will represent +the concatenation of these words: uv := u0u1 . . . ur−1v0v1 . . . vs−1, which is a word of length +r + s over X. +As in the paper [4], we take X−N +n +:= {. . . x−2x−1x0 | xi ∈ Xn} as our shift space, with +the shift operator σn defined by (xi)i∈−Nσn = (yi)i∈−N where we have yi = xi−1. We use the +characterisation of elements of Hn as strongly synchronizing transducers corresponding to +shift commuting automorphisms of X−N +n . For a finite-length word over Xn we may index this +word with negative or positive indices as seems natural at the time. When we are explicitly +thinking of a finite subword w ∈ Xk +n of a point x ∈ X−N +n +we will ordinarily index w as +w = wi−k+1wi−k+2 . . . wi for some i ∈ −N. +Suppose k is a positive integer and u = u−(k−1)u−(k−2) . . . u−1u0 ∈ Xk +n. Define uω ∈ X−N +n , +by which notation we mean the point . . . xmxm−1 . . . x−1x0 =: x where xi = ui (mod k). The +word x ∈ X−N +n +is called a periodic word. The period of x is the smallest j ∈ N such that +(x)σj +n = x. If the length |u| is the period of the word x, then u is called prime. Alternatively +u is prime if there is no smaller word γ ∈ X+ +n such that u = γi for some i ≥ 2. +Write Xk +n for the full set of prime words of length k over the alphabet Xn. +Given two words u, v ∈ X+ +n such that |u| = |v| = r, we call v a rotation of u if there is +an i ∈ N with (uω)σi +n = vω. In this case, we may refer to v as the ith-rotation of u (even if +i > |u|). +It is a well-known fact that an element φ ∈ Aut(X−N +n , σn) preserves the period of a +periodic element of X−N +n . In this way, the action of Aut(X−N +n , σn) on periodic words gives a +representation from Aut(X−N +n , σn) to the group Πk∈N Sym(Xk +n). For φ ∈ Aut(X−N +n , σn), write +5 + +φk for the action of φ on prime words of length k and write φ for the element (φk)k∈N ∈ +Πk∈N Sym(Xk +n). The map φ is the periodic point representation of Aut(X−N +n , σn), introduced +in [7]. +2.3 +Automata and transducers +An automaton, in our context, is a triple A = (XA, QA, πA), where +(a) XA is a finite set called the alphabet of A (we assume that this has cardinality n, and +identify it with Xn, for some n); +(b) QA is a finite set called the set of states of A; +(c) πA is a function XA × QA → QA, called the transition function. +The size of an automaton A is the cardinality of its state set. We use the notation |A| +for the size of the A. +We regard an automaton A as operating as follows. If it is in state q and reads symbol +a (which we suppose to be written on an input tape), it moves into state πA(a, q) before +reading the next symbol. As this suggests, we can imagine that the automaton A is in the +middle of an input word, reads the next letter and moves to the right, possibly changing +state in the process. +We can extend the notation as follows. For w ∈ Xm +n , let πA(w, q) be the final state of the +automaton which reads the word w from initial state q. Thus, if w = x0x1 . . . xm−1, then +πA(w, q) = πA(xm−1, πA(xm−2, . . . , πA(x0, q) . . .)). +By convention, we take πA(ε, q) = q. +For a given state q ∈ QA, we call the automaton A which starts in state q an initial +automaton, denoted by Aq, and say that it is initialised at q. +An automaton A can be represented by a labeled directed graph GA, whose vertex set +VA is QA. For this directed graph there is a directed edge labeled by a ∈ XA from p to q +if πA(a, p) = q. Representing this, we determine the set EA of edges of GA to be the set of +triples +EA := {(p, a, q) | ∃p, q ∈ QA, a ∈ XA, so that πA(a, p) = q}. +In what follows, the labelled directed graph GA will be referred to as the underlying digraph +for the automaton A. +A transducer is a quadruple T = (XT, QT, πT, λT), where +(a) (XT, QT, πT) is an automaton; +(b) λT : XT × QT → X∗ +T is the output function. +6 + +Formally such a transducer is an automaton which can write as well as read; after reading +symbol a in state q, it writes the string λT(a, q) on an output tape, and makes a transition +into state πT(a, q). Thus, the size of a transducer is the size of its underlying automaton. +An initial transducer Tq is simply a transducer which starts processing input from state q. +Transducers which are synchronous (i.e., which always write one letter whenever they read +one letter) are also known as Mealy machines (see [9]), although we generally will not use that +language here. Transducers which are not synchronous are described as asynchronous when +this aspect of the transducer is being highlighted. In this paper, we will only work with +synchronous transducers without an initial state, and, henceforth we simply call these +transducers. +In the same manner as for automata, we can extend the notation to allow transducers +to act on finite strings: we let πT (w, q) and λT(w, q) be, respectively, the final state and the +concatenation of all the outputs obtained when a transducer T reads a string w from a state +q. +A transducer T can also be represented as an edge-labeled directed graph. Again the +vertex set is QT ; now, if πT(a, q) = r, we put an edge with label a|λT(a, q) from q to r. In +other words, the edge label describes both the input and the output associated with that +edge. We call a the input label of the edge and λT(a, q) the output label of the edge. +For example, Figure 1 describes a synchronous transducer over the alphabet X2. +a1 +a2 +0|0 +1|0 +1|1 +0|1 +Figure 1: A transducer over X2 +In what follows, we only use the language automaton for those automata which are not +transducers. This allows us characterise a synchronous transducer T as a pair of automata +together with a directed graph isomorphism “gluing” the two automata together as a domain +automaton and a range automaton (we split any edge label ‘x|y’ of T as specifying the domain +automaton edge with label x and the range automaton edge with label y). +We can regard any state q of a transducer as acting on an infinite string from XN +n where +Xn is the alphabet. This action is given by iterating the action on a single symbol; so the +output string is given by +λT(xw, q) = λT(x, q)λT(w, πT(x, q)). +Thus Tq induces a map w �→ λT(w, q) from XN +n to itself; it is easy to see that this map is +continuous. If it is a homeomorphism, then we call the state q a homeomorphism state. We +write Im(q) for the image of the map induced by Tq. +7 + +Two states q1 and q2 are said to be ω-equivalent if the transducers Tq1 and Tq2 induce the +same continuous map. (This can be checked in finite time, see [9].) More generally, we say +that two initial transducers Tq and T ′ +q′ are ω-equivalent if they induce the same continuous +map on XN +n . +A transducer is said to be minimal if no two states are ω-equivalent. For a transducer +T, two states q1 and q2 are ω-equivalent if λT(a, q1) = λT(a, q2) for any finite word a ∈ X∗ +n. +Moreover, if q1 and q2 are ω-equivalent states of a synchronous transducer, then for any finite +word a ∈ Xp +n, πT(a, q1) and πT(a, q2) are also ω-equivalent states. +Two minimal non-initial transducers T and U are said to be ω-equal if there is a bijection +f : QT → QU, such that for any q ∈ QT , Tq is ω-equivalent to U(q)f. Two minimal initial +transducers Tp and Uq are said to be ω-equal if they are ω-equal as non-initial transducers and +there is a bijection f : QT → QU witnessing this which satisfies the equality (p)f = q. We use +the symbol ‘=’ to represent ω-equality of initial and non-initial transducers. Two non-initial +transducers T and U are said to be ω-equivalent if they have ω-equal minimal representatives, +and in this case we might instead say T and U represent the same transformation. +In the class of synchronous transducers, the ω-equivalence class of any transducer has a +unique minimal representative. +Throughout this article, as a matter of convenience, we shall not distinguish between ω- +equivalent transducers. Thus, for example, we introduce various groups as if the elements of +those groups are transducers, whereas the elements of these groups are in fact ω-equivalence +classes of transducers. +Given two transducers T = (Xn, QT, πT, λT) and U = (Xn, QU, πU, λU) with the same +alphabet Xn, we define their product T ∗ U. The intuition is that the output for T will +become the input for U. Thus we take the alphabet of T ∗ U to be Xn, the set of states to +be QT∗U = QT × QU, and define the transition and rewrite functions by the rules +πT∗U(x, (p, q)) += +(πT(x, p), πU(λT(x, p), q)), +λT∗U(x, (p, q)) += +λU(λT(x, p), q), +for x ∈ Xn, p ∈ QT and q ∈ QU. Here we use the earlier convention about extending λ and +π to the case when the transducer reads a finite string. If T and U are initial with initial +states q and p respectively then the state (q, p) is considered the initial state of the product +transducer T ∗ U. +In automata theory a synchronous (not necessarily initial) transducer T = (Xn, QT, πT, λT) +is invertible if for any state q of T, the map ρq := λT(�, q) : Xn → Xn is a bijection. In this +case the inverse of T is the transducer T −1 with state set QT −1 := {q−1 | q ∈ QT }, transition +function πT −1 : Xn ×QT −1 → QT −1 defined by (x, p−1) �→ q−1 if and only if πT((x)ρ−1 +p , p) = q, +and output function λT −1 : Xn ×QT −1 → Xn defined by (x, p) �→ (x)ρ−1 +p . Thus, in the graph +of the transducer T we simply switch the input labels with the output labels and append +‘−1’ to the state names. +We are concerned only with invertible, synchronous transducers in this article. +8 + +2.4 +Increasing alphabet size and the dual automaton +We require a couple of standard constructions in the theory of synchronous automata in this +work. +First we consider the ‘paths to letters’ construction. Let T be a transducer over the +alphabet Xn. Let m ∈ N1. Write T(m) for the transducer over the alphabet Xm +n with state +set QT and transition and output functions πT(m), λT(m) satisfying the following conditions. +For x ∈ Xm +n and q ∈ QT we set πT(m)(x, q) = p if and only if πT (x, q) = p in T; we set +λT(m)(x, q) := λT(x, q). It is clear that if T is minimal and invertible, the T(m) is also +minimal and invertible. +The other construction we require is the dual automaton (see [1, 14]). +Again let T be a transducer over the alphabet Xn. Set T ∨ = ⟨QT, Xn, π∨ +T, λ∨ +T⟩, that is the +state set of T ∨ is the set Xn, the alphabet of T ∨ is the state set QT of T, and the transition +π∨ +T and output functions λ∨ +T are defined as follows. For q ∈ QT and x ∈ Xn, π∨ +T (q, x) = y +and λ∨ +T(q, x) = p if and only if πT(x, q) = p and λT(x, q) = y. +There is a connection between the two constructions. The following is standard in the +theory of synchronous automata and provides a key insight in the analysis of [1]. +Lemma 2.1. Let T be a synchronous transducer over alphabet Xn. For positive natural m, +we have (T ∨)m = T(m)∨. +Note that to lighten our notation below, we may use the notation T ∨ +m for the transducer +T(m)∨. +Also observe that T −1∨ is obtained from T ∨ by ‘reversing the arrows’. That is if, x, y ∈ +Xn, q, p ∈ QT are such that π∨ +T(q, x) = y and λ∨(q, x) = p, then π∨ +T −1(q−1, y) = x and +λ∨(q−1, y) = p−1. +2.5 +Synchronizing automata and bisynchronizing transducers +Given a natural number k, we say that an automaton A with alphabet Xn is synchronizing +at level k if there is a map sk : Xk +n �→ QA such that, for all q and any word w ∈ Xk +n, we have +πA(w, q) = sk(w). In other words, A is synchronizing at level k if, after reading a word w of +length k from a state q, the final state depends only on w and not on q. (Again we use the +extension of πA to allow the reading of an input string rather than a single symbol.) We call +sk(w) the state of A forced by w; the map sk is called the synchronizing map at level k. An +automaton A is called strongly synchronizing if it is synchronizing at level k for some k. +We remark here that the notion of synchronization occurs in automata theory in consid- +erations around the ˇCern´y conjecture, in a weaker sense. A word w is said to be a reset word +for A if πA(w, q) is independent of q; an automaton is called synchronizing if it has a reset +word [16, 2]. Our definition of “synchonizing at level k”/“strongly synchronizing” requires +every word of length k to be a reset word for the automaton. +If the automaton A is synchronizing at level k, we define the core of A to be the maximal +sub-automaton with set of states those states in the image of the map s. +It is an easy +observation that, if A is synchronizing at level k, then its core is an automaton in its own +9 + +right using the same alphabet, and is also synchronizing at level k. We denote this automaton +by core(A). We say that an automaton or transducer is core if it is equal to its core. +Clearly, if A is synchronizing at level k, then it is synchronizing at level l for all l ≥ k. +Let Tq be an initial transducer which is invertible with inverse T −1 +q . If Tq is synchronizing +at level k, and T −1 +q +is synchronizing at level l, we say that Tq is bisynchronizing at level (k, l). +If Tq is invertible and is synchronizing at level k but not bisynchronizing, we say that it is +one-way synchronizing at level k. +For a non-initial invertible transducer T we also say T is bi-synchronizing (at level (k, l)) +if both T and its inverse T −1 are synchronizing at levels k and l respectively. +Note that if T is a strongly synchronizing transducer, then for any m ∈ N, T(m) is also +strongly synchronizing. Moreover, if k the minimal synchronizing level of T, then T(m) is +synchronizing at level 1 for any m ≥ k and, more generally, is synchronizing at level ⌈k/m⌉. +Notation 2.2. Let T be a transducer which is synchronizing at level k and let l ≥ k be +any natural number. Then for any word w ∈ Xl +n, we write qw for the state sl(w), where +sl : Xl +n → QT is the synchronizing map at level l. +The following result was proved in Bleak et al. [3]. +Proposition 2.3. Let T, U be transducers which (as automata) are synchronizing at levels +j, k respectively, Then T ∗ U is synchronizing at level j + k. +Note that in the statement of Proposition 2.3, the lowest synchronizing level of T ∗ U +might actually be less than j + k. +Let T be a transducer which (regarded as an automaton) is synchronizing at level k, then +the core of T (similarly denoted core(T)) induces a continuous map +fT : X−N +n +→ X−N +n +as follows. Let x ∈ X−N +n +and set y ∈ X−N +n +to be the sequence defined by +yi = λT(xi, qxi−kxi−(k−1)...xi−1). +Note that +πT(xi, qxi−kxi−(k−1)...xi−1) = qxi−kxi−(k−1)...xi−1. +Set +(x)fT = y. +Thus, from the point of view of the transition function of T we in fact begin processing x at +−∞ and move towards x0. (This is in keeping with our interpretation of transducer as repre- +senting machines applying sliding block codes, where here, we are thinking of Aut(X−N +n , σn) +as consisting of the sliding block code transformations that require past information only to +determine what to do with a digit.) Note, moreover, that the map fT is independent of the +(valid) synchronizing level chosen to define it. We have the following result: +10 + +Proposition 2.4. [4] Let T be a minimal transducer which is synchronizing at level k and +which is core. Then fT ∈ End(X−N +n , σn). +The transducer in Figure 1 induces the shift map on X−N +n . +In [3], the authors show that the set � +Hn of minimal finite synchronizing invertible syn- +chronous core transducers is a monoid; the monoid operation consists of taking the product of +transducers and reducing it by removing non-core states and identifying ω-equivalent states +to obtain a minimal and synchronous representative. +Let Hn be the subset of � +Hn consisting of transducers which are bi-synchronizing. A chief +result of [4] is that Aut(X−N +n , σn) ∼= Hn. +2.6 +De Bruijn graphs and folded automata +The de Bruijn graph G(n, m) can be defined as follows, for integers m ≥ 1 and n ≥ 2. The +vertex set is Xm +n , where Xn is the alphabet {0, . . . , n−1} of cardinality n. There is a directed +arc from a0 . . . am−1 to a1a2 . . . am, with label am. +Note that, in the literature, the directed edge is also from a0a1 . . . am−1 to a1 . . . am−1am +and the label on this edge is often given as the (m + 1)-tuple a0a1 . . . am−1am. However, the +labelling given above produces an isomorphic graph and is better suited for our purposes. +Figure 2 shows the de Bruijn graph G(3, 2). +Observe that the de Bruijn graph G(n, m) describes an automaton over the alphabet Xn. +Moreover, this automaton is synchronizing at level m: when it reads the string b0b1 . . . bm−1 +from any initial state, it moves into the state labeled b0b1 . . . bm−1. +The de Bruijn graph is, in a sense we now describe, the universal automaton over Xn +which is synchronizing at level m. +We define a folding of an automaton A over the alphabet Xn to be an equivalence relation +≡ on the state set of A with the property that, if a ≡ a′ and πA(x, a) = b, πA(x, a′) = b′, +then b ≡ b′. That is, reading the same letter from equivalent states takes the automaton to +equivalent states. If ≡ is a folding of A, then we can uniquely define the folded automaton +A/≡: the state set is the set of ≡-classes of states of A; and, denoting the ≡-class of a by +[a], we have πA/≡(x, [a]) = [πA(x, a)] (note that this is well-defined). +Proposition 2.5. [4] The following are equivalent for an automaton A on the alphabet Xn: +• A is synchronizing at level m, and is core; +• A is the folded automaton from a folding of the de Bruijn graph G(n, m). +We may think of a de Bruijn graph G(n, m) as determining a finite category, with ob- +jects the foldings of G(n, m) and with arrows digraph morphisms which commute with the +transition maps of the given automata. It is immediate in that point of view that all such +arrows are surjective digraph morphisms (and indeed, these are folding maps). +11 + +00 +11 +22 +01 +10 +02 +20 +12 +21 +0 +1 +2 +1 +0 +2 +2 +1 +0 +1 +0 +2 +1 +0 +2 +0 +1 +2 +0 +2 +1 +2 +0 +1 +0 +2 +1 +Figure 2: The de Bruijn graph G(3, 2). +2.7 +Automorphisms of digraphs underlying de Bruijn graphs and +Hn +In this section we describe finite order elements of Hn as automorphisms of folded de Bruijn +graphs. +Let A be a finite automaton on edge-alphabet Xn. Recall (Section 2.3) that an automaton +A may be regarded as labeled directed graph with vertex set QA, and edge set EA ⊂ QA × +Xn × QA. We let GA denote the unlabeled directed graph corresponding to an automaton +A, but we retain the triple (p, x, q) to denote the edge of GA underlying the edge (p, x, q) of +A. +Let φ be an automorphism of the directed graph GA. Let H(A, φ) be a transducer with +• state set QH(A,φ) := QA, +12 + +• alphabet set Xn, +• transition function πH(A,φ) := πA, and +• output function λH(A,φ) : Xn × QH(A,φ) → Xn, +where λH(A,φ)(x, p) = y if and only if there are edges (p, x, q) and (r, y, s) of GA so that +(p, x, q) is taken to (r, y, s) by φ. +The transducer H(A, φ) can be thought of as the result of gluing the automaton A to a +copy of itself along the map φ. That is, if p, q ∈ QA and (p, x, q) is an edge from p to q with +label x in A, and if y is the label of the edge ((p, x, q))φ in A, then the vertex p is identified +with the vertex (p)φ, the vertex q with the vertex (q)φ, the edge (p, x, q) is identified with +the edge ((p, x, q))φ and has input label x and the output label y. +Remark 2.6. We make a few observations: +(a) For each state q ∈ QH(A,φ), the map λH(A,φ)(�, q) : Xn → Xn is a bijection. +This +follows from the definition of GA: for each x ∈ Xn there is precisely one edge of the +form ((q)φ, x, p) based at the vertex (q)φ. It follows that the transducer H(A, φ) is +invertible. +(b) If A is synchronizing at level k (and so a folding of G(n, k) by Proposition 2.5) then both +H(A, φ) and H(A, φ)−1 are synchronizing at level k hence the minimal representative +H(A, φ) of H(A, φ) is an element of Hn. +(c) In fact, for a state q ∈ QA, if Wk,q is the set of words of length k, that force the state +q, i.e., +Wk,q := {a ∈ Xk +n : πH(A,φ)(a, q) = q}, +then {λH(A,φ)(a, p) | a ∈ Qk,q, p ∈ QH(A,φ)} is equal to Wk,(q)φ. +(d) An element of Hn which can be represented by a transducer H(A, φ) for some folded +de Bruijn graph A and digraph automorphism φ of GA must have finite order. +If A ∈ Hn and B is an automaton so that there is a digraph automorphism φ : GB → GB +so that A and H(B, φ) represent the same transformation then we say A is induced from +(B, φ). +2.8 +Synchronizing sequences and collapse chains +We require an algorithm given in [3] for detecting when an automaton is strongly synchro- +nizing. We state a version below. +Let A = (Xn, QA, πA) be an automaton. Define an equivalence relation ∼A on the states +of A by p ∼A q if and only if the maps πA(·, p) : QA → QA and πA(·, q) : QA → QA are +equal. For a state q ∈ QA let q represent the equivalence class of q under ∼A. Further set +QA := {q | q ∈ QA} and let πA : QA → QA be defined by πA(x, q) = p where p = πA(x, q). +13 + +Observe that πA is a well defined map. Define a new automaton A = (Xn, QA, πA) noting +that |QA| ≤ |QA| and |QA| = |QA| implies that A is isomorphic to A. +Given an automaton A, let A0 := A, A1, A2, . . . be the sequence of automata such that +Ai = Ai−1 for all i ≥ 1. We call the sequence (Ai)i∈N the synchronizing sequence of A. We +make a few observations. +By definition each term in the synchronizing sequence is a folding of the automaton +which precedes it, therefore there is a j ∈ N such that all the Ai for i ≥ j are isomorphic to +one another. By a simple induction argument, for each i, the states of Ai corresponds to a +partition of QA. We identify the states of Ai with this partition. For two states q, p ∈ QA +that belong to a state P of Ai, πA(x, q) and πA(x, p) belong to the same state of QAi for all +x ∈ Xn. We will use the language ‘two states of A are identified at level i’ if the two named +states belong to the same element of QAi. +If the automaton A is strongly synchronizing and core, then an easy induction argument +shows that all the terms in its synchronizing sequence are core and strongly synchronizing +as well (since they are all foldings of A). For example if A = G(n, m), then the first m terms +of the synchronizing sequence of A are (G(n, m), G(n, m − 1), G(n, m − 2), . . . , G(n, 1), after +this all the terms in the sequence are the single state automaton on Xn. +The result below is from [3]. +Theorem 2.7. Let A be an automaton and A0 := A, A1, A2, . . . be the sequence of automata +such that Ai = Ai−1 for all i > 1. Then +(a) a pair of states p, q ∈ QA, belong to the same element t ∈ QAi if and only if for all +words a ∈ Xi +n, πA(a, p) = πA(a, q), and +(b) A is strongly synchronizing if and only if there is a j ∈ N such that |QAj| = 1. The +minimal j for which |Aj| = 1 is the minimal synchronizing level of A. +We also require the notion of a collapse chain from [4]. Let A and B be strongly syn- +chronizing automata. Let A = A0, A1, . . . , Ak = B be a sequence such that Ai+1 is obtained +from Ai by identifying pairs of states p ∼Ai q. We note that as distinct from the synchro- +nizing sequence, we do not necessarily make all possible identifications. Such a sequence is +called a collapse chain if at each step, we make the maximal number of collapses possible +relative to the final automaton B. That is, for u, v ∈ QA belonging to the same state of +B, in the minimal Ai such that [u] ∼Ai [v], we have [u] = [v] in Ai+1. We note that this +condition means that a collapse chain is unique. Therefore, for B a strongly synchroniz- +ing automaton, we say that B belongs to a collapse chain of A if there is a collapse chain +A = A0, A1, . . . , Ak = B. In this case, we call the collapse chain A = A0, A1, . . . , Ak = B, +the the collapse chain from A to B. If B is a single state automaton, the collapse chain from +A to B is precisely the strongly synchronizing sequence of A. Thus a collapse chain can be +thought of as a synchronizing sequence relative to its end point. +The following facts are straightforward. Let A be a strongly synchronizing automaton, +and B be an automaton which is a folding of A, then there is a collapse chain from A to B. +Therefore B belongs to a collapse chain of A if and only if B is a folding of A. In particular +14 + +if B belongs to a collapse chain of A, then B is synchronizing at the minimal synchronizing +level of A. +The following result about collapse chains is proved similarly to Theorem 2.7. +Theorem 2.8. Let A be an automaton and B be a folding of A. Let A0 := A, A1, A2, . . . , Am = +B be the collapse chain from A to B. Then a pair of states p, q ∈ QA belong to the same +element t ∈ QAi if and only if p, q belong to the same state of QB and for all words a ∈ Xi +n, +πA(a, p) = πA(a, q). +3 +Minimal actions of finite order elements of Hn +For this section, we will work using facts related to dual transducers for strongly synchroniz- +ing transducers. +It has been shown in [12, 1, 14] that the dual T ∨ transducer for a synchronous transducer +T contains much information about the order of T, but implicit in those works also, much +information about the conjugacy class of T. +In [15] the dual is considered for strongly +synchronizing transducers, where it is shown that for infinite order strongly synchronizing +transducers the powers of the dual grow in size asymptotically exponentially while for finite +order transducers the dual generates a finite semigroup with a zero. In this section we bring +in some of the methods and results of those works. See [4, 15] for more details than we give +below. +3.1 +Duals and Splits +Recall our definition of the dual of a transducer from Subsection 2.4. We will mostly be +working in a power of the dual of a transducer T, below. +We introduce the following notation. Let T be a strongly synchronizing transducer, and +q ∈ QT be a state. Then we write Wq for the set of words γ ∈ X+ +n such that the map +πT(γ, ·) : QT → QT has image {q}. +Let A be an element of Hn, with synchronizing level k. +Then for r ≥ k, A∨ +r has a +split ((p1, . . . , pl), (q1 . . . , ql), Γ) if and only if the following depiction (see Figure 3) of the +transitions in A∨ +r at the state Γ is valid: +More formally, we have the following. +Definition 3.1 (Splits). Let A be an element of Hn, with synchronizing level k and let +r ≥ k. Suppose there are +• l ∈ N1, +• elements (p1, p2, . . . , pl), (q1, q2, . . . , ql), (s1, s2, . . . , sl) ∈ Ql +A, +• a word Γ ∈ Xr +n ∩ Ws1, and +• distinct states t1, t2 ∈ QA +15 + +Γ +Γ1 +Λ1 +Λ2 +Γ2 +Λl−1 +Γl−1 +Λl +Γl +p1|s1 +q1|s1 +p2|s2 +q2|s2 +ql|sl +pl|sl +∗|t1 +♯|t2 +Figure 3: A split; the symbols ∗ and ♯ represent arbitrary elements of QA. +such that when we define sequences Γ1, Γ2, . . . , Γl and Λ1, Λ2, . . . , Λl by +• Γ1 = λA(Γ, p1) and Λ1 = λA(Γ, q1), and +• for 1 < i ≤ l, Γi = λA(Γi−1, pi) and Λi = λA(Λi−1, qi), +then Γi, Λi ∈ Wsi+1 for all 1 ≤ i ≤ l − 1, Γl ∈ Wt1 and Λl ∈ Wt2. In this case we say that A∨ +r +splits. +We also say that the l-tuples (p1, . . . , pl) and (q1, . . . , ql) split A∨ +r (at Γ). We call {p1, q1} +the top of the split, {t1, t2} the bottom of the split, and the triple ((p1, . . . , pl), (q1 . . . , ql), Γ) +a split of A∨ +r (of length l). +N.B.: if we took r < k in the definition of a split above, then there is no guarantee that +some word Γ of length r would not even be synchronizing, and also no guarantee of the +existence of any synchronizing word of length r, so the definition above breaks down. +The following concept appears implicitly in the proof of Lemma 3.8. +Definition 3.2. Let A be an element of Hn, with synchronizing level k. Let r ≥ k and +((p1, . . . , pl), (q1 . . . , ql), Γ) be a split of A∨ +r . Let {t1, t2} be the bottom of this split. Then +we say that the bottom of the split ((p1, . . . , pl), (q1 . . . , ql), Γ) depends only on the top if the +following conditions hold for any other tuples U1, U2 ∈ Ql−1 +A : +• the triple ((p1, U1), (q1, U2), Γ) is also a split with bottom {t1, t2} and, +• if λAl(Γ, (p1, . . . , pl)) ∈ Wt1 and λAl(Γ, (q1, . . . , ql)) ∈ Wt2 then λAl(Γ, (p1, U1)) ∈ Wt1 +and λAl(Γ, (q1, U2)) ∈ Wt2, and vice-versa. +Observe that if, for r ≥ k, A∨ +r has a split ((p1, . . . , pl), (q1 . . . , ql), Γ) whose bottom de- +pends only on the top, then p1 ̸= q1. +Splitting length as defined below is used explicitly in Lemma 3.8. +Definition 3.3. For a transducer A, we define the r-splitting length of A (for r greater than +or equal to the minimal synchronizing length) to be minimal l such that there is a split of +A∨ +r of length l. If there is no such split then we set the r-splitting length of A to be ∞. +16 + +Note that if, for r ≥ k, A∨ +r has r-splitting length l < ∞, then any split of length l has +the property that the bottom depends only on the top as otherwise one can find a shorter +split (see [15]). +3.2 +Notational inconvenience. +We are soon to run into some collisions of notation. +Firstly, if φ ∈ Aut(X−N +n , σn), then we can represent φ by a (minimal) transducer Aφ ∈ Hn. +Secondly, if A ∈ Hn, then A represents an element φA ∈ Aut(X−N +n , σn). +Finally, if φ ∈ Aut(X−N +n , σn) has finite order, then as we will see from Theorem 3.5 there +is an automaton A (A∨ +k) and an automorphism ψ of the underlying digraph GA (A∨ +k ) so that +Aφ and H(A (A∨ +k ), ψ) represent the same element. It happens that there is a way to define +ψ from φ, and also, from Aφ. Similarly, we could have begun this paragraph with an element +A ∈ Hn, in which case ψ would be defined from both φA and from A. +In order to unify our notation here, we will simply denote ψ in the above situation as φA. +This of course means that φA will represent two different things (an automorphism of the +one-sided shift, or alternatively, an automorphism of a digraph underlying a folded de Bruijn +graph). We hope that confounding the notation in this way will not cause confusion as it +should be clear what is meant from context, noting as well that the digraph homomorphism +φA is the induced digraph homomorphism that arises on GA (A∨ +k ) by considering how φ maps +infinite paths on A (A∨ +k) to other infinite paths on A (A∨ +k). +3.3 +Finite order elements of Hn +In this subsection we build, for a finite order element A ∈ Hn and corresponding φA ∈ +Aut(X−N +n , σn), the minimal strongly synchronizing automaton A (A∨ +k ) which φA can act on as +an automorphism of the underlying directed graph with A being the minimal representative +of H(A (A∨ +k ), φA). +Note that we will retain the notation φA for both the element of Aut(X−N +n , σn) correspond- +ing to A as well as the digraph automorphism φA that is induced by this automorphism of +the shift. +Note that the process of determining Aφ ∈ Hn from a given element φ ∈ Aut(X−N +n , σn) is +not difficult: one simply relates states to different maps as determined by the fixed viewing +window (it is common for differing viewing-window strings to correspond to the same map, +the set of all such strings corresponding to “same” map can be used effectively as the name +of that state) and then one records the local letter transformations as the edge labels. For +details, see [4]. +3.3.1 +Building A (A∨ +k) from A +Let A ∈ Hn be of finite order. +In this subsubsection, we learn a process for building a strongly synchronizing automaton +A (A∨ +k) so that φA acts on the underlying digraph of A (A∨ +k) by automorphisms in such a way +17 + +that H(A (A∨ +k ), φA) has A as its minimal representative transducer. This process is essential +in the proof that follows for finding simplified elements in the conjugacy class representing +our given finite order element A. We will also given an example of the process for a specific +element. +In [15, Proposition 4.15] it is shown that there is k ∈ N such that A∨ +k is the zero of +the semigroup generated by A∨. Fix the minimal such k ∈ N so that A∨ +k is the zero of the +semigroup generated by A∨, and let A∨ +k be the minimal representative of A∨ +k. +The following is a very useful fact from [15]: for every state [γ] of the zero A∨ +k , there is +a word W([γ]) ∈ Q+ +A such that for any input word s ∈ Q+ +A, the output when s is processed +from the state [γ] of A∨ +k is the word (W([γ]))lW([γ])[1,m], where, |s| = l|W([γ])| + m and +W([γ])[1,m] is the length m prefix of (W([γ])). It follows from this that A∨ +k has the following +structure: for each state [γ] (for γ ∈ Xk +n) there is q[γ] ∈ QA so that for all p ∈ QA we have +• πA∨ +k (p, [γ]) = [γ] · A, and +• λA∨ +k (p, [γ]) = q[γ]. +We will call this the |QA|-parallel cycle structure of A∨ +k, or less formally, the cyclical structure +of A∨ +k. +Form the automaton A (A∨ +k) as follows. The states of A (A∨ +k ) are the states [γ] of A∨ +k +and the transitions are given by the rule that for x ∈ Xn, and [γ] a state of A∨ +k, we set +πA (A∨ +k )(x, [γ]) = [γ[2,|γ|]x]. (For this construction it does not matter which γ ∈ Xk +n one picks +from the class [γ], even though we use γ explicitly in the formula of the transition function: +this follows since states of A∨ +k are ω-equivalent classes of A∨ +k .) +The following lemma is immediate from the definitions: +Lemma 3.4. If A ∈ Hn be of finite order and let k ∈ N so that A∨ +k is the zero of the +semigroup generated by A∨ then the automaton A (A∨ +k ) is strongly synchronizing at level k. +We also have the following translation (into our context) of the statement of Theorem +4.5 of [4], where A(G) in that theorem corresponds to A (A∨ +k ) here, and where the group G +there is the group ⟨A⟩ here. +Theorem 3.5. Let A ∈ Hn be of finite order and let k ∈ N so that A∨ +k is the zero of the +semigroup generated by A∨. +(a) A acts as an automorphism φA of the digraph underlying A (A∨ +k) by mapping an edge +([γ], x, [γ[2,|γ|]x]) to the edge (([γ])A, λA(x, qγ), ([γ[2,|γ|]x])A) where ([γ])A = [λA(γ, q)] +for some q ∈ QA, +(b) The minimal representative of the transducer H(A (A∨ +k ), φi +A) is the transducer Ai. +Example 3.6. Consider the transducer A below. The transducer A of Figure 4 is bi-synchronizing +at the second level. The level 2 dual has 36 nodes and so we shall not give this below. How- +ever utilising the AutomGrp package [13] in GAP [8], together with (in AutomGrp) the +18 + +a0 +a1 +a2 +a3 +1|4 0|5 +2|0 +3|1 +4|2 5|3 +4|2 +5|3 +0|4 1|5 +2|1 +3|0 +1|4 +0|5 +4|2 5|3 +2|1 3|0 +1|5 +0|4 +2|0 3|1 +4|2 5|3 +Figure 4: An element A ∈ H6 of order 6. +function “MinimizationOfAutomaton( )” which returns an ω-equivalent automaton, applied +to the second power of the dual automaton, we get the result A∨ +2 , depicted in Figure 5 (which +is the zero of the semigroup generated by the dual): +q0 +q1 +q2 +∗|a1 +∗|a3 +∗|a2 +p0 +p1 +p2 +∗|a0 +∗|a3 +∗|a2 +Figure 5: The level 2 dual of A. +Considering the states {q0, q1, q2, p0, p1, p2} of A∨ +2 as a partition of the words of length 2 +over the alphabet {0, 1, 2, 3, 4, 5}, it is easy to see that +q0 = {00, 01, 10, 11, 40, 41, 50, 51} +p0 = {20, 21, 30, 31} +q1 = {24, 25, 34, 35, 44, 45, 54, 55} +p1 = {04, 05, 15, 15} +q2 = {02, 03, 12, 13, 22, 23, 32, 33} +p2 = {42, 43, 52, 53} +by verifying the 36 transitions from q0 in A using these input words, and cross-checking +state-change results against the transitions of A∨ +2 . From this we can calculate the transitions +of A (A∨ +2 ), with the resulting automaton depicted in 6. +19 + +p0 +q0 +q2 +q1 +p1 +p2 +1 +0 +2 +3 +4 +5 +4 +5 +0 +1 +2 +3 +1 +0 +4 +5 +2 +3 +1 +0 +2 +3 +4 +5 +2 +3 +5 +4 +1 +0 +0 +1 +3 +2 +4 +5 +Figure 6: The automaton A (A∨ +2 ) +Notice that both the domain and range automaton of A are foldings of A (A∨ +2 ). This +phenomenon generalises, that is, for a strongly synchronizing transducer A representing an +element of Hn of finite order, both the domain and range automata of A are foldings of +A (A∨ +k) (where k is appropriately chosen). +We revisit this example in Section 5.1 where we show that A is conjugate to a 6-cycle. +⃝ +3.3.2 +Duals, automata, and automorphisms +In this subsubsection, we will prove that for finite order A ∈ Hn and minimal k so that A∨ +k +is the zero of the semigroup generated by A∨, that A (A∨ +k ) as defined above is the minimal +(strongly synchronizing) automaton so that A can act on A (A∨ +k) as an automorphism φA, +with (A (A∨ +k ), φA) inducing A. +We first require lemmata exploring the relationship between properties of A (A∨ +k ) and of +20 + +A∨ +k. +Our first step is the following useful lemma about the automaton A (A∨ +k) constructed as +above from a finite order element A ∈ Hn. In essence, it says that if two states [δ] and [γ] +are distinct in A (A∨ +k) but their two transition functions are the same, then by following the +cycles of the level k dual (by iteratively acting by A), we will eventually get to a pair of +states which have different output letters in the level k dual, and at that pair of locations, the +states of A (A∨ +k) will still transition the same way, but the output functions of H(A (A∨ +k), φA) +at these states will disagree at the first letter. +Lemma 3.7. Let A ∈ Hn be finite order. Let γ, δ ∈ Xk +n be such that the states [γ], [δ] +of A (A∨ +k ) are distinct. Suppose moreover that the maps πA (A∨ +k )(·, [γ]) and πA (A∨ +k )(·, [δ]) co- +incide. Then there is a natural i with 0 ≤ i < o(A) and x, y, y′ ∈ Xn such that y ̸= y′, +πA (A∨ +k )(·, [γ]Ai) = πA (A∨ +k )(·, [δ]Ai) but A maps the edges +([γ]Ai, x, πA (A∨ +k )(x, [γ]Ai)) and ([δ]Ai, x, πA (A∨ +k )(x, [δ]Ai)) +respectively to the edges +([γ]Ai+1, y, πA (A∨ +k )(y, [γ]Ai+1)) and ([δ]Ai+1, y′, πA (A∨ +k )(y′, [δ]Ai+1)). +Proof. Let w = W([γ]) and v = W([δ]). Since [γ] ̸= [δ], we may find words u, w2, v2 ∈ +Q∗ +A and letters t ̸= t′ ∈ QA such that w = utw2 and v = ut′v2. Set i − 1 := |u|. We +note that for any j ∈ N with j ≤ i − 1, a straightforward induction argument shows, the +edges ([γ], a, πA (A∨ +k )(a, [γ])A) and ([δ], a, πA (A∨ +k )(q, [δ])) map respectively under Aj to edges +([γ]Aj, b, πA (A∨ +k )(a, [γ])Aj) and ([δ]Aj, b, πA (A∨ +k )(q, [δ])Aj), where b = λAj(a, u[1,j]) (if j = 0, +take b = a). In particular it follows that πA(·, t) = πA(·, t′) and so, since t ̸= t′, there is an +a ∈ Xn be such that y := λA(a, t) ̸= λA(a, t′) = y′. Let x ∈ Xn be such that λAi−1(x, u) = a. +Then it follows that the edges +([γ]Ai, x, πA (A∨ +k )(x, [γ]Ai)) and ([δ]Ai, x, πA (A∨ +k )(x, [δ]Ai)) +are mapped respectively under A, to the edges +([γ]Ai+1, y, πA (A∨ +k )(y, [γ]Ai+1)) and ([δ]Ai+1, y′, πA (A∨ +k )(y′, [δ]Ai+1)). +□ +Recall from Subsection 2.7 that if A ∈ Hn and B is an automaton so that there is a di- +graph automorphism φ : GB → GB so that A and H(B, φ) represent the same transformation +then we say A is induced from (B, φ). +Let A ∈ Hn. We say a strongly synchronizing automaton B is an automaton supporting +A if there is a digraph automorphism φ of the digraph GB, with A induced from (B, φ). In +this situation, if there is no proper folding B′ of B and digraph automorphism φ′ : GB′ → GB′ +so that A is induced from (B′, φ′), then we say B is a minimal automaton supporting A (or +simply, that B is minimal). +In the next lemma, we show that there is precisely one minimal automaton (up to iso- +morphism of automata) supporting a finite order element A of Hn. +21 + +Lemma 3.8. Let A ∈ Hn be an element of finite order and let k ∈ N be minimal such that +A∨ +k is the zero of the semigroup generated by the dual. Then (up to isomorphism of automata) +A (A∨ +k) is the minimal strongly synchronizing automaton admitting an automorphism φ of +GA (A∨ +k ) so that A is induced by (A (A∨ +k ), φ). Furthermore, φ is the automorphism φA of +Theorem 3.5. +Proof. Let A ∈ Hn be finite order of order o(A). We note that by results in [15, 4] k is +minimal such that all of the elements A, A2, . . . , Ao(A)−1 are strongly synchronizing at level +k (Ai being the product in Hn of A with itself i times). +It follows from Theorem 3.5 that A (A∨ +k) is an automaton supporting A and indeed that +(A (A∨ +k ), φA) induces A. We argue below that A (A∨ +k) is a minimal such automaton, and +further, that any minimal automaton supporting A is isomorphic to a folding of A (A∨ +k), and +hence, must actually be A (A∨ +k) up to isomorphism. +Now suppose that there was another automaton B, such that A acts as an automorphism +ψA of the underlying digraph of B so that H(B, ψA) has minimal representative A. Addi- +tionally suppose that B is a minimal strongly synchronizing transducer on which A acts as +an automorphism. We note that the minimal synchronizing level l of B is greater than or +equal to k for, H(B, ψi +A) is strongly synchronizing at level l and has minimal representative +Ai. +Suppose for a contradiction that B ̸= A (A∨ +k). There are two cases. +Firstly for any state q ∈ QB, there is a state p ∈ QA such that the set W(q, j) of words +of length j which force q is contained in the set W(p, j) of words of length j which force the +state p of A (A∨ +k ). In this case, one observes that A (A∨ +k) is a folding of B contradicting the +minimality of B. +Thus we must be in the negation of the first case. +That is, we assume that there +is a pair of words γ, δ ∈ Xk +n such that the state of B forced by γ is the same as the +state of B forced by δ but γ and δ force different states of A (A∨ +k). +We may further +assume that the states [γ], [δ] of A (A∨ +k) also satisfy πA (A∨ +k )(·, [γ]) = πA (A∨ +k )(·, [δ]). +This +is because if πA (A∨ +k )(·, [γ]) ̸= πA (A∨ +k )(·, [δ]), then we may find a word ν ∈ X+ +n such that +[γ′] := πA (A∨ +k )(ν, [γ]) ̸= πA (A∨ +k )(ν, [δ]) = [δ′], satisfy πA (A∨ +k )(·, [γ′]) = πA (A∨ +k )(·, [δ]′). Thus γν +and δν force the same state of B but force, respectively, the states [γ′] and [δ′] of A (A∨ +k). +We may then replace γ, δ with γ′, δ′. +Let z1 be the state of B forced by γ and δ and let z1, z2, . . . , zo(A) be the orbit of z1 +under the action of A. As H(B, ψA) = H(A (A∨ +k), φA), it must be the case that if a, b ∈ Xn +are such that the edge ([γ], a, [Γ]) maps to (([γ])A, b, ([Γ])A), then the edge ([δ], a, [Γ]) also +maps to (([δ])A, b, ([Γ])A). Thus we conclude that πA (A∨ +k )(·, ([γ])A) = πA (A∨ +k )(·, ([δ])A). Now +observe that since γ and δ are representatives of [γ] and [δ], respectively, and since for any +q ∈ QA, the state of B forced by λA(γ, q) is equal to the state of B forced by λA(δ, q) is equal +to z2, it follows that there are representatives of ([γ])A and ([δ])A respectively such that the +states of B forced by these representative is z2. We may thus repeat the argument in the +z1 case. By induction we therefore see that for any 1 ≤ i ≤ o(A), the points ([γ])Ai and +([δ])Ai satisfy that πA (A∨ +k )(·, ([γ])Ai) = πA (A∨ +k )(·, ([δ])Ai) and whenever there are a, b ∈ Xn, +such that (([γ])Ai, a, ν) is an edge mapping under A to the edge (([γ])Ai+1, b(ν)A, then the +22 + +edge (([δ])Ai), a, ν) also maps under A to (([δ])Ai+1), b, (ν)A). This contradicts Lemma 3.7. +□ +4 +Water for the witch – shrinking conjugacy class rep- +resentatives +Suppose we have a finite order element A ∈ Hn, induced by (B, φA) for some strongly +synchronizing B with a minimal number of states. Under certain conditions we may employ +a two-step process to find a new element C ∈ Hn, where C is conjugate to A, and C is +induced by (D, ψ) for some strongly synchronizing D with D having fewer states than B. In +what follows we describe this process of finding “smaller” conjugacy class representatives of +A. +The first (and main) step in this process is to employ “relabelling.” This is a conjugacy +which, for a pair of states that would be identified in the collapse sequence of the domain +automaton, relabels inputs and outputs on edges from this pair of states, with the goal of +making this pair of states represent the same local map. If this is possible, then we can +collapse the carrying transducer to a smaller one than we started with. +The conditions for a successful relabelling include that the orbits of these states have the +same lengths, and that for any two corresponding outgoing edges, these orbit lengths of these +edges are also the same. In the case where some of these orbit lengths differ, then in certain +circumstances we can employ the second step of the overall process. This step“fluffs up” the +carrying automaton by executing some splittings, creating what we call shadow states, and +where we can then employ relabelling to the result. In either case, after a relabelling, the +whole resultant transducer can be minimised so as to be carried by a transducer with strictly +fewer states than TA. +In the case that A is conjugate to an n-cycle this process will eventually result in single +state transducer representing an n-cycle. +4.1 +Relabellings and automata sequences +Definition 4.1. Let A be a strongly synchronizing automaton and A = A0, A1, . . . , Am be +a collapse chain of A. Let 0 ≤ k ≤ m and φk be a vertex fixing automorphism of GAk. +Define A′ to be the automaton with QA′ = QA and transition function defined as follows: +for p ∈ QA, set πA′(x′, p) = q if and only if there is an x ∈ Xn such that πA(x, p) = q with +λH(Ak,φk)(x, [p]) = x′. We call A′ the relabelling of A by (Ak, φk) or the relabelling of A by +(the transducer) H(Ak, φk). +Note that if we relabel A by (Ak, φk), then the resulting automaton A′ is strongly iso- +morphic to A in the sense that there is a natural digraph isomorphism from the underlying +digraph of A′ to the underlying digraph of A that fixes states and which maps the rela- +belled edges of A′ to the original edges in A. More precisely, if (p, x, q) is an edge of GA +and λH(Ak,φk)(x, [p]) = x′, then the natural digraph isomorphism maps the edge (p, x′, q) of +23 + +A′ to the edge (p, x, q) of A. The point of view one should have in mind is that we have +renamed/relabelled the edges of A by switching edge labels on edges which are parallel edges +in Ak. Notice that if we relabel by (A0, φ0), then all we do is switch labels on parallel edges +in A, thus the resulting underlying digraph would not change, but a “fixed” drawing of it +would be relabelled. +Lemma 4.2. Let A be a strongly synchronizing automaton and A = A0, A1, . . . , Am be a +collapse chain for A. Let 0 ≤ k ≤ m and φ be a vertex fixing automorphism of GAk. Let +A′ be the relabelling of A by (Ak, φ). Then A′ has underlying digraph strongly isomorphic +to the underlying digraph of A and Am remains a folding of A′. More specifically, writing +A′ = A′ +0, A′ +1, . . . , A′ +l for the collapse chain from A′ to Am, then l ≤ m and two states of u, v +of A belong to the same state of Ai if and only if for some i′ ≤ i we have u and v belong to +the same state of A′ +i′. +Proof. We may consider A as a non-minimal synchronizing transducer where each state +induces the identity transformation of the set Xn. Consider the core of the product A ∗ +H(Ak, φ). Let p ∈ QA, and γ ∈ X+ +n be such that the state of A forced by γ is p. Then, by +definition of Ak, the state of Ak forced by γ is the state [p] containing p. Thus the set of states +of core(A ∗ H(Ak, φ)) is the set {(p, [p]) | p ∈ QA}. Let x ∈ Xn and p, q ∈ QA such that +πA(x, p) = q. Then we have, πA(x, (p, [p])) = (q, [q]) and λA(x, (p, [p])) = λH(Ak,φ)(x, [p]). +Thus setting A′ to be the output automaton of core(A ∗ H(Ak, φ)) we see that A′ is the +relabelling of A by (Ak, φ). From this it follows that the underlying digraph of A′ is strongly +isomorphic to the underlying digraph of A. +Let u, v be two states of A which belong to the same state of Am and which transition +identically on all words of length j and suppose j is minimal for which this happens. Let +p ∈ QA be an arbitrary state and let W(p) ⊆ Xj +n consist of those words γ such that +πA(γ, u) = πA(γ, v) = p. We break into cases based on whether or not k ≥ j or k < j. +First suppose that k ≥ j. This means that in Ak, the states [u] and [v] are equal. Thus, +λH(Ak,φ)(γ, [u]) = λH(Ak,φ)(γ, [v]) for any γ ∈ X∗ +n. Therefore in A′ we see that the set of words +ν ∈ Xj +n for which πA′(ν, u) = πA′(ν, v) = p is precisely the set {λH(Ak,φ)(γ, [u]) | γ ∈ W(p)}. +Now suppose that k < j. This means that the states [u] and [v] are distinct states of Ak +such that πAk(·, [u]) and πAk(·, [v]) coincide on Xj−k +n +. Let γ ∈ W(p) be arbitrary. Set γ1 to +be the length j − k prefix of γ and set γ2 ∈ Xk +n such that γ1γ2 = γ. Set [r] = πAk(γ1, [u]) = +πAk(γ1, [v]) and set κ ∈ Xk +n such that λH(Ak,φ(κ, [r]) = γ2. For t ∈ {u, v}, set δt ∈ Xj−k +n +be +such that λH(Ak,φ)(δt, [t]) = γ1. Then since φ is a vertex fixing automorphism of Ak we notice +that πA(δu, u), πA(δv, v), {πA(γ1, t) | t ∈ {u, v}} all belong to the same state of Ak. This +means that πA(κ, πA(δu, u)) = πA(κ, πA(δv, v)) = s.(Note that [s] = [p] in Ak by the vertex +fixing property of φ.) Therefore πA∗H(Ak,φ)(δuκ, (u, [u])) = (s, [p]) = πA∗H(Ak,φ)(δvκ, (v, [v])). +Since λA∗H(Ak,φ)(δuκ, (u, [u])) = γ = λA∗H(Ak,φ)(δvκ, (v, [v])), we see that in A′, πA′(γ, u) = +πA′(γ, v). +Therefore, in A′, πA′(·, u) and πA′(·, v) coincide on the set W(p). Since p was chosen +arbitrarily and ⊔p∈QAW(p) = Xj +n we conclude that πA′(·, u) and πA′(·, v) coincide on the set +Xj +n. +The result now follows by Theorem 2.8. +24 + +Now suppose that there are states u, v of A which belong to the same state A′ +i′ for some +0 ≤ i′ ≤ l. Then since u, v belong to the same state of Am, there is an i between 0 and m +such that u and v belong to the same state of Ai. The preceding paragraph and Theorem 2.8 +show that i′ must be less than or equal to i. +□ +Let A be a strongly synchronizing automaton and A′ be the relabelling of A by (Ak, φk). +Set ι : GA → GA′ to be the natural digraph isomorphism. If ϕ is an automorphism of the +underlying digraph GA of A, then we will mean by the induced automorphism ϕ′ of GA′ +precisely the map ι−1ϕι. +Lemma 4.3. Let A ∈ Hn be an element of finite order and let B be a strongly synchronizing +automaton such that there is an automorphism φA of the underlying digraph of B with A the +minimal representative of H(B, φA). Let Bk be an element of the synchronizing sequence of +B and ψ a vertex fixing automorphism of Bk. Let B′ be the relabelling of B according to +(Bk, ψ) and ϕ be the induced isomorphism from the underlying digraph of B to the underlying +digraph of B′. Set P to be the minimal representative of the transducer H(B, B′, ϕ). Then +P −1AP is the minimal representative of H(B′, φϕ +A). +Proof. This is a straight-forward application of the definitions. +□ +In the situation of Lemma 4.3 we refer to the resulting transducer P −1AP as the trans- +ducer induced from A by the relabelling B �→ B′. +Lemma 4.4. Let A be a strongly synchronizing automaton, and let s, t be distinct states of +A. Let (Ai)1≤i≤m be a collapse chain of A such that s, t belong to the same state of Am. Let +1 ≤ k < m be minimal such that t belongs to the state [s] of the automaton Ak+1. Then +for all x, x′ ∈ Xn such that πA(x, s) = πA(x′, t) and [v] ∈ {[s], [t]}, states of Ak, we have +πAk(x, [v]) = πAk(x′, [v]). +Proof. By minimality of k, it must be the case that the states [s] and [t] of Ak are distinct +and the equality πAk(·, [s]) = πAk(·, [t]) holds. +Let x, x′ ∈ Xn and u ∈ QA be such that πA(x, s) = πA(x′, t) = u. Then by definition of +Ak, πAk(x, [s]) = [u] = πAk(x′, [t]). However, the equality πAk(·, [s]) = πAk(·, [t]), now implies +that πAk(x′, [s]) = [u] = πAk(x, [t]) also. +□ +4.1.1 +Constructing discriminant permutations disc(s, t, Q) +Let A be an automaton and let s, t ∈ QA. Set the notation: +EA(s, t) := {(s, x, t) ∈ EA}; and +LettersA(s, t) := {x ∈ Xn | (s, x, t) ∈ EA(s, t)}. +We may leave out the explicit mention of the automaton A when it is clear from context, +writing simply E(s, t) and Letters(s, t) for these sets in this case. +25 + +Let Q ⊆ QA and s ∈ QA. Set the notation +Xs,Q := +� +p∈Q +Letters(s, p). +Now, suppose s, t ∈ QA and suppose further there is a subset Q ⊆ QA so that +(a) Xs,Q = Xt,Q, and +(b) for all p ∈ Q we have | Letters(s, p)| = | Letters(t, p)| . +Then to describe this situation we say s and t distribute similarly over Q. (Note in passing +that for some choices of s and t the only possible such set Q may be empty.) For any states +s and t and set Q ⊂ QA so that s and t distribute similarly over Q, we denote by XQ the +set Xs,Q = Xt,Q. We call XQ ⊆ Xn the agreement alphabet (of s and t on Q) noting that if +Q = QA, then XQ = Xn. +Define a bijection disc(s, t, Q) : XQ → XQ as follows. +First, let p1, . . . , pr ∈ Q be a maximal sequence of distinct states such that for 1 ≤ i ≤ r +there is an x ∈ XQ with πA(x, s) = pi. Observe that the sets +{Letters(s, pi) | 1 ≤ i ≤ r} +and +{Letters(t, pi) | 1 ≤ i ≤ r} +form partitions of XQ, with equal-size corresponding parts in index i. +Now, for 1 ≤ i ≤ r, set disc(s, t, Q) to act as the identity on Letters(s, pi) ∩ Letters(t, pi). +Set +Letters(s, pi)′ := Letters(s, pi)\(Letters(s, pi) ∩ Letters(t, pi)) +and +Letters(t, pi)′ := Letters(t, pi)\(Letters(s, pi) ∩ Letters(t, pi)). +We note that | Letters(s, pi)′| = |Letters(t, pi)′| and indeed that +Ys,t := +� +1≤i≤r +Letters(s, pi)′ = +� +1≤i≤r +Letters(t, pi)′. +Order the elements of Letters(s, pi)′ and Letters(t, pi)′ with the order induced from Xn. For +1 ≤ i ≤ r and x ∈ Letters(s, pi)′ we write x′ for the corresponding element of Letters(t, pi)′, +that is, in the ordering of Letters(s, pi)′ and Letters(t, pi)′ induced from Xn, x and x′ have +the same index. +Using the definitions and facts above we extend the definition of disc(s, t, Q) over the set +Ys,t by the rule x �→ x′. One easily checks that the resulting function +disc(s, t, Q) : XQ → XQ +26 + +is a well-defined bijection. +Further, observe that for x0 ∈ Ys,t the function disc(s, t, Q) +contains a cycle (x0 x1 x2 . . . xk−1) in its disjoint cycle decomposition, where for all i we +have xi+1 = x′ +i (indices taken mod k). Recall as well that disc(s, t, Q) acts as the identity +over the set XQ\Ys,t. +For s and t satisfying points (b) and (b) for some set Q we call disc(s, t, Q) the discrim- +inant of s and t; it is a permutation that encodes the difference in transitions between s +and t amongst the set of states Q. In the case that Q = QA, we will write disc(s, t) for +the bijection disc(s, t, QA). As with the notation Letters(p, q), we often run in to situations +where we compute discriminant permutations in distinct automata sharing the same state +set, in such cases we use the notation discA(s, t, Q) and disc(s, t) to emphasise the automaton +in which the permutation is computed. +Lemma 4.5. Let A be a strongly synchronizing automaton, and let s, t be distinct states of +A. Let Q ⊆ QA be such that s and t distribute similarly over Q with agreement alphabet +XQ. Let (Ai)1≤i≤m be a collapse chain of A such that s, t belong to the same state of Am. +Let 1 ≤ k < m be minimal such that πAk(·, [t]) and πAk(·, [s]) are equal on XQ. Then for +x, y ∈ XQ which belong to the same disjoint cycle of disc(s, t, Q), +πAk(x, [s]) = πAk(y, [s]) = πAk(y, [t]) = πAk(x, [t]). +Proof. By assumption πAk(·, [s]) = πAk(·, [t]). +An easy induction argument using the definition of disc(s, t, Q) now shows that for any +x, y ∈ Xn such that y belongs to the orbit of x under the action of disc(s, t, Q), +πAk(x, [s]) = πAk(y, [s]) = πAk(x, [t]) = πAk(y, [t]). +This follows since for any x ∈ Xn, πA(x, s) = πA((x) disc(s, t, Q), t). +□ +4.1.2 +Discriminant permutations and amalgamation sequences +Let B be an automaton and G := GB be the underlying digraph of B. Define a sequence +G := G0, G1, . . . as follows. Assuming Gi is defined, Gi+1 is obtained from G in the following +manner. Let ∼ be the equivalence relation on the vertices QGi of Gi that relates two vertices +p, q precisely when for every vertex t ∈ QGi the number of edges from q to t is precisely +the number of edges from p to t. +If p ∈ QGi write [p]i+1 for the equivalence class of p +under the relation ∼. +Set QGi+1 = {[p]i+1 | p ∈ QGi}. +Now suppose p, q ∈ QGi and +enumerate those elements of [q]i+1 which have an incoming edge from a vertex in [p]i+1 in +some order as q1, q2, . . . , qr. For 1 ≤ j ≤ r, let kj be the number of edges from p to qj and +set ec(i + 1, p, q) := � +1≤j≤r kj. Set Gi+1 to be the directed graph with vertices QGi+1 and +with ec(i + 1, p, q) many edges from [p]i+1 to [q]i+1 for each [p]i+1, [q]i+1 ∈ QGi+1. +We refer to the resulting sequence G0, G1, . . . , as defined above as the amalgamation +sequence of G (see [17]). Note that for each natural i the construction above induces an +identification of the states of Gi with a partition of B. It follows that after finitely many +steps, the amalgamation sequence stabilises to a fixed digraph. +27 + +The lemma below says that for a given automaton B, there is a relabelling of B such +that the synchronizing sequence coincides with the amalgamation sequence. +Lemma 4.6. Let B be an automaton with underlying digraph G and synchronizing sequence +B = B0, B1, . . .. Let G = G0, G1, . . . be the amalgamation sequence of G. Then there is +a relabelling D of B such that if D = D0, D1, . . . is the synchronizing sequence of D, the +underlying digraph of Di is Gi; in particular, the partition of the state set of B induced by +Di is the same partition induced by Gi. +Proof. Since B is strongly synchronizing there is a minimal l ∈ N for which Gl = Gl+1 and +both have a single vertex with n loops. We proceed by induction on the amalgamation +sequence. +We begin with the base case. Let s, t ∈ QB be distinct such that s and t belong to +the same state of G1. This means that s and t distribute similarly over QB. Suppose that +disc(s, t) is not trivial. +Let k ∈ N be minimal such that s and t belong to the same state of Bk+1. Note that +since disc(s, t) is not trivial, then k ≥ 1. By Lemma 4.5, for any x, y which belong to the +same orbit under disc(s, t), +πBk(x, [s]) = πBk(y, [s]) = πBk(x, [t]) = πBk(x, [t]). +Let λBk be defined such that λBk(·, [q]) : Xn → Xn is trivial whenever [q] is not equal to +[t]. We set λBk(·, [t]) = disc(s, t)−1. We note that the transducer Bk is induced by a vertex +fixing automorphism of Bk. Furthermore, for any pair (u, v) ̸= (s, t) such that u, v belong +to the same state of G1 and disc(u, v) is trivial, [u] = [v] in Bk. +Let E be the relabelling of B by the transducer Bk. Note that QE = QB and G remains +the underlying digraph of E. It therefore follows that for any pair u, v in B which distribute +similarly over QB, u, v still distribute similarly over QB in E. If moreover, discB(u, v) is +trivial, then construction of λBk, means discE(u, v) remains trivial. Lastly we note that s, t +distribute similarly over QB in E, and discE(s, t) is trivial. +Applying an induction argument, there is an automaton E1 a relabelling of B such that +for any pair s, t ∈ QB which belong to the same state of G1, discE1(s, t) is trivial. +In +particular, such s, t satisfy, πE1(·, s) = πE1(·, t). +Now assume by induction that there is a relabelling E of B with synchronizing sequence +E = E0, E1, . . . possessing the following property: for 0 ≤ i ≤ k < l two states s, t ∈ QB +belonging to the same state of Gi belong to the same state of Ei. +Let s, t ∈ QB and suppose s and t belong to the same state of Gk+1 but do not belong +to the same state of Ek+1. We note that since the underlying digraph of Ek is the same as +Gk and they induce the same partition of the state set QB, then s and t belong to distinct +states of Ek and so to distinct states of Gk. The fact that s and t belong to the same state +of Gk+1 means that [s] and [t] distribute similarly over QEk but discEk([s], [t]) is not trivial. +Let k < j ≤ l be minimal such that s and t belong to the same state of Ej+1. By +Lemma 4.5 once more, we have the equalities: for any x, y which belong to the same orbit +under discEk([s], [t]), +πEj(x, [s]) = πEj(y, [s]) = πEj(x, [t]) = πEj(x, [t]). +28 + +Let λEj be defined such that λEj(·, [q]) : Xn → Xn is trivial whenever [q] is not equal +to [t]. We set λEj(·, [t]) = discEk([s], [t])−1. We note that the transducer Ej is induced by +a vertex fixing automorphism of Ej. Furthermore, for any pair (u, v) ̸= (s, t) such that u, v +belong to the same state of Gk+1 and discEk([u], [v]) is trivial, [u] = [v] in Ej. +Let F be the relabelling of E by the transducer Ej. Let F1, F2 . . . be the synchronizing +sequence of F. +Let u, v ∈ QB belong to the same state of Gi for some 0 ≤ i ≤ k < l. Then by the +inductive assumption and Lemma 4.2, u, v belong to the same state of Fi. +Let u, v ∈ QB belong to the same state of Gk+1 and suppose that discEk(u, v) is trivial. +Note that since discEk(u, v) and [u], [v] distribute similarly over QEk, [u] = [v] in Ek+1. +Therefore, Lemma 4.2 implies that [u] = [v] in Fk+1 as well. +Lastly observe that [s] = [t] in Fk+1 by construction of λEk and the fact that states which +are identified in Ek remain identified in Fk+1. +The result now follows by induction. +□ +4.2 +Relabellings along orbits +For lemma below, we give stronger hypotheses than appear to be required as per the following +observation. +Let B be a strongly synchronizing automaton, φ an automorphism of the +underlying digraph GB of B, and, s and p states of B. Every edge from s to a state in the +orbit of p (under the action of φ) is on an orbit of length N (when such an edge exists) if and +only if every edge from any state in the orbit of s to a state in the orbit of p is on an orbit +of length N (when such an edge exists). We state the lemma with the stronger hypotheses +below to ease understanding. +Lemma 4.7. Let B be a strongly synchronizing automaton and φ an automorphism of the +underlying digraph GB of B. Let s, p be vertices of GB so that LettersB(s, p) is non-empty. +Suppose +• there is N ∈ N1 so that for every edge e from a vertex in the orbit of s to a vertex in +the orbit of p, the orbit length of e is N, and secondly +• there is r ∈ N1 so that if (sφi, y, pφj) is any edge from the orbit of s to the orbit of p, +then we have Letters(sφi, pφj) = Letters(sφi+r, pφj). +Then there is a relabelling of B′ of B such that the induced automorphism φ′ of GB′ satisfies +the following: for any i, j ∈ N, +• LettersB′(sφi, pφj) = LettersB′(sφi+r, pφj), and, +• if x ∈ LettersB′(sφi, pφj), then the labels of the edges (sφ′i, x, pφ′j)φ′ and (sφ′i+r, x, pφ′j)φ′ +are equal. +Proof. We first set up some notational convenience. +Given an edge (u, x, v) of B, with +respect to this edge, we shall write xφi for the label of its image (u, x, v)φi so that we have +29 + +the equality (u, x, v)φi = (uφi, xφi, vφi). Note that in general we do not have an induced +action of φ on Xn, but the notation will be well-defined in the context of a base edge (u, x, v) +being understood. +Let B, φ, s, p, N and r be as in the hypotheses, and assume r is minimal. It follows that +r divides the orbit length of s (by minimality). Write mr for the orbit length of s. Write +s1, s2, . . . , smr for the orbit of s = s1 (we note that mr|N and we make this explicit below). +Let �P be those states p′ in the orbit of p which have Letters(s, p′) non-empty. The set +�P can be partitioned according to the orbits under the action of φmr, that is, two elements +of �P belong to the same part if they belong to the same orbit. Choose T ⊂ �P so that T +has exactly one representative from each block of this partition. Note, by definition, for any +edge (s(φr)i, x′, pφj), i, j ∈ N, there is a unique p′′ ∈ T and an element x ∈ Letters(s, p′′) so +that the orbit of (s, x, p′′) under φr contains (s(φr)i, x′, pφj). +We inductively define a map λB (i.e., induced by a vertex fixing automorphism of B) along +the orbit of an edge (s, x, q) for some q ∈ T. The map λB will then determine a transducer +(Xn, QB, πB, λB) which can be used (as in Definition 4.1) to carry out the required relabelling. +To this end fix q ∈ T and set k = |{qφimr|i ∈ N}|. +For 0 ≤ a ≤ m − 1, partition Letters(sar+1, q) via the equivalence relation relating two +edge labels whose corresponding edges are in the same orbit under φkmr. Recall there is +an order on the elements of each equivalence class induced from the standard ≤ ordering +on Xn. Use this ordering to determine a transversal for the equivalence classes, choosing as +representative of each class the least element in that class. Write β(sar+1) for this transversal. +For b ∈ β(sar+1) we use the phrase the equivalence class of b at sar+1 to mean the edge labels +in Letters(sar+1, q) which are orbit equivalent to b. +Let 0 ≤ a, a′ ≤ m − 1. We note that since N is the orbit length of any edge from a state +in the orbit of s to a state in the orbit of q we have |β(sar+1)| = |β(sa′r+1)|. Let α ∈ N +such that for b ∈ β(sar+1) and b′ ∈ β(sa′r+1), the size of the equivalence class of b at sar+1 +is equal to the size of the equivalence class of b′ at sa′r+1 is equal to α. We fix a bijection +between the sets β(sar+1) and β(sa′r+1) induced by the ordering of the elements. We note +that αkmr = N. +For 1 ≤ j ≤ r, and 0 ≤ a ≤ m − 1 we write β(sar+j) for the set {bφj−1|b ∈ β(sar+1)}. We +note that the orbit equivalence class of bφj−1 at sar+j, b ∈ β(sar+1), is precisely the image +of the equivalence class of b at sar+1 under the image of φj−1. We transport using φj−1 the +orderings of β(sar+1), and the equivalence classes of elements b ∈ β(sar+1) to the set β(sar+j) +and the equivalence classes of its elements. That is, for instance, if b < b′ ∈ β(sar+1), then +bφj−1 < b′φj−1 in β(sar+j). +Let 1 ≤ l ≤ m be minimal such that {qφimr|i ∈ N}φrl = {qφimr|i ∈ N}. We note that by +minimality l|m since: {qφimr|i ∈ N}φmr = {qφimr|i ∈ N}. Moreover φlr : {qφimr|i ∈ N} → +{qφimr|i ∈ N} is a k-cycle since φmr : {qφimr|i ∈ N} → {qφimr|i ∈ N} is a k-cycle and φmr is +a power of φlr. Let M ∈ N be such that Ml = m so that αkMlr = N. +Further observe that if {qφimr|i ∈ N}φrd ∩ {qφimr|i ∈ N} ̸= ∅ for some d ∈ N, then +{qφimr|i ∈ N}φrd = {qφimr|i ∈ N}. +For suppose qφfmr ∈ {qφimr|i ∈ N}φrd for some +1 ≤ f ≤ k. Then there is some 1 ≤ j ≤ k such that qφjmrφdr = qφfmr, this now means that +30 + +qφdrφjmr ∈ {qφimr|i ∈ N}. However since φmr is a k-cycle on the set {qφimr|i ∈ N}, then +qφdr ∈ {qφimr|i ∈ N}. +Thus we conclude that the sets {qφimr|i ∈ N}φar for 0 ≤ a ≤ l − 1 are pairwise disjoint. +We define a relabelling map λB inductively as follows. +Let b = b0 ∈ β(s1) be the smallest element such that λB(b, s) is undefined and for all +1 ≤ i ≤ l − 1 λB(bi, sir+1) is undefined for the element bi of β(sir+1) corresponding to b. +(Note that bi is the least element of β(sir+1) such that λB(bi, sir+1) that is undefined.) In +the inductive process which follows, we will define λB(bi, sir+1) for all 0 ≤ i ≤ l − 1 in order. +Define a kl-by-r matrix r with entries tuples of size α as follows. Set +r0,0 = (b = b1,1, b1,2, . . . , b1,α) +where (b1,1, . . . , b1,α) is the ordered tuple of element β(s1). For 0 ≤ i < kl and 0 ≤ j < r set +ri,j = (b1,1, b1,2, . . . , b1,α)φir+j = (b1,1φir+j, b1,2φir+j, . . . , b1,αφir+j). +Define a matrix R of dimension Mkl-by-r such that Ri,j for 0 ≤ i < Mkl and 0 ≤ j < r +has entry +((b1,1, s), . . . , (b1,α, s))φir+j := ((b1,1φir+j, sφir+j), . . . , (b1,αφir+j, sφir+j)). +For 0 ≤ d < M, set R(d) to be the kl-by-r matrix corresponding to rows dkl to row +(d + 1)kl − 1. For 0 ≤ d < M, 0 ≤ i < kl and 0 ≤ j < r we set λBR(d)i,j = ri,j, where we +extend λB naturally to act on tuples (Xn × QB)α to produce tuples in Xα +n. +Let 1 ≤ i < l. We note that for the element b′ ∈ βsir+1, the function λB(b′, sir+1) remains +undefined. Let the matrix r be exactly as above and define the matrix R as above but with +b′ playing the role of b and sir+1 playing the role of s1 = s. For 0 ≤ d < M define the +component R(d) as above. Then once more for 0 ≤ d < αM, 0 ≤ i < kl and 0 ≤ j < r we +set λBR(d)i,j = ri,j. +Continuing on in this way across the set T, we define λB on all pairs (x, sφi) where i ∈ N +and there is a j ∈ N such that (sφi, x, pφj) is an edge. We set λB to be projection onto the +first coordinate on all other pairs in Xn × QB. +By construction λB is induced by a vertex fixing automorphism and induces the required +relabelling of B. +□ +Remark 4.8. Note that the relabelling B′ of B given by Lemma 4.7 is in fact isomorphic +as an automaton to B, since the relabelling is by a vertex fixing automorphism of B. This +means we may instead write (B, φ) for the pair (B′, φ′). +Lemma 4.9. Let B be a strongly synchronizing automaton and φ an automorphism of the +underlying digrpah GB of B. Let s, t, p be states of B such that there is an x ∈ Xn with +πB(x, s) = p. Suppose +• for i, j ∈ N, πB(x, sφi) = pφj if and only if πB(x, tφi) = pφj; +• the orbits of s and t are distinct and have equal length l; +31 + +• there is an N ∈ N such that for any j ∈ N, all edges (s, x, pφj) and (t, x, pφj) are on +orbits of length N. +Then there is a relabelling of B′ of B such that the induced automorphism φ′ of GB′ satisfies: +for any i, j ∈ N Letters(sφi, pφj) = Letters(tφi, pφj), and for any x ∈ Letters(sφi, pφj), the +labels of the edges (sφi, x, pφj)φ′ and (tφi, x, pφj)φ′ coincide. +Proof. This is a more straight-forward relabelling operation than the previous case. +We +simply match the orbits of t along p with those of s along p. We define the relabelling map +λB inductively. As before, throughout we observe the following notation. Let u, v ∈ QB and +x ∈ Xn such that (u, x, v) is an edge. For i ∈ N we write xφi, whenever there is no ambiguity, +for the label of the edge (u, x, v)φi. +First, for any pair (c, d) ∈ Xn × QB such that (d, c, π(c, d)) is not an edge from a state +in the orbit of t to a state in the orbit of p, set λB(c, d) = c. +Let x ∈ Xn be smallest such that (t, x, pφi) is an edge for some i and λB(x, t) is not +defined. Let y ∈ Xn be minimal such that (s, y, pφi) is an edge and y is not equal to λB(z, t) +for (t, z, pφi) an edge. For 0 ≤ j < N set λB(xφj, tφj) = yφj where yφj is the label of the +edge (s, y, pφi)φj. +This inductively defined relabelling map λB is given by a vertex fixing automorphism +and induces the required relabelling of B. +□ +Remark 4.10. We note that, once more, B′ and B are isomorphic as automata and so we +may write (B, φ′) for (B′, φ′). +4.3 +Shadow states +In this second part of our process, we find new states to add to the transducer via splitting +operations, to provide more room for relabelling. +Let A ∈ Hn have finite order and let B be a minimal strongly synchronizing automaton +such that there is an automorphism φA of the underlying digraph GB of B with A the +minimal representative of H(B, φA). +The following definition is motivated by considering paths into a vertex t that might +provide an obstruction to a collapse through relabelling of B, as described in the next +paragraph. +Suppose there is a state q of B so that there is a minimal length r so that all paths of +length r that end on q have orbits of length n under the action of φA, and for this choice +of q we have r > 1. Let P = e1e2 . . . er be a path of length r terminating at q, where the +orbit of P has length n but the orbit of e2e3 . . . er is of size c for some c < n. For indices +1 ≤ i ≤ j ≤ r set Pi,j := eiei+1 . . . ej. By construction, the least common multiple of the +orbit size of the edge e1 and of c is n, and further, the orbit length of P2,r−1 must divide +c < n. As will become clear later, if this situation arises, it may be an obstruction to collapse +of a transducer through a relabelling process. +32 + +In the definition that follows, the state t corresponds to the target of e1 from the path +mentioned above, while b is some integer multiple of the orbit length of t, but which still +properly divides n. +Definition 4.11. Let A ∈ Hn, B be a minimal strongly synchronizing automaton such +that there is an automorphism φA of the underlying digraph GB of B with A the minimal +representative of H(B, φA), and t ∈ QB. We say t is heavy (for the pair (B, φA)) if the +following conditions hold: +• there is a proper divisor b of n, where b is divisible by the length of the orbit of t; +• there is at least one pair (x, s) ∈ Xn × QB such that (s, x, t) is an edge; +• for any x ∈ Xn and any s ∈ QB such that (s, x, t) is an edge of B, the lowest common +multiple of b and the length of the orbit of (s, x, t) under φA is n. +In this case, we call the value b above a divisibility constant for t and observe that the set of +valid divisibility constants for t might have more than one element. +In our overall process, we will apply the Lemma 4.13 (directly below) in a situation where +we cannot simplify a transducer H(B, φA) directly by a relabelling operation. Specifically, +this lemma is useful in situations where we can carry out an in-split of the domain automaton +B along the orbit of a heavy state t to create a new automaton B′ with automorphism ψA so +that A is a minimal representative of both H(B, φA) and of H(B′, ψA), and where the new +pair (B′, ψA) has a reduced obstruction to the existence of a helpful relabelling. Note that +here, we mean “in-split” in the normal sense of that operation for edge-shift equivalences, +see, e.g. [11]. +The following lemma characterises how to perform an in-split along the orbit of a heavy +state t. The new automaton that is created has all of the old states, together with new states +which we call shadow states (from the orbit of t). +The following two lemmas address the same set of hypotheses, but we split the results +into two statements as the lemma of primary interest is the second one. +Any such number n′ which arises as in Lemma 4.12 below will be referred to as a valid +splitting length for (the heavy state) t with respect to divisibility constant b. +Lemma 4.12. Let A ∈ Hn and let B be a minimal strongly synchronizing automaton such +that there is an automorphism φA of the underlying digraph GB of B with A the minimal +representative of H(B, φA). Suppose there are b ∈ N and t ∈ QB so that t is heavy for the +pair (B, φA) with b a divisibility constant for t, and where +��{tφp +A|p ∈ Z} +�� = r. +In these circumstances, there is n′ ∈ N a number which divides the lengths of orbits of +all edges (s, x, t) and satisfies the following conditions: +i) the lowest common multiple of n′ and b is n, +ii) there is m > 1 so that n′ = mr. +33 + +Proof. Let N be the greatest common divisor of the orbit lengths of all edges (s, x, t) in +B. Let (s, x, t) be an edge of B with orbit length k under the action of ⟨φA⟩. Now, by the +third bullet point of the definition of the state t being heavy we see that lcm(k, b) = n. It +follows, as (s, x, t) is an arbitrary incoming edge for t, that lcm(N, b) = n as well. Since r +is the orbit length of t we see that r|k and since (s, x, t) is an arbitrary incoming edge for t +we therefore have r|N. By assumption, r|b, so if r = N we would have lcm(N, b) = b < n +which is a contradiction. It then follows that N = kr for some integer k > 1. Thus the set +of numbers N which divide the orbit lengths of all edges (s, x, t) and satisfy points i) and ii) +is non-empty. Now let n′ be an element of this set and determine m ∈ N so that mr = n′ +(noting that 1 < m by construction). +□ +Lemma 4.13. Let A ∈ Hn and let B be a minimal strongly synchronizing automaton such +that there is an automorphism φA of the underlying digraph GB of B with A the minimal +representative of H(B, φA). Suppose there is b ∈ N and t ∈ QB so that t is heavy for the pair +(B, φA) with b a divisibility constant for t. Set t0,0 = t and let t0,0, t0,1 . . . , t0,r−1 be the orbit +of t under iteration by φA. Let n′ be a valid splitting length of t and let m > 1 be determined +by n′ = mr. +In these circumstances we may form a new strongly synchronizing automaton B′ with +QB′ = QB ⊔ {ta,0, . . . , ta,r−1 | 1 ≤ a < m} +such that we have +i) πB′(x, s) := πB(x, s) for those pairs (x, s) ∈ Xn ×QB where πB(x, s) is not in the orbit +of t; +ii) πB′(·, ta,i) := πB′(·, t0,i) for all 0 ≤ a < m, 0 ≤ i < r; +iii) The incoming transitions of B′ to the set of vertices {ta,i | 0 ≤ a < m, 0 ≤ i < r} are +determined by the above rules, and by an automorphism ψA of the underlying digraph +GB′ of B′ satisfying: (t0,0)ψar+i +A += ta,i for 0 ≤ a < m, 0 ≤ i < r, (t0,0)ψn′ +A = t0,0, and +H(B′, ψA) = A. +Proof. Let n′ be a valid splitting length for the heavy state t0,0 with divisibility constant b, +and let m > 1 be an integer so that n′ = mr. +Set +T := {t0,0, t0,1, . . . , t0,r−1} +and build a set of new objects (the extra “shadow states” arising from the splitting along +the orbit of t0,0) +T ′ = {ta,1, ta,2, . . . , ta,r | 1 ≤ a < m}. +Note that |T ∪ T ′| = n′. +We will define an action ψA on +QB′ := QB ∪ T ′ +34 + +as follows. +For s ∈ QB′, set +sψA = + + + + + + + +sφA +if +s ∈ QB\T +ta,i+1 +if +s = ta,i and i < r − 1 +ta+1,0 +if +s = ta,r−1 and a < m − 1 +t0,0 +if +s = tm−1,r−1. +It is immediate by construction that this is an action, and also that the orbit of t0,0 has size +n′. We will specify transitions for B′ by steadily expanding the definition of the underlying +digraph GB′ of B′ through adding edges of the form (p, x, q) for p, q ∈ QB′ and x ∈ Xn (thus +adding the transition (x, p)πB′ = q to B′), while simultaneously extending the function ψA on +the corresponding edges of GB′. Ultimately, B′ will be a strongly synchronizing automaton +and ψA will be an automorphism of the digraph GB′ with H(B′, ψA) being equivalent to A. +Important in what follows will be a graph homomorphism ι : GB′ → GB, which we will +automatically extend to the (new) edges of GB′ whenever they are added. On the set QB′, ι +is defined as follows: for s ∈ QB′\(T ∪T ′) set sι := s, and for any ta,i ∈ T ∪T ′ set ta,iι := t0,i. +Below, whenever we extend GB′ by adding new edges, we also extend the graph homomor- +phisms ι : GB′ → GB and ψA : GB′ → GB′ so as to maintain rsc, the rule of semi-conjugacy, +which we define here. +rsc: +(a) for all q ∈ QB′ we have qψAι = qιφA, and +(b) for all edges e of GB′ we further require eψAι = eιφA. +Of course we have part (a) of the rule because we have already defined ι and ψA over QB′ +to satisfy this rule. +In the above construction of ι if eι = (r, x, s) then we will identify e as (p, x, q) where p +is the source of e and q is the target of e, so after any extension we can always think of the +new GB′ as an edge-labelled directed graph with edge labels “lifted” from GB by the map ι. +Note that below we will sometimes add a large collection of edges at one go, but in this +case, there is always a well defined triple (p, x, q) for each new edge, as we add in edges along +an orbit under ψA which always contains a well-defined edge (r, y, s), from which we can +detect the correct letter labelling of all edges along the orbit by using rsc. +It follows that if B′ is a strongly synchronizing automaton then H(B, φA) will represent +the same element of Hn as H(B′, ψA), since the map ι never changes edge labels, and the +map ψA will have to change edge labels in the corresponding fashion as φA in order to uphold +rsc. +We now begin to specify the edges of GB′, and hence the transition function πB′. Recall +below that QB\T = QB′\(T ∪ T ′). +35 + +Partition the edges of GB into the following four sets. +NT :={(p, x, q) | p, q ̸∈ T}, +BT :={(p, x, q) | p, q ∈ T}, +DT :={(p, x, q) | p ∈ T, q ̸∈ T}, and +RT :={(p, x, q) | p ̸∈ T, q ∈ T}. +We observe in passing that φA acts on each of the sets NT, BT, DT, and RT . +For (p, x, q) an edge in NT, let (p, x, q) also be an edge of GB′ (and so (x, p)πB′ = q as +well) and set (p, x, q)ψA := (p, x, q)φA. +Recall that for a group H acting on a set X, a traversal for the orbits is a subset Y ⊂ X +so that each orbit under the group action has a unique representative in the set Y . +Let YB be a traversal for the orbits of the edges in BT such that each edge in YB is of +the form (t0,0, x, t0,i). Similarly set YD to be a traversal for the orbits of the edges in DT so +that each edge of YD is of the form (t0,0, x, s) for some s ∈ QB\T. Finally set YR to be a +traversal for the orbits of the edges in RT so that each edge of YR is of the form (s, x, t0,0) +for some s ∈ Q\T. +Extend GB′ to include YD ∪ YR ∪ YB as edges incident on t0,0 (we will add more edges +incident on t0,0 later). Furthermore, use the action of ψA on the set QB′ together with the +map ι to uniquely determine new edges (of the form (p, x, q) for x ∈ Xn) that must be added +to GB′ so that the resulting digraph is closed under the action of ψA, contains the transversal +edges YD ∪ YR ∪ YB and satisfies rsc. Note that this process extends the definition of ι and +ψA to these new edges as well, but these extensions are inductively well defined. Now we may +use the new edges of GB′ in the obvious way to also extend the definition of the transition +function πB′ so as to create a correspondingly larger automaton B′. +Observe that for any state s ∈ QB′\(T ∪T ′) = QB\T, the process above now has created +a unique edge of the form (s, x, q) for each x ∈ Xn (which are the “lifts” of NT and RT +to GB′ by ι). For edges in NT this is simply by definition. For an edge (s, x, t0,i) ∈ RT, +there is an edge (s′, x′, t0,0) ∈ YR so that there is a minimal non-negative integer k with +(s′, x′, t0,0)φk +A = (s, x, t0,i). +It follows that (s′, x′, t0,0)ψk +A = (s, x, ta,i) for the unique non- +negative a so that k = ar + i. Now suppose there is an edge (s, x, tb,j) of GB′. By rsc, we +see that (s, x, tb,j)ι = (s, x, t0,j) but as there is a unique outgoing edge in GB from s with +letter x we see that t0,j = t0,i and in particular, i = j. We assume without meaningful loss +of generality that b ≥ a and that |b−a| is minimal amongst all such differences. Thus by rsc +the orbit length of the edge (s, x, t0,i) under φA is precisely (b − a)r or else b = a. However, +(b − a)r < n′ and n′ divides the length of the orbit of (s, x, t0,i) by the definition of n′. It +follows that (s, x, ta,i) is the unique pre-image of (s, x, t0,i) under ι. +There remains a special concern that we must address. Specifically, there are now pairs +(x, ta,i) ∈ Xn×(T ∪T ′) so that there are no edges of the form (ta,i, x, q) in GB′. This happens +as the orbit of t0,0 has length r under φA but length n′ = mr > r under ψA. Also, to verify +the coherence of the rsc condition for edges in YB, recall that n′ divides the orbit length of +these edges as they are in the orbit of an edge incident to t. +36 + +Let us now deal with the “missing edges” issue. Observe that for an edge (t0,0, x, s) ∈ +YD ∪ YB, its orbit under φA may contain multiple edges of the form (t0,0, y, q) (for various +y ∈ Xn and q ∈ QB). +Let us organise these as the sequence of pairwise distinct edges +(e0, e1, . . . , ek) where ei = (t0,0, x, s)φri +A, and with ekφr +A = e0 = (t0,0, x, s). In this context, +the orbit of (t0,0, x, s) under φA has length (k + 1)r. Let us set notation ei =: (t0,0, xi, qi) so +we can understand the letter xi associated to ei for each valid index i. The concern is that +in our current graph GB′ we see for any index 0 < i ≤ k that there is no edge of the form +(t0,0, xi, q) ∈ GB′. For the letter xi observe that there is an edge of the form (ta,0, xi, qi) of GB′ +for a = ir mod n′ and some state qi. The rule of modification is, add the edge (t0,0, xi, qi) +to GB′, for all indices 0 < i ≤ k. Repeat this same procedure across all of the transversal +elements (t0,0, x′, s′) ∈ YD ∪YB and as a consequence, for each letter y ∈ Xn, we see that the +vertex t0,0 now has a unique outgoing edge of the form (t0,0, y, q). Finally, we again use the +action of ψA on vertices and the action of φA on GB along with the rsc condition to extend +the definitions of ψA and ι to the necessary edges we have to add to GB′ in order to complete +the orbits of our newly-added edges based at t0,0, and to discern what letters needed to be +associated to these new edges. Now induce from GB′ the enlarged automaton B′. +One observes that for any valid indices a and b and fixed index i, the states ta,i and tb,i +of B′ have all the same outgoing transitions, and indeed, that the automaton B′ collapses +back down to B by identifying these states for each fixed i. In particular GB′ is strongly +synchronizing as it admits a collapse sequence to the n-leafed rose. Further, the rsc condi- +tion implies that ψA acts as an automorphism of the directed graph GB′ in fashion locally +emulating how φA acts on GB so that H(GB′, ψA) represents A. +□ +Let A, B, t = t0,0 be as in the statement of Lemma 4.13. Assume we applied Lemma 4.13 +to lengthen the orbit of t as in the lemma statement to create automaton B′ with automor- +phism ψA so that H(B′, ψA) has minimal representative A and where the orbit of t in GB′ +under the action of ψA is the set {ta,i | 0 ≤ a < m, 0 ≤ i < r}. Now for each state t0,i for +0 ≤ i < r (these states are in the original orbit of t in GB under the action of φA), we call +the set of states +{ta,i | 0 < a ≤ m − 1} ⊊ QB′ +the shadow states for t0,i (in QB′). Note that these are precisely the states of H(B′, ψA) +with local maps equivalent to the local map at t0,i for the transducer H(B, φA). If we apply +Lemma 4.13 inductively and perhaps repeatedly on states on the now extended orbit of t, +we extend the definition of the shadow states of t0,i to be the union of the sets of states +added in each round of applying Lemma 4.13 which have local maps equivalent to the local +map at t0,i for the transducer H(B, φA). Note that this process cannot go on forever as each +application of Lemma 4.13 lengthens the orbit of t0,0 with n an upper bound on the length +of this orbit. Also note that each added state will be a shadow state of one of the original +states t0,j after any number of iterated applications of Lemma 4.13. +Lemma 4.14. Let A ∈ Hn and let B be the minimal strongly synchronizing automaton +such that there is an automorphism φA of the underlying digraph of B with A the minimal +37 + +representative of H(B, φA). Suppose that all circuits in B are on orbits of length n under +the action of A. Then there is a strongly synchronizing transducer �B such that A acts as an +automorphism �ψA of �B and all edges of �B are on orbits of length n under the action of A. +Proof. We first observe that all states of B must be on orbits of length dividing n as the +underlying digraph of B is strongly synchronizing and therefore each state is visited by some +circuit which is on an orbit of length n. Also, we may assume that |B| > 1 otherwise all +loops at the state of B will be on orbits of length n and we would be done. +Let s ∈ QB be such that s is on an orbit of length strictly less than n under φA. (If all +states of QB were on orbits of length n then all edges of B would be on orbits of length n as +well and we would be done.) +Inductively define states as follows. +Set QB(0, s) := {s}. Assume QB(i, s) is defined for some i ∈ N. We now define QB(i + +1, s) ⊆ QB. An element q ∈ QB belongs to QB(i + 1, s) if the following conditions hold: +(a) there are elements x1, x2, . . . , xi+1 ∈ Xn, such that, for all 1 ≤ j ≤ i + 1, we have +πB(xi+1 . . . xj, q) ∈ QB(j − 1, s); and +(b) the path (q, xi+1xi . . . x1, s) is on an orbit of length strictly less than n under φA. +We observe that for any state q in a set QB(i+1, s), the orbit of q under φA has size properly +dividing n. If q ∈ QB(i + 1, s) and xi+1, xi, . . . , x1 ∈ Xn satisfies points (a) and (b) then we +call the path (q, xi+1xi . . . x1, s) conformant for QB(i + 1, s). +Let k ∈ N be minimal so that QB(k + 1, s) = ∅. If such k did not exist then there would +a long path (as in point (b) of the definition of the sets QB(i, s)) which is long enough that +it must contain a circuit in B. Any such circuit would be on an orbit of length strictly less +than n under the action of φA, which is a contradiction. +From the argument directly above it also follows that whenever j > 0, s /∈ QB(j, s). +We now apply an induction argument using Lemma 4.13 to reduce k to 0. +Let t ∈ QB(k, s) and fix xk, xk−1, . . . , x1 ∈ Xn, such that the path (t, xkxk−1 · · ·x1, s) is +conformant for QB(k, s) and where the orbit of this path under the action of φA is of length +b < n (note that b|n and also that the length of the orbit of t divides b). Moreover, by choice +of k, for any pair (x, p) ∈ Xn × QB with πB(x, p) = t, the lowest common multiple of the +length of the orbit of the edge (p, x, t) and b is n. +Therefore t is heavy for the pair (B, φA) and b is a divisibility constant for t, so we may +apply Lemma 4.13 with the state t and constant b to add states in the orbit of t and necessary +edges to form a new strongly synchronizing automaton B′ with ψA an automorphism of GB′, +and so that H(B′, ψA) still represents A (and so in particular, all circuits of GB′ are still on +orbits of length n under the action of ψA). +Recall that by the construction of B′, the orbit of t, which includes all of its shadow +states, is now of larger size n′ +t under the action of the resulting digraph automorphism ψA. +Thus, for any t′ in the orbit of t (including the shadow states we have just added), if there +is a path from t′ to s which is conformant for QB′(k, s), then t′ (and therefore t) is heavy +for (QB′, ψA), so we may again inductively increase the length of the orbit of t by adding +38 + +more shadow states until there are no paths from a point in the orbit of t to s which are +conformant for QB′(k, s) (note this happens to be a consequence of t no longer being heavy +for (B′, ψA) which must happen eventually as the orbit of t is getting longer and is bounded +above by n). Note that if p ∈ QB′(k, s) but p is not in the orbit of t, then p ∈ QB(k, s). In +particular, we have |QB′(k, s)| < |QB(k, s)| as this count of states for B′ no longer includes +the state t nor any state in its orbit. +We can now inductively repeat this process for s until |QB′(k, s)| = 0. +Note that we can repeat this process for any state p with |QB′(k, p)| ̸= 0. Thus we may +now proceed inductively in this fashion until finally we have constructed an automaton B′ +and an automorphism ψA of GB′ so that B′ folds onto B and H(B′, ψA) represents A, and +where if q is any state and j is minimal so that QB′(j, q) = ∅, then j = 0. +We set �B = B′ and �ψA = ψA in this final case, noting that the orbit of every edge of �B +under the action of �ψA is of length n. +□ +Remark 4.15. Let A ∈ Hn be an element of finite order and suppose that every point in +X−N +n +is on an orbit of length n under the action of A. By lemma 4.14, there is a minimal (in +size) strongly synchronizing automaton B such that A acts as an automorphism φA of the +underlying digraph of B and all edges of B are on orbits of length n under the action of A. +4.4 +Relabelling through shadows +In this section we make use of Lemma 4.13 to deal with situations in which we are unable +to directly apply Lemma 4.7 or Lemma 4.9. +The lemma below says that applying Lemma 4.13 and then suitably relabelling does not +move us out of the conjugacy class of the considered finite order element A ∈ Hn. +Definition 4.16. Let A ∈ Hn and let B be the minimal strongly synchronizing automaton +such that there is an automorphism φA of the underlying digraph of B with A the minimal +representative of H(B, φA). Let D be a strongly synchronizing automaton which is obtained +by repeated applications of Lemma 4.13 to the automaton B. Let ψA be the automorphism +of the underlying digraph of D with A the minimal representative of H(D, ψA). A relabelling +D′ of D is called a relabelling through shadows if, for the induced automorphism ψ′ +A of the +underlying digraph of D′, for any state q ∈ QB the set Sq,D of shadow states of q are all +ω-equivalent in H(D′, ψ′ +A) to the state q. +Lemma 4.17. Let A ∈ Hn and let B be the minimal strongly synchronizing automaton +such that there is an automorphism φA of the underlying digraph of B with A the minimal +representative of H(B, φA). Let D be a strongly synchronizing automaton which is obtained +by repeated applications of Lemma 4.13 to the automaton B. Let ψA be the automorphism +of underlying digraph of D with A the minimal representative of H(D, ψA). Let D′ be a +relabelling through shadows of D, ψA′ be the induced automorphism of the underlying digraph +of D′, and A′ be the minimal representative of H(D′, ψA′). Then A′ is conjugate to A and +there is a strongly synchronizing automaton B′ and an automorphism φA′ of the underlying +39 + +digraph GB′ so that A′ is the minimal representative of H(B′, φA′), with GB′ equal to the +underlying digraph of B. +Thus, the minimal strongly synchronizing automaton C on which A′ acts is carried by a +digraph GC that is a graph quotient of GB and we have |GC| ≤ |GB|. +Proof. As the relabelling process employed is a relabelling through shadows, it preserves the +equivalence of the local maps induced by the graph automorphism across all shadow states +shadowing any particular original state of B. In particular the collapse of each state with all +of its shadow states results in an automaton B′ which still admits an automorphism ψA′ of its +underlying graph (which graph is isomorphic to GB) so that A′ is the minimal representative +of H(B′, ψA′). The result now follows. +□ +Lemma 4.18. Let A ∈ Hn and let B be the minimal strongly synchronizing automaton +such that there is an automorphism φA of the underlying digraph of B with A the minimal +representative of H(B, φA). Suppose that all circuits in B are on orbits of length n under +the action of A. Let p ∈ QB, i ∈ N be less than or equal to the orbit length of p, and γ ∈ X∗ +n +be such that (p, γ, pφi) is a path in B from p to pφi, then (p, γ, pφi) is on an orbit of length +n under φ. +Proof. Let k be the orbit length of p and let r be the order of i in the additive group Zk. +For 1 ≤ a < r write γa for the label of the path (p, γ, pφi)φia and set γ0 = γ. +Write +Γ = γ0γ1 . . . γr−1, then the circuit (p, Γ, p) is on an orbit of length n by assumption. From +this it follows that the path (p, γ, pφi) is also on an orbit of length n. +□ +Lemma 4.19. Let A ∈ Hn and suppose there is a minimal strongly synchronizing automa- +ton B such that there is an automorphism φA of the underlying digraph of B with A the +minimal representative of H(B, φA) and so that all circuits in B are on orbits of length n +under φA. Let s, t, p ∈ QB be such that Letters(s, p) = Letters(t, p) ̸= ∅. Then by repeated +application of Lemma 4.13 one may obtain from B a strongly synchronizing automaton D +with automorphism ψA so that A is the minimal representative of H(D, ψA) and where the +pair (D, ψA) satisfies the following conditions: +(a) there are no shadow states of any element in the orbit of p in D (that is, each application +of Lemma 4.13 creates no shadow states for elements in the orbit of p), +(b) +LettersD(s, p) = LettersD(t, p) = LettersB(s, p), +(c) for u ∈ {s, t}, for any shadow state u′ of u we have, +LettersD(u, p) = LettersD(s, p) = LettersB(s, p), +(d) for any x ∈ LettersB(s, p), the length of the orbits of the edges (s, x, p), (t, x, p) under +the action of ψA on D is n. +40 + +Proof. We proceed in a similar way to Lemma 4.14. If |B| = 1, then we are done. Therefore +we assume that |B| > 1. +Inductively define subsets of QB as follows. +Set QB(0, p) = {p}. Assume that QB(i, p) is defined for some i ∈ N. Define QB(i + 1, p) +as follows. A state q ∈ QB belongs to QB(i + 1, p), if there are elements x0, x1, . . . , xi ∈ Xn, +such that πB(xixi−1 . . . xj, q) ∈ QB(j, p) for 0 ≤ j ≤ i, and the path (q, xixi−1 . . . x0, p) is on +an orbit of length strictly less than n under φA. +As in the proof of Lemma 4.14, there is a k ∈ N such that QB(k + 1, p) = ∅. Set k ∈ N +to be minimal such that QB(k + 1, p) = ∅. If QB(1, p) ∩ {s, t} = ∅, then we are done. Thus +we may assume that at least one of s, t belongs to QB(1, p) (and so k ≥ 1). +Observe that by Lemma 4.18 for any j ∈ N, pφj is not an element of QB(i, p) for any +1 ≤ i ≤ k. +We now repeatedly apply Lemma 4.13, as in the proof of Lemma 4.14, until we have an +automaton D such that QD(1, p)∩{s, t} = ∅. We note that since p is always the single element +of QD(0, p), then, as for any j ∈ N, pφj is not an element of QB(i, p) for any 1 ≤ i ≤ k, we do +not create a shadow state of p or elements in the orbit of p in an application of Lemma 4.13. +We prove the base case to illustrate how the proof goes. Let q ∈ QB(k, p). We may +find x0, x1, . . . , xk such that for any 0 ≤ j ≤ k, πB(q, xkxk−1 . . . xj) ∈ QB(j, p). Let b be the +length of the orbit of the path (q, xkxk−1 . . . x0, p). Then for any state u ∈ QB for which there +is an edge (u, x, q), the lowest common multiple of the length of the orbit of (u, x, q) and b +is n. We may now apply Lemma 4.13 to form a new transducer B′ by adding shadow states +of q. We may define the sets QB′(i, p) as before, noting that QB′(k + 1, p) = ∅. This follows +as for any edge (u, x, q) in B, the orbit of the edge (u′, x, q′) in B′ (for u′ and q′ either equal +to u and q or shadow states of u and q respectively) is equal to the orbit of the edge (u, x, q). +Moreover, as in the proof of Lemma 4.14, the number of paths (q′, x0x1 . . . xk, p), where q′ +is either q or one of its shadow states, witnessing that q′ ∈ QB′(k) is strictly fewer than the +witness paths for q in B. Thus inductively applying Lemma 4.13, we find an automaton, +which we again denote B′, in which |QB′(k, p)| < |QB(k, p)| since neither q nor any of its +shadow states belong to QB′(k, p). Thus, replacing B′ with B we may repeat the process. +Eventually we reach an automaton D′ such that QD(1, p) ∩ {s, t} = ∅. Moreover, since, +by Lemma 4.13, shadow states transition identically to their original counterparts on edges +into states which have no shadow states added (and this transition mirrors the transition in +B), the automaton D′ satisfies the requirements of the lemma. +□ +In what follows, set notation olτ(⋆) to represent the orbit length of ⋆ under the action of +τ a digraph automorphism of some digraph G, where ⋆ is a vertex, edge, or path in G. +For an automaton C over alphabet Y with a, b ∈ QC, recall that EC(a, b) represents the +set of edges of GC from a to b, while +LettersC(a, b) := {y ∈ Y | ∃(a, y, b) ∈ EC(a, b)} +represents the set of letters from Y which are the labels of these edges. +41 + +Lemma 4.20. Let A ∈ Hn and let B be the minimal strongly synchronizing automaton +such that there is an automorphism φA of the underlying digraph of B with A the minimal +representative of H(B, φA). Suppose that all circuits in B are on orbits of length n under +φA. Let s, t, p ∈ QB be such that s and t belong to distinct orbits but have the same orbit +length under φA, and so that Letters(s, p) = Letters(t, p). Then there is A′ ∈ Hn and an +automorphism φA′ of the underlying digraph of B so that the following hold: +(a) the element A is conjugate to A′ in Hn, where A′ is the minimal representative of +H(B, φA′); and, +(b) there is an N ∈ N such that for all x ∈ Letters(s, p) the edges (s, x, p) and (t, x, p) have +orbit length N under φA′. +Proof. The strategy is to apply Lemma 4.19 to add shadow states to B to obtain an automa- +ton D and an automorphism ψA of D, such that H(D, ψA) has minimal representative A, +ψA preserves the orbit length of the state p, and, the orbit length of the edges from s and t +into p is n. We then apply a relabelling through shadows of D by Lemma 4.17 to obtain a +conjugate element A′ to A represented by a transducer H(B, φA′) for φA′ an automorphism +of the underlying digraph GB. The key ingredient is that the relabelling of D is chosen such +that the orbits of the edges of s and t into p now all have the same length under φA′. In order +to find a relabelling achieving this goal, we will need to track numerous integer constants. +Set r = olφA(p) and determine m so that mr = lcm(olφA(s), olφA(p)) = lcm(olφA(t), olφA(p)). +For any edge (s, x, p) ∈ EB(s, p) we have (s, x, p)φk +A ∈ EB(s, p) if and only if k is an integer +multiple of mr. For each x ∈ LettersB(s, p) = LettersB(t, p), determine integers ux, vx so that +uxmr = olφA((s, x, p)) and vxmr = olφA((t, x, p)). In particular, for any x ∈ LettersB(s, p), +the number of edges in EB(s, p) which belong to the orbit of (s, x, p) under φA is precisely +ux, while vx defines the analogous number for (t, x, p). It follows that there are permutations +θs : LettersB(s, p) → LettersB(s, p) and θt : LettersB(t, p) → LettersB(t, p) induced from the +permutations of edges from s to p (and from t to p respectively) achieved by applying φmr +A . +In particular, for x ∈ LettersB(s, p)(= LettersB(t, p)), we have the cycle of θs containing x +has length ux and the cycle of θt containing x has length vx. +Adding shadows: +Apply Lemma 4.19 to the quadruple (s, t, p, B) to obtain a strongly synchronizing au- +tomaton D and a corresponding automorphism ψA of the digraph GD underlying D so that +A is the minimal representative of H(D, ψA), and the conclusions of Lemma 4.19 are satisfied. +In particular, for any x ∈ LettersB(s, p) we have olψA((s, x, p)) = olψA((t, x, p)) = n. +We now determine various constants arising from the construction so far. +By construction, r remains the length of the orbit of p in D; however, the orbit lengths +of s and t have possibly been padded out with shadow states. +Determine e, f so that +emr = lcm(olψA(s), olψA(p)) and fmr = lcm(olψA(t), olψA(p)). +As LettersB(s, p) = LettersB(t, p) = LettersD(t, p) = LettersD(s, p) we will often use the +notation Letters (s||t, p) for this set, although, we might use one of the other names if we +42 + +specifically wish to emphasise that we are considering the action of φA or of ψA in that case. +Set as well ζ := | Letters (s||t, p)|. +Determining constants and orbit blocks from LettersD(s, p) and LettersD(t, p): +As above, consider the permutation of LettersD(s, p) induced by applying ψemr +A +to GD. +Note that this permutation is θe +s. Similarly ψfmr +A +induces θf +t on LettersD(t, p). Set qe to be the +order of θe +s, so that θqee +s +is the identity permutation. Analogously define qf to be the order of +θf +t . Note in passing that qe = n/(emr) is the number of times the orbit of an edge of the form +(s, x, p) intersects ED(s, p) under the action of ⟨ψA⟩ (and that this number is independent +from the choice of such edge), and similarly, qf = n/(fmr) counts the cardinality of the +intersection of the orbit of an edge of the form (t, x, p) with ED(t, p) under the action of ⟨ψA⟩ +(again, independent of the choice of such an edge). It follows that all cycles of θe +s have length +qe and that all the cycles of θf +t have length qf. +We note that for all x ∈ LettersD(s, p) we have qee = lcm(ux, e) and qff = lcm(vx, f). +Therefore, for any x ∈ Letters (s||t, p), we have qe|ux and qf|vx. Further, as qeemr = qffmr = +n we have qee = qff. +Let u = gcd{ux : x ∈ Letters (s||t, p)} and v = gcd{vx : x ∈ Letters (s||t, p)}. As qe|ux +and qf|vx for all x ∈ Letters (s||t, p), we see that qe|u and qf|v. Let u and v be such that +uqe = u and vqf = v. It also follows that lcm(u, e) = qee = qff = lcm(v, f). +Let τ ∈ {s, t} and set ∼τ,p to be the equivalence relation on the set of edges from τ to +p, where two edges are equivalent under ∼τ,p if they are in the same orbit under ψB. Let +X(τ, p) be a transversal for this equivalence relation. It follows that +� +(s,x,p)∈X(s,p) +ux = +� +(t,x,p)∈X(t,p) +vx = | Letters (s||t, p)| = ζ +so in particular we see that both u and v divide ζ. +Set w := lcm(u, v). Since u and v divide ζ we have that w|ζ. Let α be such that αw = ζ. +Further, as u|w and v|w there are µ, ν ∈ N such that w = µu = µuqe and w = νv = νvqf. +Further, qee = lcm(u, e)| lcm(w, e) and qff = lcm(v, f)| lcm(w, f). Since both u and v divide +qee = qff, it follows that w|qee and so lcm(w, e) = qee = qff = lcm(w, f). In particular, +wmr|qeemr = n and wmr|qffmr = n and we have lcm(wmr, emr) = n = lcm(wmr, fmr). +Building the relabelling: +Our relabelling will be a relabelling through shadows which will create a pattern of +labels along an edge orbit that repeats after every wmr steps under iteration of the new +automorphism ψA′ of GD. +To be a relabelling through shadows we will need that for any 0 ≤ i < mr and integer k +that the local map at sψi+kmr +A′ +agrees with the local map at sψi +A′ (and similarly for the orbit +of t). +We will now define a new labelling λD : Xn × QD → Xn. +Recall that a relabelling +function is always induced from a vertex fixing automorphism of the underlying digraph of +an automaton in the collapse sequence of the original. In our particular construction of λD, +43 + +the reader will see that the automorphism employed in its creation is simply a vertex fixing +automorphism of the underlying digraph GD of D. +Suppose (l, x′, p′) is an edge of D which does not belong to the orbit of an edge (τ, x, p) +for any pair (τ, x) ∈ {s, t} × LettersD(τ, p). In this case we set (x′, l)λB = x′. +For the moment we focus on edges in the orbit of some edge (s, x, p). The edges in the +orbit of (t, x, p) are dealt with analogously. +As | Letters (s||t, p)| = ζ = wα we may partition Letters (s||t, p) into α blocks of size w, +which partition we organise through some labelling of the set’s elements as follows: +Letters (s||t, p) = {xc +i | 0 ≤ i < w, 0 ≤ c < α}, +where here, the cth part, denoted xc, has w elements arranged as the ordered sequence +(xc +i)0≤i 1 (as otherwise we are done). Since B is strongly synchronising, we +may find a pair of distinct states s, t which distribute similarly over QB. +We consider two cases. +First suppose that s and t belong to the same orbit. Fix a state p for which there is an +edge from s (and so from t) into p. +We apply Lemma 4.19 to the triple (s, t, p) to obtain a transducer D′′ and automorphism +ψ′ +A such that there are no shadow states for elements in the orbit of p, edges from s and t +into p are on orbits of length n. +Now since there are no shadow states for elements in the orbit of p in D′′, we may +repeatedly apply Lemma 4.19 to s, t and states in the orbits of p in turn, until we obtain an +automaton D′ and automorphism ψA which has no shadow states for elements in the orbit +of p, such that LettersD′(s, pψi +A) = LettersD′(t, pψi +A) = LettersB(s, pφi +A) for all i ∈ N, and +such that any edge from s or t into a state in the orbit of p has orbit length n. +45 + +Now we apply Lemma 4.7 to the automaton D′. This results in an automaton D and +conjugate automorphism ψA′ with the following properties. For r minimal such that, for +any i, j ∈ N, LettersD(sψi +A′, pψj +A′) = LettersD(sψi+r +A′ , pψj+r +A′ ), then, the labels of the edges +(sψi +A′, x, pψj +A′)ψA′ and (sψi+r +A′ , x, pψj +A′)ψA′ coincide. Notice that, by minimality of r, Shadow +states remain ω-equivalent to the state they shadow and t is an element of the orbit of s +under the action of ψr +A′. +We now apply Lemma 4.17 to collapse down to the automaton B with a conjugate +automorphism φA′. +The conjugate automorphism φA′ has the following properties. +For +edges not belonging to the orbit of an edge from s or t into a state not in the orbit of p, the +action of φA′ coincides with the action of A. By construction of D, the label of the edges +(sφi +A′, x, pφj +A′)φA′ and (tφi +A′, x, tφj +A′)φA′ coincide for any i, j ∈ N. +We now repeat this process across all states of B which have an edge from s. Thus we +end up up with a conjugate automorphism ψC such that the label of the edges (sψi +C, x, q)ψC +and (tφi +C, x, q)ψC coincide, for any state q and any incoming edge from s and t into q labelled +x. +Let B′ be the automaton obtained from B by identifying the pair of states (sψi +C, tψi +C) for +all i ∈ N. Since ψC induces the same action on labels for corresponding edges from elements +sψi +C and tψi +C, then there is an induced action φC of ψC on the underlying digraph of B′. +That is there is an element C ∈ Hn which is a conjugate of A such that C is the minimal +representative of H(B′, φC) an H(B, ψC). +Now consider the case that s, t belong to distinct orbits. We may assume that the orbit +lengths of s and t coincide otherwise we may find a τ ′ distinct from s and t which has the +same orbit length and agrees with one of s or t on QB. Whereby we apply the previous case +to the pair (s, τ) or (t, τ). +Fix a state p ∈ QB with an edge from s (and so from t). We apply Lemma 4.20 to obtain +a conjugate automorphism φA′′ of the underlying digraph of B such that all edges from s +and t into p have the same orbit length. +We repeat the process along all states of B with an incoming edge from s. This yields a +conjugate automorphism φA′ of B, whereby, for a given state q ∈ QB all edges from s and t +into q have the same orbit length under φA′. +We now repeatedly apply Lemma 4.9 to the triple (s, t, B, φA′) to obtain a conjugate +automorphism ψC of B which satisfies the following. For any pair of edges (s, x, q) and +(t, x, q), (s, x, q)ψC an (t, x, q)ψC have the same labels. +This means, we may once more +identify the pair of states (sψi +C, tψi +C) to obtain an action φC of C, the minimal representative +of H(B, ψC), on a smaller automaton B′. +If |C| > 1, then as C is conjugate to A we may now repeat the process with C instead of +A. +Eventually we end up with the single state transducer. +□ +We recall that by Theorem 3.5, for an element A ∈ Hn of finite order, there is a strongly +synchronizing automaton B on which A acts as an automorphism φA of the underlying +digraph of B so that H(B, φA) has minimal representative A. +46 + +Theorem 5.3. Let A ∈ Hn be an element of finite order. Then A is conjugate to an n-cycle +if and only if every element of X−N +n +is on an orbit of length n under the action of A if and +only if for any strongly synchronizing automaton B on which A acts as an automorphism φA +of the underlying digraph of B, every circuit of B is on an orbit of length n under the action +of A. +Proof. The equivalences follow from lemmas 5.1 and 5.2. +□ +5.1 +An Example +In this section we work through an example that illustrates the key ideas of the proof. +Consider the automaton A of Figure 7 which we encountered already in Example 3.6. This +a0 +a1 +a2 +a3 +1|4 0|5 +2|0 +3|1 +4|2 5|3 +4|2 +5|3 +0|4 1|5 +2|1 +3|0 +1|4 +0|5 +4|2 5|3 +2|1 3|0 +1|5 +0|4 +2|0 3|1 +4|2 5|3 +Figure 7: The element A +is an element of H6 of order 6, where every point in the Cantor space X−N +6 +is on an orbit of +length 6 under the action of A. Following the construction in Subsection 3.3.1 (see Example +3.6), the minimal strongly synchronising automaton B which admits an automorphism φA +of GB such that H(B, φA) has minimal representative A, is as depicted in Figure 8, where +each drawn edge represents two edges with labels as listed; the map φA on the vertices on +GB is the permutation which in cycle notation is +(p0 p1 p2)(q0 q1 q2); +the action of φA on the vertices and edges of GB is uniquely determined from the fact that A +is the minimal representative of H(GB, φA). We refer to the vertices q0, q1, q2 as the vertices +of the “inner triangle” and the vertices p0, p1, p2 as the vertices of the “outer triangle”. +47 + +p0 +q0 +q2 +q1 +p1 +p2 +1 +0 +2 +3 +4 +5 +4 +5 +0 +1 +2 +3 +1 +0 +4 +5 +2 +3 +1 +0 +2 +3 +4 +5 +2 +3 +5 +4 +1 +0 +0 +1 +3 +2 +4 +5 +Figure 8: Minimal automaton witnessing finite order of A. +We notice that the automaton B has the property that its synchronizing and amalgama- +tion sequences cohere. In particular both reduce to the single vertex with 6 looped edges +after 2 steps. +The fact that every circuit of GB is on orbit of length 6 can be seen as follows. A circuit of +GB which is not formed by repeating the circuit (or a cyclic rotation of it) p0 → p1 → p2 → p0 +a finite number of times, must have an edge leaving a vertex in the inner triangle — any +such edge has orbit length 6. +Therefore A satisfies the hypothesis of Theorem 5.3. We work through Lemma 5.2 to +find an element of H6 which conjugates A to a 6-cycle. +In the first step we find two states of GB which can be collapsed i.e. which distribute +similarly over QB. We may take the pair (p0, q0) (any other valid pair belongs to the orbit +of this one). The orbits of p0 and q0 are distinct so we are in the second case of Lemma 5.2. +Now all edges leaving any vertex in the orbit of q0 have orbit length 6, whereas the edges +48 + +edges from p0 to q0 and p0 to q2 have orbit length 3 while the edges from p0 to p1 have orbit +length 6. Thus we apply Lemma 4.20. We add shadow states using Lemma4.19. Focusing +on the vertex q0 as our vertex p (in the notation of Lemma4.19), we see that +Q(0, q0) = {q0} +Q(1, q0) = {p0, p1} +Q(2, q0) = ∅. +The last follows since any incoming edge to a vertex on the outer triangle is on an orbit of +length 6. +We may take either p0 or p1 as the heavy state (since they belong to the same orbit). Our +divisibility constant is 3 (the orbit lengths of the two orbits of edges from the outer triangle +into the inner triangle, i.e. the edge from p0 into q0 represents one such orbit, and the edge +from p1 into q0 represents the other); the number n′ is precisely 6 – since every incoming +edge into either p0 or p1 has orbit length 6. (We note as an aside that since the edge from +p1 to q0 is in the orbit of the edge from p0 to q2, we only need one round of adding shadow +states in order to fix the orbit lengths of both of these edges, using more words, we need not +consider p = q2 as a separate case). +Our new automaton B′ will have shadow states p′ +0, p′ +1 and p′ +2 as is as depicted in Figure 9. +There is a lift ψA of φA to GB′. The action of ψA′ is uniquely determined by the facts that +the orbit of p0 under ψA′ is (p0 p1 p2 p′ +0 p′ +1 p′ +2) and H(B′, ψA) has minimal representative A. +49 + +p′ +0 +p0 +q0 +q2 +q1 +p1 +p′ +1 +p2 +p′ +2 +1 +0 +2 +3 +4 +5 +1 +0 +2 +3 +4 +5 +4 +5 +0 +1 +2 +3 +1 +0 +4 +5 +2 +3 +1 +0 +2 +3 +4 +5 +2 +3 +5 +4 +1 +0 +2 +3 +5 +4 +1 +0 +1 +0 +3 +2 +4 +5 +1 +0 +3 +2 +4 +5 +Figure 9: Adding shadows to form B′. +We can now apply Lemma 4.20 to the orbit of the edge from p2 to q0 and from p1 to q0. +Notice that since the orbit lengths of edges leaving the inner triangle is 6, the relabelling +map of Lemma 4.20 will simply wrap around the orbits of the relevant edges from p2 and p1 +to increase their orbit lengths after re-identifying shadow states to 6. This can be achieved +by relabelling such that the actions on letters of orbits in the edge (p0, {1, 0}, q0) mirrors +the action on the corresponding edge in the orbit of (q0, {0, 1}, q0) (similarly for the pair +(p1, {1, 0}, q0} and (q1, {1, 0}, q0). One such relabelling is that induced by the vertex fixing +50 + +automorphism of GB′ that swaps the edges from p1 to q1, the edges from p2 to q2, the edges +from p′ +0 to q0; the edges from p1 to q0, from p2 to q1 and from p′ +0 to q2. This gives rise to the +element C in figure 10. +c0 +c1 +c3 +c5 +c4 +c2 +1|0 +0|1 +2|3 +3|2 +4|4 +5|5 +4|4 +0|0 1|1 +2|2 +3|3 +5|5 +0|0 +4|4 +5|5 +2|2 3|3 +1|1 +1|1 +0|0 +2|2 +4|4 +5|5 +3|3 +2|2 +3|3 +5|4 +4|5 +1|0 +0|1 +0|0 +1|1 +3|2 +2|3 +4|5 +5|4 +Figure 10: Conjugator C. +The reader can verify that the conjugate of A by C is the automaton D to the left of +Figure 11. +The automaton E to the right of Figure 11 admits an automorphism φD of its under- +lying digraph such that H(E, φD) has minimal representative D. The map φD is uniquely +determined by the fact that H(E, φD) has minimal representative D. Notice that all edges +of GD are on orbits of length 6 and the collapse and amalgamation sequences of GD coincide. +Following Lemma 5.2, we find a pair of vertices which distribute similarly over QD, any pair +of distinct vertices works — we choose (a1, a3). Now we are in the first case of Lemma 5.2 +and the relabelling protocol we apply is that given by Lemma 4.7. Essentially we want to +relabel such that the action of a1 and a3 on X6 coincide along their orbits. A relabelling +that achieves this is obtained by swapping the edged between a1 and a5 and between a5 and +a3. This relabelling gives rise to the conjugator F in figure 12 +The reader can verify that conjugating D by F results in the single state transducer +corresponding to the 6-cycle (0 4 2 1 5 3). +51 + +a1 +a3 +a5 +4|2 +5|3 +0|4 1|5 +2|1 +3|0 +1|4 +0|5 +4|2 +5|3 +2|1 3|0 +1|5 +0|4 +2|0 +3|1 +4|2 +5|3 +a1 +a3 +a5 +4 +5 +0 +1 +2 +3 +1 +0 +4 +5 +2 +3 +1 +0 +2 +3 +4 +5 +Figure 11: Conjugate of A by relabelling map C. +a1 +a3 +a5 +4|5 +5|4 +0|0 1|1 +2|3 +3|3 +1|1 +0|0 +4|4 +5|5 +2|2 3|3 +1|1 +0|0 +2|3 +3|2 +4|4 +5|5 +Figure 12: Conjugator F. +Therefore, the element CF of H6 conjugates A to the 6-cycle (0 4 2 1 5 3). +References +[1] +Ali Akhavi, Ines Klimann, Sylvain Lombardy, Jean Mairesse, and Matthieu Picantin. +“On the finiteness problem for automaton (semi)groups”. In: Internat. J. 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American Mathematical Society, Providence, RI, 2005, pp. xii+231. isbn: +0-8218-3831-8. doi: 10.1090/surv/117. url: https://doi.org/10.1090/surv/117. +[15] +Feyishayo Olukoya. The growth rates of automaton groups generated by reset automata. +submitted. 2017. +[16] +Mikhail V. Volkov. “Language and Automata Theory and Applications”. In: ed. by +Carlos Mart´ın-Vide, Friedrich Otto, and Henning Fernau. Berlin, Heidelberg: Springer- +Verlag, 2008. Chap. Synchronizing Automata and the ČErný Conjecture, +pp. 11–27. isbn: 978-3-540-88281-7. doi: 10.1007/978-3-540-88282-4_4. url: http://dx.doi.org/10.1007/978-3-540-88282-4_4. +[17] +R. F. Williams. “Classification of subshifts of finite type”. In: Ann. of Math. (2) 98 +(1973), 120–153; errata, ibid. (2) 99 (1974), 380–381. issn: 0003-486X. +53 + diff --git a/qNFRT4oBgHgl3EQfeDf8/content/tmp_files/load_file.txt b/qNFRT4oBgHgl3EQfeDf8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c53b35bc498fd9a71de3338a64f1523368863e7a --- /dev/null +++ b/qNFRT4oBgHgl3EQfeDf8/content/tmp_files/load_file.txt @@ -0,0 +1,2249 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf,len=2248 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13570v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='GR] 31 Jan 2023 Conjugacy for certain automorphisms of the one-sided shift via transducers Collin Bleak and Feyishayo Olukoya Abstract We address the following open problem, implicit in the 1990 article Automorphisms of one-sided subshifts of finite type of Boyle, Franks and Kitchens (BFK): Does there exists an element ψ in the group of automorphisms of the one- sided shift Aut({0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , n − 1}N, σn) so that all points of {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , n − 1}N have orbits of length n under ψ and ψ is not conjugate to a permutation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Here, by a permutation we mean an automorphism of one-sided shift dynamical system induced by a permutation of the symbol set {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We resolve this question by showing that any ψ with properties as above must be conjugate to a permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Our techniques naturally extend those of BFK using the strongly synchronizing automata technology developed here and in several articles of the authors and collab- orators (although, this article has been written to be largely self-contained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Preliminaries 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 The natural numbers and some of its subsets .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 47 1 Introduction Let n be a positive integer and set Xn := {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We will use Xn to represent our standard alphabet of size n and we will denote by σn the usual shift map on XN n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The group Aut(XN n , σn) of homeomorphisms of XN n which commute with the shift map is called the group of automorphisms of the shift dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This is a well-studied group in symbolic dynamics, with the special property (first given by Hedlund in [10]) that if φ ∈ Aut(XN n , σn) has (x0x1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' )φ = y0y1y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' then there is an integer k so that for all indices i, the value yi is determined by the finite word xixi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' xi+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The paper [6] characterises all of the finite subgroups of the group Aut(XN n , σn), shows that this group contains non-abelian free groups whenever n ≥ 3, and investigates other algebraic structures of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The papers [7, 5] develop a conjugacy invariant for the group Aut(XN n , σn), arising from the action of the group on periodic words, which for an automorphism φ we will denote as Sp(φ) (this invariant consists of a tuple: the well-known gyration and sign functions, together with first return data: bundled data associated to the permutation representation on prime words of length k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This article resolves the following open problem, implicit in [6], which Mike Boyle sug- gested to us for its own sake, and, as a test of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let Σn represent the group of permutations of the set Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By a mild abuse of language, we say φ ∈ Aut(XN n , σn) is a permutation if there is a fixed permutation α ∈ Σn so that if (x0x1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' )φ = y0y1y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' then we have yi = (xi)α for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We say a permutation is a rotation if the permutation from Σn is an n-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We can now state the problem: Does there exist an automorphism ψ ∈ Aut(XN n , σn) of order n so that all points of XN n travel on orbits of size n, where ψ is not conjugate to a rotation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In [6] Boyle, Franks and Kitchens show that if n is prime then any such ψ is in fact conjugate to a rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We show that the Boyle, Franks, and Kitchens result holds for general n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 2 We have written this article so that it is essentially self-contained for general researchers working with automorphisms of the shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular, we gather definitions and key con- structions from [15] and [4] here to simplify the presentation without insisting the reader peruse those articles to follow our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We use the highlighted technology to enhance the key method in the article [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The paper [4] shows how to represent any automorphism φ of the one-sided shift by a particularly nice family of transducers (finite state machines that transform inputs sequentially) while [15] investigates the order problem for that same family of transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A key idea of [4] is that any such transducer T representing φ can be thought of as a triple (D, R, φ∗), where D and R are strongly synchronizing automata (edge-labelled directed graphs with the particularly nice property of having a synchronizing sequence) with D representing the domain and R representing the range, and where φ∗ is an isomorphism of the underlying digraphs Γ(D) and Γ(R) of D and R determined by the action of φ on periodic words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the case of a finite order element, the domain and range automata can also be chosen to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the article [6] the central method for studying finite subgroups of Aut(XN n , σn) is firstly to find an action of the group on the underlying digraph of an automaton (now understood to be a strongly synchronizing automaton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Once the first step is accomplished, the group is decomposed as a composition series where each composition factor is isomorphic to a subgroup of the symmetric group Σn on n-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This is accomplished by pushing the action down along what is called an “amalgamation sequence” (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2 here) of the digraph until one has an action by automorphisms on a particularly nice digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The construction typically requires passing through the automorphism groups of various one-sided shifts of finite type via topological conjugations induced by the amalgamations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Our first step simplifies this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular we show that we can always find an action of a finite subgroup of Aut(XN n , σn) on the underlying digraph of a strongly synchro- nizing automaton whose amalgamation and synchronizing sequences cohere (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2), thus we can push down along the synchronizing sequence of that automaton without needing to possibly change alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This is already enough, when n is prime, to show that every element of order p in Aut(XN n , σn) is conjugate in Aut(XN n , σn) to a rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' However, to answer the open problem above, we need to go beyond this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose φ ∈ Aut(XN n , σn) has order n and with the condition (⋆) that all points of XN n travel on orbits of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It turns out that (⋆) is equivalent to the condition that for any transducer (A, A, φ∗) representing φ, the action of φ∗ on Γ(A) has the property that for every (based) circuit C of Γ(A) the orbit length of C under this action is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (We are using based circuits here to avoid a circuit returning to itself with some non-trivial rotation as counting as completing the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') When n is a prime p, it is not hard to see that the action of φ∗ on the underlying digraph is limited in orbit lengths for edges and vertices to 1 and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' When n is not prime, orbit lengths of edges and vertices can be any divisor of n even though all circuits have orbit length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This last issue creates problems when trying to implement the approach successfully carried out by Boyle et al for n prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We overcome this issue for such a φ with representative transducer (A, A, φ∗) with several technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' These aim to show that the automaton A can be “fluffed up” by adding 3 shadow states (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3) to create a new strongly synchronizing automaton B with an induced and more informative action ψ∗ on Γ(B) so that (B, B, ψ∗) still represents φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By ‘more informative’ we mean that the correct addition of shadow states results in states and edges originally on orbits of length < n having resulting orbits of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This new action makes it possible to find a conjugate action of φ on a strongly synchronizing automaton of strictly smaller size than A (where the conjugacy occurs entirely with Aut(XN n , σn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Our approach can now be summarised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' First we conjugate to get a (conjugate) action of φ on a strongly synchronizing automaton whose synchronizing sequence coheres with the amalgamation sequence of its underlying digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then we have a series of “fluffing up” moves followed by reductions via conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Eventually, these processes result in a conjugate action given by a transducer over a single state automaton with n labelled loops, where each edge is on an orbit of length n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' our original element φ must then be conjugate to a rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The example in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 might prove helpful to the reader as an illustration of our approach and of the difficulties discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The property of being a strongly synchronizing automaton is equivalent to that of being a folded de Bruijn graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Crucial to the approach we have sketched out is the process: given a finite order element φ ∈ Aut(XN n , σn), find the minimal folded de Bruijn graph Γ so that φ acts faithfully on Γ by automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following is essentially a result from [4] stated in our context (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='4 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5, below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let n ≥ 2 be an integer and suppose φ ∈ Aut(XN n , σn) is a finite order element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' There is an effective process for determining Γφ, the minimal folded de Bruijn graph on an n letter alphabet, so that φ induces a natural automorphism of Γφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Finally, we can state the theorem which answers the question of Boyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let φ ∈ Aut(XN n , σn) be an element of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following are equiv- alent: φ is conjugate to a rotation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' every element of XN n is on an orbit of length n under the action of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' and for any folded de Bruijn graph Γφ admitting a faithful action by φ via an automorphism φ∗, every (based) circuit of Γφ is on an orbit of length n under φ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It is unclear at the moment how much our approach depends on the condition that “all circuits are on orbits of length n”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In work in progress we aim to extend our current ideas towards resolving the conjugacy problem for finite order elements of Aut(XN n , σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Acknowledgements The authors are grateful for partial support from EPSRC research grant EP/R032866/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The second author is additionally grateful for support from Leverhulme Trust Research Project Grant RPG-2017-159 and LMS ECF grant ECF-1920-105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Finally, we are also grateful to Mike Boyle for conversations on the question we address here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 4 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 The natural numbers and some of its subsets We use the notation N for the set {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' for j ∈ N we write Nj for the set {i ∈ N : 1 ≥ j} of all natural numbers which are bigger than or equal to j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2 Words and infinite sequences In this subsection we set up necessary notation for words and sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Firstly, we employ all of the usual notation around finitary words over the alphabet X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Namely, for a base set X, and natural n, Xn is the set of ordered n-tuples with coordinates from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We call these the words of length n (over alphabet X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By convention, we set X0 := {ε} and we refer to ε as the empty word or empty string, proclaiming this to be the same object, independent of the (non-empty) set X used as our alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We set X∗ := ∪n∈NXn, the words of finite length over X (this is the Kleene-star operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We also set X+ := X∗\\{ε}, the non-trivial/non-empty finite length words over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If w ∈ X∗ we set |w| = n where w ∈ Xn, and we call |w| the length of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If X has a linear order <, then we give X∗ the induced dictionary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If u ∈ Xn then we implicitly set values ui ∈ X for 0 ≤ i < n so that u = (u0, ui, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , un−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this context, from here forward we will simply write u = u0u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' un−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For u ∈ Xn and i ≤ |u|, we write u[1,i] for the prefix u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' ui of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Finally, if u, v ∈ X∗, so that u = u0u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' ur−1 and v = v0v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' vs−1 then uv will represent the concatenation of these words: uv := u0u1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' ur−1v0v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' vs−1, which is a word of length r + s over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As in the paper [4], we take X−N n := {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' x−2x−1x0 | xi ∈ Xn} as our shift space, with the shift operator σn defined by (xi)i∈−Nσn = (yi)i∈−N where we have yi = xi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We use the characterisation of elements of Hn as strongly synchronizing transducers corresponding to shift commuting automorphisms of X−N n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For a finite-length word over Xn we may index this word with negative or positive indices as seems natural at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' When we are explicitly thinking of a finite subword w ∈ Xk n of a point x ∈ X−N n we will ordinarily index w as w = wi−k+1wi−k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' wi for some i ∈ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose k is a positive integer and u = u−(k−1)u−(k−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' u−1u0 ∈ Xk n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define uω ∈ X−N n , by which notation we mean the point .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' xmxm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' x−1x0 =: x where xi = ui (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The word x ∈ X−N n is called a periodic word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The period of x is the smallest j ∈ N such that (x)σj n = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If the length |u| is the period of the word x, then u is called prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Alternatively u is prime if there is no smaller word γ ∈ X+ n such that u = γi for some i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Write Xk n for the full set of prime words of length k over the alphabet Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Given two words u, v ∈ X+ n such that |u| = |v| = r, we call v a rotation of u if there is an i ∈ N with (uω)σi n = vω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this case, we may refer to v as the ith-rotation of u (even if i > |u|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It is a well-known fact that an element φ ∈ Aut(X−N n , σn) preserves the period of a periodic element of X−N n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this way, the action of Aut(X−N n , σn) on periodic words gives a representation from Aut(X−N n , σn) to the group Πk∈N Sym(Xk n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For φ ∈ Aut(X−N n , σn), write 5 φk for the action of φ on prime words of length k and write φ for the element (φk)k∈N ∈ Πk∈N Sym(Xk n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The map φ is the periodic point representation of Aut(X−N n , σn), introduced in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3 Automata and transducers An automaton, in our context, is a triple A = (XA, QA, πA), where (a) XA is a finite set called the alphabet of A (we assume that this has cardinality n, and identify it with Xn, for some n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (b) QA is a finite set called the set of states of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (c) πA is a function XA × QA → QA, called the transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The size of an automaton A is the cardinality of its state set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We use the notation |A| for the size of the A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We regard an automaton A as operating as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If it is in state q and reads symbol a (which we suppose to be written on an input tape), it moves into state πA(a, q) before reading the next symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As this suggests, we can imagine that the automaton A is in the middle of an input word, reads the next letter and moves to the right, possibly changing state in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We can extend the notation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For w ∈ Xm n , let πA(w, q) be the final state of the automaton which reads the word w from initial state q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus, if w = x0x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' xm−1, then πA(w, q) = πA(xm−1, πA(xm−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , πA(x0, q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By convention, we take πA(ε, q) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For a given state q ∈ QA, we call the automaton A which starts in state q an initial automaton, denoted by Aq, and say that it is initialised at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' An automaton A can be represented by a labeled directed graph GA, whose vertex set VA is QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For this directed graph there is a directed edge labeled by a ∈ XA from p to q if πA(a, p) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Representing this, we determine the set EA of edges of GA to be the set of triples EA := {(p, a, q) | ∃p, q ∈ QA, a ∈ XA, so that πA(a, p) = q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In what follows, the labelled directed graph GA will be referred to as the underlying digraph for the automaton A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A transducer is a quadruple T = (XT, QT, πT, λT), where (a) (XT, QT, πT) is an automaton;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (b) λT : XT × QT → X∗ T is the output function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 6 Formally such a transducer is an automaton which can write as well as read;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' after reading symbol a in state q, it writes the string λT(a, q) on an output tape, and makes a transition into state πT(a, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus, the size of a transducer is the size of its underlying automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' An initial transducer Tq is simply a transducer which starts processing input from state q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Transducers which are synchronous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=', which always write one letter whenever they read one letter) are also known as Mealy machines (see [9]), although we generally will not use that language here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Transducers which are not synchronous are described as asynchronous when this aspect of the transducer is being highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this paper, we will only work with synchronous transducers without an initial state, and, henceforth we simply call these transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the same manner as for automata, we can extend the notation to allow transducers to act on finite strings: we let πT (w, q) and λT(w, q) be, respectively, the final state and the concatenation of all the outputs obtained when a transducer T reads a string w from a state q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A transducer T can also be represented as an edge-labeled directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Again the vertex set is QT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' now, if πT(a, q) = r, we put an edge with label a|λT(a, q) from q to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In other words, the edge label describes both the input and the output associated with that edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We call a the input label of the edge and λT(a, q) the output label of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For example, Figure 1 describes a synchronous transducer over the alphabet X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' a1 a2 0|0 1|0 1|1 0|1 Figure 1: A transducer over X2 In what follows, we only use the language automaton for those automata which are not transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This allows us characterise a synchronous transducer T as a pair of automata together with a directed graph isomorphism “gluing” the two automata together as a domain automaton and a range automaton (we split any edge label ‘x|y’ of T as specifying the domain automaton edge with label x and the range automaton edge with label y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We can regard any state q of a transducer as acting on an infinite string from XN n where Xn is the alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This action is given by iterating the action on a single symbol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' so the output string is given by λT(xw, q) = λT(x, q)λT(w, πT(x, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus Tq induces a map w �→ λT(w, q) from XN n to itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' it is easy to see that this map is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If it is a homeomorphism, then we call the state q a homeomorphism state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We write Im(q) for the image of the map induced by Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 7 Two states q1 and q2 are said to be ω-equivalent if the transducers Tq1 and Tq2 induce the same continuous map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (This can be checked in finite time, see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') More generally, we say that two initial transducers Tq and T ′ q′ are ω-equivalent if they induce the same continuous map on XN n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A transducer is said to be minimal if no two states are ω-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For a transducer T, two states q1 and q2 are ω-equivalent if λT(a, q1) = λT(a, q2) for any finite word a ∈ X∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Moreover, if q1 and q2 are ω-equivalent states of a synchronous transducer, then for any finite word a ∈ Xp n, πT(a, q1) and πT(a, q2) are also ω-equivalent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Two minimal non-initial transducers T and U are said to be ω-equal if there is a bijection f : QT → QU, such that for any q ∈ QT , Tq is ω-equivalent to U(q)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Two minimal initial transducers Tp and Uq are said to be ω-equal if they are ω-equal as non-initial transducers and there is a bijection f : QT → QU witnessing this which satisfies the equality (p)f = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We use the symbol ‘=’ to represent ω-equality of initial and non-initial transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Two non-initial transducers T and U are said to be ω-equivalent if they have ω-equal minimal representatives, and in this case we might instead say T and U represent the same transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the class of synchronous transducers, the ω-equivalence class of any transducer has a unique minimal representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Throughout this article, as a matter of convenience, we shall not distinguish between ω- equivalent transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus, for example, we introduce various groups as if the elements of those groups are transducers, whereas the elements of these groups are in fact ω-equivalence classes of transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Given two transducers T = (Xn, QT, πT, λT) and U = (Xn, QU, πU, λU) with the same alphabet Xn, we define their product T ∗ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The intuition is that the output for T will become the input for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus we take the alphabet of T ∗ U to be Xn, the set of states to be QT∗U = QT × QU, and define the transition and rewrite functions by the rules πT∗U(x, (p, q)) = (πT(x, p), πU(λT(x, p), q)), λT∗U(x, (p, q)) = λU(λT(x, p), q), for x ∈ Xn, p ∈ QT and q ∈ QU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Here we use the earlier convention about extending λ and π to the case when the transducer reads a finite string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If T and U are initial with initial states q and p respectively then the state (q, p) is considered the initial state of the product transducer T ∗ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In automata theory a synchronous (not necessarily initial) transducer T = (Xn, QT, πT, λT) is invertible if for any state q of T, the map ρq := λT(�, q) : Xn → Xn is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this case the inverse of T is the transducer T −1 with state set QT −1 := {q−1 | q ∈ QT }, transition function πT −1 : Xn ×QT −1 → QT −1 defined by (x, p−1) �→ q−1 if and only if πT((x)ρ−1 p , p) = q, and output function λT −1 : Xn ×QT −1 → Xn defined by (x, p) �→ (x)ρ−1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus, in the graph of the transducer T we simply switch the input labels with the output labels and append ‘−1’ to the state names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We are concerned only with invertible, synchronous transducers in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='4 Increasing alphabet size and the dual automaton We require a couple of standard constructions in the theory of synchronous automata in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' First we consider the ‘paths to letters’ construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let T be a transducer over the alphabet Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let m ∈ N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Write T(m) for the transducer over the alphabet Xm n with state set QT and transition and output functions πT(m), λT(m) satisfying the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For x ∈ Xm n and q ∈ QT we set πT(m)(x, q) = p if and only if πT (x, q) = p in T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' we set λT(m)(x, q) := λT(x, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It is clear that if T is minimal and invertible, the T(m) is also minimal and invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The other construction we require is the dual automaton (see [1, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Again let T be a transducer over the alphabet Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set T ∨ = ⟨QT, Xn, π∨ T, λ∨ T⟩, that is the state set of T ∨ is the set Xn, the alphabet of T ∨ is the state set QT of T, and the transition π∨ T and output functions λ∨ T are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For q ∈ QT and x ∈ Xn, π∨ T (q, x) = y and λ∨ T(q, x) = p if and only if πT(x, q) = p and λT(x, q) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' There is a connection between the two constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following is standard in the theory of synchronous automata and provides a key insight in the analysis of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let T be a synchronous transducer over alphabet Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For positive natural m, we have (T ∨)m = T(m)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that to lighten our notation below, we may use the notation T ∨ m for the transducer T(m)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Also observe that T −1∨ is obtained from T ∨ by ‘reversing the arrows’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' That is if, x, y ∈ Xn, q, p ∈ QT are such that π∨ T(q, x) = y and λ∨(q, x) = p, then π∨ T −1(q−1, y) = x and λ∨(q−1, y) = p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5 Synchronizing automata and bisynchronizing transducers Given a natural number k, we say that an automaton A with alphabet Xn is synchronizing at level k if there is a map sk : Xk n �→ QA such that, for all q and any word w ∈ Xk n, we have πA(w, q) = sk(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In other words, A is synchronizing at level k if, after reading a word w of length k from a state q, the final state depends only on w and not on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (Again we use the extension of πA to allow the reading of an input string rather than a single symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') We call sk(w) the state of A forced by w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' the map sk is called the synchronizing map at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' An automaton A is called strongly synchronizing if it is synchronizing at level k for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We remark here that the notion of synchronization occurs in automata theory in consid- erations around the ˇCern´y conjecture, in a weaker sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A word w is said to be a reset word for A if πA(w, q) is independent of q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' an automaton is called synchronizing if it has a reset word [16, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Our definition of “synchonizing at level k”/“strongly synchronizing” requires every word of length k to be a reset word for the automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If the automaton A is synchronizing at level k, we define the core of A to be the maximal sub-automaton with set of states those states in the image of the map s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It is an easy observation that, if A is synchronizing at level k, then its core is an automaton in its own 9 right using the same alphabet, and is also synchronizing at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We denote this automaton by core(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We say that an automaton or transducer is core if it is equal to its core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Clearly, if A is synchronizing at level k, then it is synchronizing at level l for all l ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let Tq be an initial transducer which is invertible with inverse T −1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If Tq is synchronizing at level k, and T −1 q is synchronizing at level l, we say that Tq is bisynchronizing at level (k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If Tq is invertible and is synchronizing at level k but not bisynchronizing, we say that it is one-way synchronizing at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For a non-initial invertible transducer T we also say T is bi-synchronizing (at level (k, l)) if both T and its inverse T −1 are synchronizing at levels k and l respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that if T is a strongly synchronizing transducer, then for any m ∈ N, T(m) is also strongly synchronizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Moreover, if k the minimal synchronizing level of T, then T(m) is synchronizing at level 1 for any m ≥ k and, more generally, is synchronizing at level ⌈k/m⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let T be a transducer which is synchronizing at level k and let l ≥ k be any natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then for any word w ∈ Xl n, we write qw for the state sl(w), where sl : Xl n → QT is the synchronizing map at level l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following result was proved in Bleak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let T, U be transducers which (as automata) are synchronizing at levels j, k respectively, Then T ∗ U is synchronizing at level j + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that in the statement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3, the lowest synchronizing level of T ∗ U might actually be less than j + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let T be a transducer which (regarded as an automaton) is synchronizing at level k, then the core of T (similarly denoted core(T)) induces a continuous map fT : X−N n → X−N n as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let x ∈ X−N n and set y ∈ X−N n to be the sequence defined by yi = λT(xi, qxi−kxi−(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='xi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that πT(xi, qxi−kxi−(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='xi−1) = qxi−kxi−(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='xi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set (x)fT = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus, from the point of view of the transition function of T we in fact begin processing x at −∞ and move towards x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (This is in keeping with our interpretation of transducer as repre- senting machines applying sliding block codes, where here, we are thinking of Aut(X−N n , σn) as consisting of the sliding block code transformations that require past information only to determine what to do with a digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') Note, moreover, that the map fT is independent of the (valid) synchronizing level chosen to define it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We have the following result: 10 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' [4] Let T be a minimal transducer which is synchronizing at level k and which is core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then fT ∈ End(X−N n , σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The transducer in Figure 1 induces the shift map on X−N n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In [3], the authors show that the set � Hn of minimal finite synchronizing invertible syn- chronous core transducers is a monoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' the monoid operation consists of taking the product of transducers and reducing it by removing non-core states and identifying ω-equivalent states to obtain a minimal and synchronous representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let Hn be the subset of � Hn consisting of transducers which are bi-synchronizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A chief result of [4] is that Aut(X−N n , σn) ∼= Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='6 De Bruijn graphs and folded automata The de Bruijn graph G(n, m) can be defined as follows, for integers m ≥ 1 and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The vertex set is Xm n , where Xn is the alphabet {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , n−1} of cardinality n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' There is a directed arc from a0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' am−1 to a1a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' am, with label am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that, in the literature, the directed edge is also from a0a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' am−1 to a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' am−1am and the label on this edge is often given as the (m + 1)-tuple a0a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' am−1am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' However, the labelling given above produces an isomorphic graph and is better suited for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Figure 2 shows the de Bruijn graph G(3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Observe that the de Bruijn graph G(n, m) describes an automaton over the alphabet Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Moreover, this automaton is synchronizing at level m: when it reads the string b0b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' bm−1 from any initial state, it moves into the state labeled b0b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' bm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The de Bruijn graph is, in a sense we now describe, the universal automaton over Xn which is synchronizing at level m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We define a folding of an automaton A over the alphabet Xn to be an equivalence relation ≡ on the state set of A with the property that, if a ≡ a′ and πA(x, a) = b, πA(x, a′) = b′, then b ≡ b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' That is, reading the same letter from equivalent states takes the automaton to equivalent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If ≡ is a folding of A, then we can uniquely define the folded automaton A/≡: the state set is the set of ≡-classes of states of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' and, denoting the ≡-class of a by [a], we have πA/≡(x, [a]) = [πA(x, a)] (note that this is well-defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' [4] The following are equivalent for an automaton A on the alphabet Xn: A is synchronizing at level m, and is core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A is the folded automaton from a folding of the de Bruijn graph G(n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may think of a de Bruijn graph G(n, m) as determining a finite category, with ob- jects the foldings of G(n, m) and with arrows digraph morphisms which commute with the transition maps of the given automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It is immediate in that point of view that all such arrows are surjective digraph morphisms (and indeed, these are folding maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 11 00 11 22 01 10 02 20 12 21 0 1 2 1 0 2 2 1 0 1 0 2 1 0 2 0 1 2 0 2 1 2 0 1 0 2 1 Figure 2: The de Bruijn graph G(3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7 Automorphisms of digraphs underlying de Bruijn graphs and Hn In this section we describe finite order elements of Hn as automorphisms of folded de Bruijn graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be a finite automaton on edge-alphabet Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Recall (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3) that an automaton A may be regarded as labeled directed graph with vertex set QA, and edge set EA ⊂ QA × Xn × QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We let GA denote the unlabeled directed graph corresponding to an automaton A, but we retain the triple (p, x, q) to denote the edge of GA underlying the edge (p, x, q) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let φ be an automorphism of the directed graph GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let H(A, φ) be a transducer with state set QH(A,φ) := QA, 12 alphabet set Xn, transition function πH(A,φ) := πA, and output function λH(A,φ) : Xn × QH(A,φ) → Xn, where λH(A,φ)(x, p) = y if and only if there are edges (p, x, q) and (r, y, s) of GA so that (p, x, q) is taken to (r, y, s) by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The transducer H(A, φ) can be thought of as the result of gluing the automaton A to a copy of itself along the map φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' That is, if p, q ∈ QA and (p, x, q) is an edge from p to q with label x in A, and if y is the label of the edge ((p, x, q))φ in A, then the vertex p is identified with the vertex (p)φ, the vertex q with the vertex (q)φ, the edge (p, x, q) is identified with the edge ((p, x, q))φ and has input label x and the output label y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We make a few observations: (a) For each state q ∈ QH(A,φ), the map λH(A,φ)(�, q) : Xn → Xn is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This follows from the definition of GA: for each x ∈ Xn there is precisely one edge of the form ((q)φ, x, p) based at the vertex (q)φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that the transducer H(A, φ) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (b) If A is synchronizing at level k (and so a folding of G(n, k) by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5) then both H(A, φ) and H(A, φ)−1 are synchronizing at level k hence the minimal representative H(A, φ) of H(A, φ) is an element of Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (c) In fact, for a state q ∈ QA, if Wk,q is the set of words of length k, that force the state q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=', Wk,q := {a ∈ Xk n : πH(A,φ)(a, q) = q}, then {λH(A,φ)(a, p) | a ∈ Qk,q, p ∈ QH(A,φ)} is equal to Wk,(q)φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (d) An element of Hn which can be represented by a transducer H(A, φ) for some folded de Bruijn graph A and digraph automorphism φ of GA must have finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If A ∈ Hn and B is an automaton so that there is a digraph automorphism φ : GB → GB so that A and H(B, φ) represent the same transformation then we say A is induced from (B, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='8 Synchronizing sequences and collapse chains We require an algorithm given in [3] for detecting when an automaton is strongly synchro- nizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We state a version below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A = (Xn, QA, πA) be an automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define an equivalence relation ∼A on the states of A by p ∼A q if and only if the maps πA(·, p) : QA → QA and πA(·, q) : QA → QA are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For a state q ∈ QA let q represent the equivalence class of q under ∼A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Further set QA := {q | q ∈ QA} and let πA : QA → QA be defined by πA(x, q) = p where p = πA(x, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 13 Observe that πA is a well defined map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define a new automaton A = (Xn, QA, πA) noting that |QA| ≤ |QA| and |QA| = |QA| implies that A is isomorphic to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Given an automaton A, let A0 := A, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' be the sequence of automata such that Ai = Ai−1 for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We call the sequence (Ai)i∈N the synchronizing sequence of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We make a few observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By definition each term in the synchronizing sequence is a folding of the automaton which precedes it, therefore there is a j ∈ N such that all the Ai for i ≥ j are isomorphic to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By a simple induction argument, for each i, the states of Ai corresponds to a partition of QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We identify the states of Ai with this partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For two states q, p ∈ QA that belong to a state P of Ai, πA(x, q) and πA(x, p) belong to the same state of QAi for all x ∈ Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We will use the language ‘two states of A are identified at level i’ if the two named states belong to the same element of QAi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If the automaton A is strongly synchronizing and core, then an easy induction argument shows that all the terms in its synchronizing sequence are core and strongly synchronizing as well (since they are all foldings of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For example if A = G(n, m), then the first m terms of the synchronizing sequence of A are (G(n, m), G(n, m − 1), G(n, m − 2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , G(n, 1), after this all the terms in the sequence are the single state automaton on Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The result below is from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be an automaton and A0 := A, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' be the sequence of automata such that Ai = Ai−1 for all i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then (a) a pair of states p, q ∈ QA, belong to the same element t ∈ QAi if and only if for all words a ∈ Xi n, πA(a, p) = πA(a, q), and (b) A is strongly synchronizing if and only if there is a j ∈ N such that |QAj| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The minimal j for which |Aj| = 1 is the minimal synchronizing level of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We also require the notion of a collapse chain from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A and B be strongly syn- chronizing automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A = A0, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Ak = B be a sequence such that Ai+1 is obtained from Ai by identifying pairs of states p ∼Ai q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that as distinct from the synchro- nizing sequence, we do not necessarily make all possible identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Such a sequence is called a collapse chain if at each step, we make the maximal number of collapses possible relative to the final automaton B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' That is, for u, v ∈ QA belonging to the same state of B, in the minimal Ai such that [u] ∼Ai [v], we have [u] = [v] in Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that this condition means that a collapse chain is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Therefore, for B a strongly synchroniz- ing automaton, we say that B belongs to a collapse chain of A if there is a collapse chain A = A0, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Ak = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this case, we call the collapse chain A = A0, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Ak = B, the the collapse chain from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If B is a single state automaton, the collapse chain from A to B is precisely the strongly synchronizing sequence of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus a collapse chain can be thought of as a synchronizing sequence relative to its end point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following facts are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be a strongly synchronizing automaton, and B be an automaton which is a folding of A, then there is a collapse chain from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Therefore B belongs to a collapse chain of A if and only if B is a folding of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular 14 if B belongs to a collapse chain of A, then B is synchronizing at the minimal synchronizing level of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following result about collapse chains is proved similarly to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be an automaton and B be a folding of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A0 := A, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Am = B be the collapse chain from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then a pair of states p, q ∈ QA belong to the same element t ∈ QAi if and only if p, q belong to the same state of QB and for all words a ∈ Xi n, πA(a, p) = πA(a, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 3 Minimal actions of finite order elements of Hn For this section, we will work using facts related to dual transducers for strongly synchroniz- ing transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It has been shown in [12, 1, 14] that the dual T ∨ transducer for a synchronous transducer T contains much information about the order of T, but implicit in those works also, much information about the conjugacy class of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In [15] the dual is considered for strongly synchronizing transducers, where it is shown that for infinite order strongly synchronizing transducers the powers of the dual grow in size asymptotically exponentially while for finite order transducers the dual generates a finite semigroup with a zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this section we bring in some of the methods and results of those works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' See [4, 15] for more details than we give below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 Duals and Splits Recall our definition of the dual of a transducer from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We will mostly be working in a power of the dual of a transducer T, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let T be a strongly synchronizing transducer, and q ∈ QT be a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then we write Wq for the set of words γ ∈ X+ n such that the map πT(γ, ·) : QT → QT has image {q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be an element of Hn, with synchronizing level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then for r ≥ k, A∨ r has a split ((p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pl), (q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ql), Γ) if and only if the following depiction (see Figure 3) of the transitions in A∨ r at the state Γ is valid: More formally, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 (Splits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be an element of Hn, with synchronizing level k and let r ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose there are l ∈ N1, elements (p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pl), (q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ql), (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , sl) ∈ Ql A, a word Γ ∈ Xr n ∩ Ws1, and distinct states t1, t2 ∈ QA 15 Γ Γ1 Λ1 Λ2 Γ2 Λl−1 Γl−1 Λl Γl p1|s1 q1|s1 p2|s2 q2|s2 ql|sl pl|sl ∗|t1 ♯|t2 Figure 3: A split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' the symbols ∗ and ♯ represent arbitrary elements of QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' such that when we define sequences Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Γl and Λ1, Λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Λl by Γ1 = λA(Γ, p1) and Λ1 = λA(Γ, q1), and for 1 < i ≤ l, Γi = λA(Γi−1, pi) and Λi = λA(Λi−1, qi), then Γi, Λi ∈ Wsi+1 for all 1 ≤ i ≤ l − 1, Γl ∈ Wt1 and Λl ∈ Wt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this case we say that A∨ r splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We also say that the l-tuples (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pl) and (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ql) split A∨ r (at Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We call {p1, q1} the top of the split, {t1, t2} the bottom of the split, and the triple ((p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pl), (q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ql), Γ) a split of A∨ r (of length l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' : if we took r < k in the definition of a split above, then there is no guarantee that some word Γ of length r would not even be synchronizing, and also no guarantee of the existence of any synchronizing word of length r, so the definition above breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following concept appears implicitly in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be an element of Hn, with synchronizing level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let r ≥ k and ((p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pl), (q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ql), Γ) be a split of A∨ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let {t1, t2} be the bottom of this split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then we say that the bottom of the split ((p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pl), (q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ql), Γ) depends only on the top if the following conditions hold for any other tuples U1, U2 ∈ Ql−1 A : the triple ((p1, U1), (q1, U2), Γ) is also a split with bottom {t1, t2} and, if λAl(Γ, (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pl)) ∈ Wt1 and λAl(Γ, (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ql)) ∈ Wt2 then λAl(Γ, (p1, U1)) ∈ Wt1 and λAl(Γ, (q1, U2)) ∈ Wt2, and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Observe that if, for r ≥ k, A∨ r has a split ((p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pl), (q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ql), Γ) whose bottom de- pends only on the top, then p1 ̸= q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Splitting length as defined below is used explicitly in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For a transducer A, we define the r-splitting length of A (for r greater than or equal to the minimal synchronizing length) to be minimal l such that there is a split of A∨ r of length l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If there is no such split then we set the r-splitting length of A to be ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 16 Note that if, for r ≥ k, A∨ r has r-splitting length l < ∞, then any split of length l has the property that the bottom depends only on the top as otherwise one can find a shorter split (see [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2 Notational inconvenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We are soon to run into some collisions of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Firstly, if φ ∈ Aut(X−N n , σn), then we can represent φ by a (minimal) transducer Aφ ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Secondly, if A ∈ Hn, then A represents an element φA ∈ Aut(X−N n , σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Finally, if φ ∈ Aut(X−N n , σn) has finite order, then as we will see from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5 there is an automaton A (A∨ k) and an automorphism ψ of the underlying digraph GA (A∨ k ) so that Aφ and H(A (A∨ k ), ψ) represent the same element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It happens that there is a way to define ψ from φ, and also, from Aφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Similarly, we could have begun this paragraph with an element A ∈ Hn, in which case ψ would be defined from both φA and from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In order to unify our notation here, we will simply denote ψ in the above situation as φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This of course means that φA will represent two different things (an automorphism of the one-sided shift, or alternatively, an automorphism of a digraph underlying a folded de Bruijn graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We hope that confounding the notation in this way will not cause confusion as it should be clear what is meant from context, noting as well that the digraph homomorphism φA is the induced digraph homomorphism that arises on GA (A∨ k ) by considering how φ maps infinite paths on A (A∨ k) to other infinite paths on A (A∨ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3 Finite order elements of Hn In this subsection we build, for a finite order element A ∈ Hn and corresponding φA ∈ Aut(X−N n , σn), the minimal strongly synchronizing automaton A (A∨ k ) which φA can act on as an automorphism of the underlying directed graph with A being the minimal representative of H(A (A∨ k ), φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that we will retain the notation φA for both the element of Aut(X−N n , σn) correspond- ing to A as well as the digraph automorphism φA that is induced by this automorphism of the shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that the process of determining Aφ ∈ Hn from a given element φ ∈ Aut(X−N n , σn) is not difficult: one simply relates states to different maps as determined by the fixed viewing window (it is common for differing viewing-window strings to correspond to the same map, the set of all such strings corresponding to “same” map can be used effectively as the name of that state) and then one records the local letter transformations as the edge labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For details, see [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 Building A (A∨ k) from A Let A ∈ Hn be of finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this subsubsection, we learn a process for building a strongly synchronizing automaton A (A∨ k) so that φA acts on the underlying digraph of A (A∨ k) by automorphisms in such a way 17 that H(A (A∨ k ), φA) has A as its minimal representative transducer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This process is essential in the proof that follows for finding simplified elements in the conjugacy class representing our given finite order element A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We will also given an example of the process for a specific element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In [15, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='15] it is shown that there is k ∈ N such that A∨ k is the zero of the semigroup generated by A∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Fix the minimal such k ∈ N so that A∨ k is the zero of the semigroup generated by A∨, and let A∨ k be the minimal representative of A∨ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following is a very useful fact from [15]: for every state [γ] of the zero A∨ k , there is a word W([γ]) ∈ Q+ A such that for any input word s ∈ Q+ A, the output when s is processed from the state [γ] of A∨ k is the word (W([γ]))lW([γ])[1,m], where, |s| = l|W([γ])| + m and W([γ])[1,m] is the length m prefix of (W([γ])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows from this that A∨ k has the following structure: for each state [γ] (for γ ∈ Xk n) there is q[γ] ∈ QA so that for all p ∈ QA we have πA∨ k (p, [γ]) = [γ] · A, and λA∨ k (p, [γ]) = q[γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We will call this the |QA|-parallel cycle structure of A∨ k, or less formally, the cyclical structure of A∨ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Form the automaton A (A∨ k) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The states of A (A∨ k ) are the states [γ] of A∨ k and the transitions are given by the rule that for x ∈ Xn, and [γ] a state of A∨ k, we set πA (A∨ k )(x, [γ]) = [γ[2,|γ|]x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (For this construction it does not matter which γ ∈ Xk n one picks from the class [γ], even though we use γ explicitly in the formula of the transition function: this follows since states of A∨ k are ω-equivalent classes of A∨ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') The following lemma is immediate from the definitions: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If A ∈ Hn be of finite order and let k ∈ N so that A∨ k is the zero of the semigroup generated by A∨ then the automaton A (A∨ k ) is strongly synchronizing at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We also have the following translation (into our context) of the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5 of [4], where A(G) in that theorem corresponds to A (A∨ k ) here, and where the group G there is the group ⟨A⟩ here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn be of finite order and let k ∈ N so that A∨ k is the zero of the semigroup generated by A∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (a) A acts as an automorphism φA of the digraph underlying A (A∨ k) by mapping an edge ([γ], x, [γ[2,|γ|]x]) to the edge (([γ])A, λA(x, qγ), ([γ[2,|γ|]x])A) where ([γ])A = [λA(γ, q)] for some q ∈ QA, (b) The minimal representative of the transducer H(A (A∨ k ), φi A) is the transducer Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Consider the transducer A below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The transducer A of Figure 4 is bi-synchronizing at the second level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The level 2 dual has 36 nodes and so we shall not give this below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' How- ever utilising the AutomGrp package [13] in GAP [8], together with (in AutomGrp) the 18 a0 a1 a2 a3 1|4 0|5 2|0 3|1 4|2 5|3 4|2 5|3 0|4 1|5 2|1 3|0 1|4 0|5 4|2 5|3 2|1 3|0 1|5 0|4 2|0 3|1 4|2 5|3 Figure 4: An element A ∈ H6 of order 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' function “MinimizationOfAutomaton( )” which returns an ω-equivalent automaton, applied to the second power of the dual automaton, we get the result A∨ 2 , depicted in Figure 5 (which is the zero of the semigroup generated by the dual): q0 q1 q2 ∗|a1 ∗|a3 ∗|a2 p0 p1 p2 ∗|a0 ∗|a3 ∗|a2 Figure 5: The level 2 dual of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Considering the states {q0, q1, q2, p0, p1, p2} of A∨ 2 as a partition of the words of length 2 over the alphabet {0, 1, 2, 3, 4, 5}, it is easy to see that q0 = {00, 01, 10, 11, 40, 41, 50, 51} p0 = {20, 21, 30, 31} q1 = {24, 25, 34, 35, 44, 45, 54, 55} p1 = {04, 05, 15, 15} q2 = {02, 03, 12, 13, 22, 23, 32, 33} p2 = {42, 43, 52, 53} by verifying the 36 transitions from q0 in A using these input words, and cross-checking state-change results against the transitions of A∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' From this we can calculate the transitions of A (A∨ 2 ), with the resulting automaton depicted in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 19 p0 q0 q2 q1 p1 p2 1 0 2 3 4 5 4 5 0 1 2 3 1 0 4 5 2 3 1 0 2 3 4 5 2 3 5 4 1 0 0 1 3 2 4 5 Figure 6: The automaton A (A∨ 2 ) Notice that both the domain and range automaton of A are foldings of A (A∨ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This phenomenon generalises, that is, for a strongly synchronizing transducer A representing an element of Hn of finite order, both the domain and range automata of A are foldings of A (A∨ k) (where k is appropriately chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We revisit this example in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 where we show that A is conjugate to a 6-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' ⃝ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2 Duals, automata, and automorphisms In this subsubsection, we will prove that for finite order A ∈ Hn and minimal k so that A∨ k is the zero of the semigroup generated by A∨, that A (A∨ k ) as defined above is the minimal (strongly synchronizing) automaton so that A can act on A (A∨ k) as an automorphism φA, with (A (A∨ k ), φA) inducing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We first require lemmata exploring the relationship between properties of A (A∨ k ) and of 20 A∨ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Our first step is the following useful lemma about the automaton A (A∨ k) constructed as above from a finite order element A ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In essence, it says that if two states [δ] and [γ] are distinct in A (A∨ k) but their two transition functions are the same, then by following the cycles of the level k dual (by iteratively acting by A), we will eventually get to a pair of states which have different output letters in the level k dual, and at that pair of locations, the states of A (A∨ k) will still transition the same way, but the output functions of H(A (A∨ k), φA) at these states will disagree at the first letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn be finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let γ, δ ∈ Xk n be such that the states [γ], [δ] of A (A∨ k ) are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose moreover that the maps πA (A∨ k )(·, [γ]) and πA (A∨ k )(·, [δ]) co- incide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then there is a natural i with 0 ≤ i < o(A) and x, y, y′ ∈ Xn such that y ̸= y′, πA (A∨ k )(·, [γ]Ai) = πA (A∨ k )(·, [δ]Ai) but A maps the edges ([γ]Ai, x, πA (A∨ k )(x, [γ]Ai)) and ([δ]Ai, x, πA (A∨ k )(x, [δ]Ai)) respectively to the edges ([γ]Ai+1, y, πA (A∨ k )(y, [γ]Ai+1)) and ([δ]Ai+1, y′, πA (A∨ k )(y′, [δ]Ai+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let w = W([γ]) and v = W([δ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Since [γ] ̸= [δ], we may find words u, w2, v2 ∈ Q∗ A and letters t ̸= t′ ∈ QA such that w = utw2 and v = ut′v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set i − 1 := |u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that for any j ∈ N with j ≤ i − 1, a straightforward induction argument shows, the edges ([γ], a, πA (A∨ k )(a, [γ])A) and ([δ], a, πA (A∨ k )(q, [δ])) map respectively under Aj to edges ([γ]Aj, b, πA (A∨ k )(a, [γ])Aj) and ([δ]Aj, b, πA (A∨ k )(q, [δ])Aj), where b = λAj(a, u[1,j]) (if j = 0, take b = a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular it follows that πA(·, t) = πA(·, t′) and so, since t ̸= t′, there is an a ∈ Xn be such that y := λA(a, t) ̸= λA(a, t′) = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let x ∈ Xn be such that λAi−1(x, u) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then it follows that the edges ([γ]Ai, x, πA (A∨ k )(x, [γ]Ai)) and ([δ]Ai, x, πA (A∨ k )(x, [δ]Ai)) are mapped respectively under A, to the edges ([γ]Ai+1, y, πA (A∨ k )(y, [γ]Ai+1)) and ([δ]Ai+1, y′, πA (A∨ k )(y′, [δ]Ai+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Recall from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7 that if A ∈ Hn and B is an automaton so that there is a di- graph automorphism φ : GB → GB so that A and H(B, φ) represent the same transformation then we say A is induced from (B, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We say a strongly synchronizing automaton B is an automaton supporting A if there is a digraph automorphism φ of the digraph GB, with A induced from (B, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this situation, if there is no proper folding B′ of B and digraph automorphism φ′ : GB′ → GB′ so that A is induced from (B′, φ′), then we say B is a minimal automaton supporting A (or simply, that B is minimal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the next lemma, we show that there is precisely one minimal automaton (up to iso- morphism of automata) supporting a finite order element A of Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 21 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn be an element of finite order and let k ∈ N be minimal such that A∨ k is the zero of the semigroup generated by the dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then (up to isomorphism of automata) A (A∨ k) is the minimal strongly synchronizing automaton admitting an automorphism φ of GA (A∨ k ) so that A is induced by (A (A∨ k ), φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Furthermore, φ is the automorphism φA of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn be finite order of order o(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that by results in [15, 4] k is minimal such that all of the elements A, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Ao(A)−1 are strongly synchronizing at level k (Ai being the product in Hn of A with itself i times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5 that A (A∨ k) is an automaton supporting A and indeed that (A (A∨ k ), φA) induces A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We argue below that A (A∨ k) is a minimal such automaton, and further, that any minimal automaton supporting A is isomorphic to a folding of A (A∨ k), and hence, must actually be A (A∨ k) up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now suppose that there was another automaton B, such that A acts as an automorphism ψA of the underlying digraph of B so that H(B, ψA) has minimal representative A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Addi- tionally suppose that B is a minimal strongly synchronizing transducer on which A acts as an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that the minimal synchronizing level l of B is greater than or equal to k for, H(B, ψi A) is strongly synchronizing at level l and has minimal representative Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose for a contradiction that B ̸= A (A∨ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' There are two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Firstly for any state q ∈ QB, there is a state p ∈ QA such that the set W(q, j) of words of length j which force q is contained in the set W(p, j) of words of length j which force the state p of A (A∨ k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this case, one observes that A (A∨ k) is a folding of B contradicting the minimality of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus we must be in the negation of the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' That is, we assume that there is a pair of words γ, δ ∈ Xk n such that the state of B forced by γ is the same as the state of B forced by δ but γ and δ force different states of A (A∨ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may further assume that the states [γ], [δ] of A (A∨ k) also satisfy πA (A∨ k )(·, [γ]) = πA (A∨ k )(·, [δ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This is because if πA (A∨ k )(·, [γ]) ̸= πA (A∨ k )(·, [δ]), then we may find a word ν ∈ X+ n such that [γ′] := πA (A∨ k )(ν, [γ]) ̸= πA (A∨ k )(ν, [δ]) = [δ′], satisfy πA (A∨ k )(·, [γ′]) = πA (A∨ k )(·, [δ]′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus γν and δν force the same state of B but force, respectively, the states [γ′] and [δ′] of A (A∨ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may then replace γ, δ with γ′, δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let z1 be the state of B forced by γ and δ and let z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , zo(A) be the orbit of z1 under the action of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As H(B, ψA) = H(A (A∨ k), φA), it must be the case that if a, b ∈ Xn are such that the edge ([γ], a, [Γ]) maps to (([γ])A, b, ([Γ])A), then the edge ([δ], a, [Γ]) also maps to (([δ])A, b, ([Γ])A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus we conclude that πA (A∨ k )(·, ([γ])A) = πA (A∨ k )(·, ([δ])A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now observe that since γ and δ are representatives of [γ] and [δ], respectively, and since for any q ∈ QA, the state of B forced by λA(γ, q) is equal to the state of B forced by λA(δ, q) is equal to z2, it follows that there are representatives of ([γ])A and ([δ])A respectively such that the states of B forced by these representative is z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may thus repeat the argument in the z1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By induction we therefore see that for any 1 ≤ i ≤ o(A), the points ([γ])Ai and ([δ])Ai satisfy that πA (A∨ k )(·, ([γ])Ai) = πA (A∨ k )(·, ([δ])Ai) and whenever there are a, b ∈ Xn, such that (([γ])Ai, a, ν) is an edge mapping under A to the edge (([γ])Ai+1, b(ν)A, then the 22 edge (([δ])Ai), a, ν) also maps under A to (([δ])Ai+1), b, (ν)A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This contradicts Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ 4 Water for the witch – shrinking conjugacy class rep- resentatives Suppose we have a finite order element A ∈ Hn, induced by (B, φA) for some strongly synchronizing B with a minimal number of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Under certain conditions we may employ a two-step process to find a new element C ∈ Hn, where C is conjugate to A, and C is induced by (D, ψ) for some strongly synchronizing D with D having fewer states than B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In what follows we describe this process of finding “smaller” conjugacy class representatives of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The first (and main) step in this process is to employ “relabelling.” This is a conjugacy which, for a pair of states that would be identified in the collapse sequence of the domain automaton, relabels inputs and outputs on edges from this pair of states, with the goal of making this pair of states represent the same local map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If this is possible, then we can collapse the carrying transducer to a smaller one than we started with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The conditions for a successful relabelling include that the orbits of these states have the same lengths, and that for any two corresponding outgoing edges, these orbit lengths of these edges are also the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the case where some of these orbit lengths differ, then in certain circumstances we can employ the second step of the overall process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This step“fluffs up” the carrying automaton by executing some splittings, creating what we call shadow states, and where we can then employ relabelling to the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In either case, after a relabelling, the whole resultant transducer can be minimised so as to be carried by a transducer with strictly fewer states than TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the case that A is conjugate to an n-cycle this process will eventually result in single state transducer representing an n-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 Relabellings and automata sequences Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be a strongly synchronizing automaton and A = A0, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Am be a collapse chain of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let 0 ≤ k ≤ m and φk be a vertex fixing automorphism of GAk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define A′ to be the automaton with QA′ = QA and transition function defined as follows: for p ∈ QA, set πA′(x′, p) = q if and only if there is an x ∈ Xn such that πA(x, p) = q with λH(Ak,φk)(x, [p]) = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We call A′ the relabelling of A by (Ak, φk) or the relabelling of A by (the transducer) H(Ak, φk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that if we relabel A by (Ak, φk), then the resulting automaton A′ is strongly iso- morphic to A in the sense that there is a natural digraph isomorphism from the underlying digraph of A′ to the underlying digraph of A that fixes states and which maps the rela- belled edges of A′ to the original edges in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' More precisely, if (p, x, q) is an edge of GA and λH(Ak,φk)(x, [p]) = x′, then the natural digraph isomorphism maps the edge (p, x′, q) of 23 A′ to the edge (p, x, q) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The point of view one should have in mind is that we have renamed/relabelled the edges of A by switching edge labels on edges which are parallel edges in Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Notice that if we relabel by (A0, φ0), then all we do is switch labels on parallel edges in A, thus the resulting underlying digraph would not change, but a “fixed” drawing of it would be relabelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be a strongly synchronizing automaton and A = A0, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , Am be a collapse chain for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let 0 ≤ k ≤ m and φ be a vertex fixing automorphism of GAk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A′ be the relabelling of A by (Ak, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then A′ has underlying digraph strongly isomorphic to the underlying digraph of A and Am remains a folding of A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' More specifically, writing A′ = A′ 0, A′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , A′ l for the collapse chain from A′ to Am, then l ≤ m and two states of u, v of A belong to the same state of Ai if and only if for some i′ ≤ i we have u and v belong to the same state of A′ i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may consider A as a non-minimal synchronizing transducer where each state induces the identity transformation of the set Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Consider the core of the product A ∗ H(Ak, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let p ∈ QA, and γ ∈ X+ n be such that the state of A forced by γ is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then, by definition of Ak, the state of Ak forced by γ is the state [p] containing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus the set of states of core(A ∗ H(Ak, φ)) is the set {(p, [p]) | p ∈ QA}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let x ∈ Xn and p, q ∈ QA such that πA(x, p) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then we have, πA(x, (p, [p])) = (q, [q]) and λA(x, (p, [p])) = λH(Ak,φ)(x, [p]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus setting A′ to be the output automaton of core(A ∗ H(Ak, φ)) we see that A′ is the relabelling of A by (Ak, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' From this it follows that the underlying digraph of A′ is strongly isomorphic to the underlying digraph of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let u, v be two states of A which belong to the same state of Am and which transition identically on all words of length j and suppose j is minimal for which this happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let p ∈ QA be an arbitrary state and let W(p) ⊆ Xj n consist of those words γ such that πA(γ, u) = πA(γ, v) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We break into cases based on whether or not k ≥ j or k < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' First suppose that k ≥ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This means that in Ak, the states [u] and [v] are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus, λH(Ak,φ)(γ, [u]) = λH(Ak,φ)(γ, [v]) for any γ ∈ X∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Therefore in A′ we see that the set of words ν ∈ Xj n for which πA′(ν, u) = πA′(ν, v) = p is precisely the set {λH(Ak,φ)(γ, [u]) | γ ∈ W(p)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now suppose that k < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This means that the states [u] and [v] are distinct states of Ak such that πAk(·, [u]) and πAk(·, [v]) coincide on Xj−k n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let γ ∈ W(p) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set γ1 to be the length j − k prefix of γ and set γ2 ∈ Xk n such that γ1γ2 = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set [r] = πAk(γ1, [u]) = πAk(γ1, [v]) and set κ ∈ Xk n such that λH(Ak,φ(κ, [r]) = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For t ∈ {u, v}, set δt ∈ Xj−k n be such that λH(Ak,φ)(δt, [t]) = γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then since φ is a vertex fixing automorphism of Ak we notice that πA(δu, u), πA(δv, v), {πA(γ1, t) | t ∈ {u, v}} all belong to the same state of Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This means that πA(κ, πA(δu, u)) = πA(κ, πA(δv, v)) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='(Note that [s] = [p] in Ak by the vertex fixing property of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') Therefore πA∗H(Ak,φ)(δuκ, (u, [u])) = (s, [p]) = πA∗H(Ak,φ)(δvκ, (v, [v])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Since λA∗H(Ak,φ)(δuκ, (u, [u])) = γ = λA∗H(Ak,φ)(δvκ, (v, [v])), we see that in A′, πA′(γ, u) = πA′(γ, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Therefore, in A′, πA′(·, u) and πA′(·, v) coincide on the set W(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Since p was chosen arbitrarily and ⊔p∈QAW(p) = Xj n we conclude that πA′(·, u) and πA′(·, v) coincide on the set Xj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The result now follows by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 24 Now suppose that there are states u, v of A which belong to the same state A′ i′ for some 0 ≤ i′ ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then since u, v belong to the same state of Am, there is an i between 0 and m such that u and v belong to the same state of Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The preceding paragraph and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='8 show that i′ must be less than or equal to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Let A be a strongly synchronizing automaton and A′ be the relabelling of A by (Ak, φk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set ι : GA → GA′ to be the natural digraph isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If ϕ is an automorphism of the underlying digraph GA of A, then we will mean by the induced automorphism ϕ′ of GA′ precisely the map ι−1ϕι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn be an element of finite order and let B be a strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph of B with A the minimal representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let Bk be an element of the synchronizing sequence of B and ψ a vertex fixing automorphism of Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let B′ be the relabelling of B according to (Bk, ψ) and ϕ be the induced isomorphism from the underlying digraph of B to the underlying digraph of B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set P to be the minimal representative of the transducer H(B, B′, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then P −1AP is the minimal representative of H(B′, φϕ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This is a straight-forward application of the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ In the situation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3 we refer to the resulting transducer P −1AP as the trans- ducer induced from A by the relabelling B �→ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be a strongly synchronizing automaton, and let s, t be distinct states of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let (Ai)1≤i≤m be a collapse chain of A such that s, t belong to the same state of Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let 1 ≤ k < m be minimal such that t belongs to the state [s] of the automaton Ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then for all x, x′ ∈ Xn such that πA(x, s) = πA(x′, t) and [v] ∈ {[s], [t]}, states of Ak, we have πAk(x, [v]) = πAk(x′, [v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By minimality of k, it must be the case that the states [s] and [t] of Ak are distinct and the equality πAk(·, [s]) = πAk(·, [t]) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let x, x′ ∈ Xn and u ∈ QA be such that πA(x, s) = πA(x′, t) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then by definition of Ak, πAk(x, [s]) = [u] = πAk(x′, [t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' However, the equality πAk(·, [s]) = πAk(·, [t]), now implies that πAk(x′, [s]) = [u] = πAk(x, [t]) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1 Constructing discriminant permutations disc(s, t, Q) Let A be an automaton and let s, t ∈ QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set the notation: EA(s, t) := {(s, x, t) ∈ EA};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' and LettersA(s, t) := {x ∈ Xn | (s, x, t) ∈ EA(s, t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may leave out the explicit mention of the automaton A when it is clear from context, writing simply E(s, t) and Letters(s, t) for these sets in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 25 Let Q ⊆ QA and s ∈ QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set the notation Xs,Q := � p∈Q Letters(s, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now, suppose s, t ∈ QA and suppose further there is a subset Q ⊆ QA so that (a) Xs,Q = Xt,Q, and (b) for all p ∈ Q we have | Letters(s, p)| = | Letters(t, p)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then to describe this situation we say s and t distribute similarly over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (Note in passing that for some choices of s and t the only possible such set Q may be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') For any states s and t and set Q ⊂ QA so that s and t distribute similarly over Q, we denote by XQ the set Xs,Q = Xt,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We call XQ ⊆ Xn the agreement alphabet (of s and t on Q) noting that if Q = QA, then XQ = Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define a bijection disc(s, t, Q) : XQ → XQ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' First, let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , pr ∈ Q be a maximal sequence of distinct states such that for 1 ≤ i ≤ r there is an x ∈ XQ with πA(x, s) = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Observe that the sets {Letters(s, pi) | 1 ≤ i ≤ r} and {Letters(t, pi) | 1 ≤ i ≤ r} form partitions of XQ, with equal-size corresponding parts in index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now, for 1 ≤ i ≤ r, set disc(s, t, Q) to act as the identity on Letters(s, pi) ∩ Letters(t, pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set Letters(s, pi)′ := Letters(s, pi)\\(Letters(s, pi) ∩ Letters(t, pi)) and Letters(t, pi)′ := Letters(t, pi)\\(Letters(s, pi) ∩ Letters(t, pi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that | Letters(s, pi)′| = |Letters(t, pi)′| and indeed that Ys,t := � 1≤i≤r Letters(s, pi)′ = � 1≤i≤r Letters(t, pi)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Order the elements of Letters(s, pi)′ and Letters(t, pi)′ with the order induced from Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 1 ≤ i ≤ r and x ∈ Letters(s, pi)′ we write x′ for the corresponding element of Letters(t, pi)′, that is, in the ordering of Letters(s, pi)′ and Letters(t, pi)′ induced from Xn, x and x′ have the same index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Using the definitions and facts above we extend the definition of disc(s, t, Q) over the set Ys,t by the rule x �→ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' One easily checks that the resulting function disc(s, t, Q) : XQ → XQ 26 is a well-defined bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Further, observe that for x0 ∈ Ys,t the function disc(s, t, Q) contains a cycle (x0 x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' xk−1) in its disjoint cycle decomposition, where for all i we have xi+1 = x′ i (indices taken mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Recall as well that disc(s, t, Q) acts as the identity over the set XQ\\Ys,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For s and t satisfying points (b) and (b) for some set Q we call disc(s, t, Q) the discrim- inant of s and t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' it is a permutation that encodes the difference in transitions between s and t amongst the set of states Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the case that Q = QA, we will write disc(s, t) for the bijection disc(s, t, QA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As with the notation Letters(p, q), we often run in to situations where we compute discriminant permutations in distinct automata sharing the same state set, in such cases we use the notation discA(s, t, Q) and disc(s, t) to emphasise the automaton in which the permutation is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A be a strongly synchronizing automaton, and let s, t be distinct states of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let Q ⊆ QA be such that s and t distribute similarly over Q with agreement alphabet XQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let (Ai)1≤i≤m be a collapse chain of A such that s, t belong to the same state of Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let 1 ≤ k < m be minimal such that πAk(·, [t]) and πAk(·, [s]) are equal on XQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then for x, y ∈ XQ which belong to the same disjoint cycle of disc(s, t, Q), πAk(x, [s]) = πAk(y, [s]) = πAk(y, [t]) = πAk(x, [t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By assumption πAk(·, [s]) = πAk(·, [t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' An easy induction argument using the definition of disc(s, t, Q) now shows that for any x, y ∈ Xn such that y belongs to the orbit of x under the action of disc(s, t, Q), πAk(x, [s]) = πAk(y, [s]) = πAk(x, [t]) = πAk(y, [t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This follows since for any x ∈ Xn, πA(x, s) = πA((x) disc(s, t, Q), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2 Discriminant permutations and amalgamation sequences Let B be an automaton and G := GB be the underlying digraph of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define a sequence G := G0, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Assuming Gi is defined, Gi+1 is obtained from G in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let ∼ be the equivalence relation on the vertices QGi of Gi that relates two vertices p, q precisely when for every vertex t ∈ QGi the number of edges from q to t is precisely the number of edges from p to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If p ∈ QGi write [p]i+1 for the equivalence class of p under the relation ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set QGi+1 = {[p]i+1 | p ∈ QGi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now suppose p, q ∈ QGi and enumerate those elements of [q]i+1 which have an incoming edge from a vertex in [p]i+1 in some order as q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 1 ≤ j ≤ r, let kj be the number of edges from p to qj and set ec(i + 1, p, q) := � 1≤j≤r kj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set Gi+1 to be the directed graph with vertices QGi+1 and with ec(i + 1, p, q) many edges from [p]i+1 to [q]i+1 for each [p]i+1, [q]i+1 ∈ QGi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We refer to the resulting sequence G0, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , as defined above as the amalgamation sequence of G (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that for each natural i the construction above induces an identification of the states of Gi with a partition of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that after finitely many steps, the amalgamation sequence stabilises to a fixed digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 27 The lemma below says that for a given automaton B, there is a relabelling of B such that the synchronizing sequence coincides with the amalgamation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let B be an automaton with underlying digraph G and synchronizing sequence B = B0, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='. Let G = G0, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' be the amalgamation sequence of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then there is a relabelling D of B such that if D = D0, D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' is the synchronizing sequence of D, the underlying digraph of Di is Gi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' in particular, the partition of the state set of B induced by Di is the same partition induced by Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Since B is strongly synchronizing there is a minimal l ∈ N for which Gl = Gl+1 and both have a single vertex with n loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We proceed by induction on the amalgamation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We begin with the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let s, t ∈ QB be distinct such that s and t belong to the same state of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This means that s and t distribute similarly over QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose that disc(s, t) is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let k ∈ N be minimal such that s and t belong to the same state of Bk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that since disc(s, t) is not trivial, then k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5, for any x, y which belong to the same orbit under disc(s, t), πBk(x, [s]) = πBk(y, [s]) = πBk(x, [t]) = πBk(x, [t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let λBk be defined such that λBk(·, [q]) : Xn → Xn is trivial whenever [q] is not equal to [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We set λBk(·, [t]) = disc(s, t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that the transducer Bk is induced by a vertex fixing automorphism of Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Furthermore, for any pair (u, v) ̸= (s, t) such that u, v belong to the same state of G1 and disc(u, v) is trivial, [u] = [v] in Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let E be the relabelling of B by the transducer Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that QE = QB and G remains the underlying digraph of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It therefore follows that for any pair u, v in B which distribute similarly over QB, u, v still distribute similarly over QB in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If moreover, discB(u, v) is trivial, then construction of λBk, means discE(u, v) remains trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lastly we note that s, t distribute similarly over QB in E, and discE(s, t) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Applying an induction argument, there is an automaton E1 a relabelling of B such that for any pair s, t ∈ QB which belong to the same state of G1, discE1(s, t) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular, such s, t satisfy, πE1(·, s) = πE1(·, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now assume by induction that there is a relabelling E of B with synchronizing sequence E = E0, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' possessing the following property: for 0 ≤ i ≤ k < l two states s, t ∈ QB belonging to the same state of Gi belong to the same state of Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let s, t ∈ QB and suppose s and t belong to the same state of Gk+1 but do not belong to the same state of Ek+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that since the underlying digraph of Ek is the same as Gk and they induce the same partition of the state set QB, then s and t belong to distinct states of Ek and so to distinct states of Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The fact that s and t belong to the same state of Gk+1 means that [s] and [t] distribute similarly over QEk but discEk([s], [t]) is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let k < j ≤ l be minimal such that s and t belong to the same state of Ej+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='5 once more, we have the equalities: for any x, y which belong to the same orbit under discEk([s], [t]), πEj(x, [s]) = πEj(y, [s]) = πEj(x, [t]) = πEj(x, [t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 28 Let λEj be defined such that λEj(·, [q]) : Xn → Xn is trivial whenever [q] is not equal to [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We set λEj(·, [t]) = discEk([s], [t])−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that the transducer Ej is induced by a vertex fixing automorphism of Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Furthermore, for any pair (u, v) ̸= (s, t) such that u, v belong to the same state of Gk+1 and discEk([u], [v]) is trivial, [u] = [v] in Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let F be the relabelling of E by the transducer Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let F1, F2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' be the synchronizing sequence of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let u, v ∈ QB belong to the same state of Gi for some 0 ≤ i ≤ k < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then by the inductive assumption and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2, u, v belong to the same state of Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let u, v ∈ QB belong to the same state of Gk+1 and suppose that discEk(u, v) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that since discEk(u, v) and [u], [v] distribute similarly over QEk, [u] = [v] in Ek+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Therefore, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2 implies that [u] = [v] in Fk+1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lastly observe that [s] = [t] in Fk+1 by construction of λEk and the fact that states which are identified in Ek remain identified in Fk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The result now follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='2 Relabellings along orbits For lemma below, we give stronger hypotheses than appear to be required as per the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let B be a strongly synchronizing automaton, φ an automorphism of the underlying digraph GB of B, and, s and p states of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Every edge from s to a state in the orbit of p (under the action of φ) is on an orbit of length N (when such an edge exists) if and only if every edge from any state in the orbit of s to a state in the orbit of p is on an orbit of length N (when such an edge exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We state the lemma with the stronger hypotheses below to ease understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let B be a strongly synchronizing automaton and φ an automorphism of the underlying digraph GB of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let s, p be vertices of GB so that LettersB(s, p) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose there is N ∈ N1 so that for every edge e from a vertex in the orbit of s to a vertex in the orbit of p, the orbit length of e is N, and secondly there is r ∈ N1 so that if (sφi, y, pφj) is any edge from the orbit of s to the orbit of p, then we have Letters(sφi, pφj) = Letters(sφi+r, pφj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then there is a relabelling of B′ of B such that the induced automorphism φ′ of GB′ satisfies the following: for any i, j ∈ N, LettersB′(sφi, pφj) = LettersB′(sφi+r, pφj), and, if x ∈ LettersB′(sφi, pφj), then the labels of the edges (sφ′i, x, pφ′j)φ′ and (sφ′i+r, x, pφ′j)φ′ are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We first set up some notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Given an edge (u, x, v) of B, with respect to this edge, we shall write xφi for the label of its image (u, x, v)φi so that we have 29 the equality (u, x, v)φi = (uφi, xφi, vφi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that in general we do not have an induced action of φ on Xn, but the notation will be well-defined in the context of a base edge (u, x, v) being understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let B, φ, s, p, N and r be as in the hypotheses, and assume r is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that r divides the orbit length of s (by minimality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Write mr for the orbit length of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Write s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , smr for the orbit of s = s1 (we note that mr|N and we make this explicit below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let �P be those states p′ in the orbit of p which have Letters(s, p′) non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The set �P can be partitioned according to the orbits under the action of φmr, that is, two elements of �P belong to the same part if they belong to the same orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Choose T ⊂ �P so that T has exactly one representative from each block of this partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note, by definition, for any edge (s(φr)i, x′, pφj), i, j ∈ N, there is a unique p′′ ∈ T and an element x ∈ Letters(s, p′′) so that the orbit of (s, x, p′′) under φr contains (s(φr)i, x′, pφj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We inductively define a map λB (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=', induced by a vertex fixing automorphism of B) along the orbit of an edge (s, x, q) for some q ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The map λB will then determine a transducer (Xn, QB, πB, λB) which can be used (as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='1) to carry out the required relabelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' To this end fix q ∈ T and set k = |{qφimr|i ∈ N}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 0 ≤ a ≤ m − 1, partition Letters(sar+1, q) via the equivalence relation relating two edge labels whose corresponding edges are in the same orbit under φkmr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Recall there is an order on the elements of each equivalence class induced from the standard ≤ ordering on Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Use this ordering to determine a transversal for the equivalence classes, choosing as representative of each class the least element in that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Write β(sar+1) for this transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For b ∈ β(sar+1) we use the phrase the equivalence class of b at sar+1 to mean the edge labels in Letters(sar+1, q) which are orbit equivalent to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let 0 ≤ a, a′ ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that since N is the orbit length of any edge from a state in the orbit of s to a state in the orbit of q we have |β(sar+1)| = |β(sa′r+1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let α ∈ N such that for b ∈ β(sar+1) and b′ ∈ β(sa′r+1), the size of the equivalence class of b at sar+1 is equal to the size of the equivalence class of b′ at sa′r+1 is equal to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We fix a bijection between the sets β(sar+1) and β(sa′r+1) induced by the ordering of the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that αkmr = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 1 ≤ j ≤ r, and 0 ≤ a ≤ m − 1 we write β(sar+j) for the set {bφj−1|b ∈ β(sar+1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that the orbit equivalence class of bφj−1 at sar+j, b ∈ β(sar+1), is precisely the image of the equivalence class of b at sar+1 under the image of φj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We transport using φj−1 the orderings of β(sar+1), and the equivalence classes of elements b ∈ β(sar+1) to the set β(sar+j) and the equivalence classes of its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' That is, for instance, if b < b′ ∈ β(sar+1), then bφj−1 < b′φj−1 in β(sar+j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let 1 ≤ l ≤ m be minimal such that {qφimr|i ∈ N}φrl = {qφimr|i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that by minimality l|m since: {qφimr|i ∈ N}φmr = {qφimr|i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Moreover φlr : {qφimr|i ∈ N} → {qφimr|i ∈ N} is a k-cycle since φmr : {qφimr|i ∈ N} → {qφimr|i ∈ N} is a k-cycle and φmr is a power of φlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let M ∈ N be such that Ml = m so that αkMlr = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Further observe that if {qφimr|i ∈ N}φrd ∩ {qφimr|i ∈ N} ̸= ∅ for some d ∈ N, then {qφimr|i ∈ N}φrd = {qφimr|i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For suppose qφfmr ∈ {qφimr|i ∈ N}φrd for some 1 ≤ f ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then there is some 1 ≤ j ≤ k such that qφjmrφdr = qφfmr, this now means that 30 qφdrφjmr ∈ {qφimr|i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' However since φmr is a k-cycle on the set {qφimr|i ∈ N}, then qφdr ∈ {qφimr|i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus we conclude that the sets {qφimr|i ∈ N}φar for 0 ≤ a ≤ l − 1 are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We define a relabelling map λB inductively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let b = b0 ∈ β(s1) be the smallest element such that λB(b, s) is undefined and for all 1 ≤ i ≤ l − 1 λB(bi, sir+1) is undefined for the element bi of β(sir+1) corresponding to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (Note that bi is the least element of β(sir+1) such that λB(bi, sir+1) that is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') In the inductive process which follows, we will define λB(bi, sir+1) for all 0 ≤ i ≤ l − 1 in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define a kl-by-r matrix r with entries tuples of size α as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set r0,0 = (b = b1,1, b1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , b1,α) where (b1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , b1,α) is the ordered tuple of element β(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 0 ≤ i < kl and 0 ≤ j < r set ri,j = (b1,1, b1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , b1,α)φir+j = (b1,1φir+j, b1,2φir+j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , b1,αφir+j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define a matrix R of dimension Mkl-by-r such that Ri,j for 0 ≤ i < Mkl and 0 ≤ j < r has entry ((b1,1, s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , (b1,α, s))φir+j := ((b1,1φir+j, sφir+j), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , (b1,αφir+j, sφir+j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 0 ≤ d < M, set R(d) to be the kl-by-r matrix corresponding to rows dkl to row (d + 1)kl − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 0 ≤ d < M, 0 ≤ i < kl and 0 ≤ j < r we set λBR(d)i,j = ri,j, where we extend λB naturally to act on tuples (Xn × QB)α to produce tuples in Xα n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let 1 ≤ i < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that for the element b′ ∈ βsir+1, the function λB(b′, sir+1) remains undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let the matrix r be exactly as above and define the matrix R as above but with b′ playing the role of b and sir+1 playing the role of s1 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 0 ≤ d < M define the component R(d) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then once more for 0 ≤ d < αM, 0 ≤ i < kl and 0 ≤ j < r we set λBR(d)i,j = ri,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Continuing on in this way across the set T, we define λB on all pairs (x, sφi) where i ∈ N and there is a j ∈ N such that (sφi, x, pφj) is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We set λB to be projection onto the first coordinate on all other pairs in Xn × QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By construction λB is induced by a vertex fixing automorphism and induces the required relabelling of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that the relabelling B′ of B given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7 is in fact isomorphic as an automaton to B, since the relabelling is by a vertex fixing automorphism of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This means we may instead write (B, φ) for the pair (B′, φ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let B be a strongly synchronizing automaton and φ an automorphism of the underlying digrpah GB of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let s, t, p be states of B such that there is an x ∈ Xn with πB(x, s) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose for i, j ∈ N, πB(x, sφi) = pφj if and only if πB(x, tφi) = pφj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' the orbits of s and t are distinct and have equal length l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 31 there is an N ∈ N such that for any j ∈ N, all edges (s, x, pφj) and (t, x, pφj) are on orbits of length N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then there is a relabelling of B′ of B such that the induced automorphism φ′ of GB′ satisfies: for any i, j ∈ N Letters(sφi, pφj) = Letters(tφi, pφj), and for any x ∈ Letters(sφi, pφj), the labels of the edges (sφi, x, pφj)φ′ and (tφi, x, pφj)φ′ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This is a more straight-forward relabelling operation than the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We simply match the orbits of t along p with those of s along p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We define the relabelling map λB inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As before, throughout we observe the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let u, v ∈ QB and x ∈ Xn such that (u, x, v) is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For i ∈ N we write xφi, whenever there is no ambiguity, for the label of the edge (u, x, v)φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' First, for any pair (c, d) ∈ Xn × QB such that (d, c, π(c, d)) is not an edge from a state in the orbit of t to a state in the orbit of p, set λB(c, d) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let x ∈ Xn be smallest such that (t, x, pφi) is an edge for some i and λB(x, t) is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let y ∈ Xn be minimal such that (s, y, pφi) is an edge and y is not equal to λB(z, t) for (t, z, pφi) an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 0 ≤ j < N set λB(xφj, tφj) = yφj where yφj is the label of the edge (s, y, pφi)φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This inductively defined relabelling map λB is given by a vertex fixing automorphism and induces the required relabelling of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that, once more, B′ and B are isomorphic as automata and so we may write (B, φ′) for (B′, φ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='3 Shadow states In this second part of our process, we find new states to add to the transducer via splitting operations, to provide more room for relabelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn have finite order and let B be a minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph GB of B with A the minimal representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following definition is motivated by considering paths into a vertex t that might provide an obstruction to a collapse through relabelling of B, as described in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose there is a state q of B so that there is a minimal length r so that all paths of length r that end on q have orbits of length n under the action of φA, and for this choice of q we have r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let P = e1e2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' er be a path of length r terminating at q, where the orbit of P has length n but the orbit of e2e3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' er is of size c for some c < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For indices 1 ≤ i ≤ j ≤ r set Pi,j := eiei+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By construction, the least common multiple of the orbit size of the edge e1 and of c is n, and further, the orbit length of P2,r−1 must divide c < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As will become clear later, if this situation arises, it may be an obstruction to collapse of a transducer through a relabelling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 32 In the definition that follows, the state t corresponds to the target of e1 from the path mentioned above, while b is some integer multiple of the orbit length of t, but which still properly divides n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn, B be a minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph GB of B with A the minimal representative of H(B, φA), and t ∈ QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We say t is heavy (for the pair (B, φA)) if the following conditions hold: there is a proper divisor b of n, where b is divisible by the length of the orbit of t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' there is at least one pair (x, s) ∈ Xn × QB such that (s, x, t) is an edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' for any x ∈ Xn and any s ∈ QB such that (s, x, t) is an edge of B, the lowest common multiple of b and the length of the orbit of (s, x, t) under φA is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this case, we call the value b above a divisibility constant for t and observe that the set of valid divisibility constants for t might have more than one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In our overall process, we will apply the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 (directly below) in a situation where we cannot simplify a transducer H(B, φA) directly by a relabelling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Specifically, this lemma is useful in situations where we can carry out an in-split of the domain automaton B along the orbit of a heavy state t to create a new automaton B′ with automorphism ψA so that A is a minimal representative of both H(B, φA) and of H(B′, ψA), and where the new pair (B′, ψA) has a reduced obstruction to the existence of a helpful relabelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that here, we mean “in-split” in the normal sense of that operation for edge-shift equivalences, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following lemma characterises how to perform an in-split along the orbit of a heavy state t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The new automaton that is created has all of the old states, together with new states which we call shadow states (from the orbit of t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The following two lemmas address the same set of hypotheses, but we split the results into two statements as the lemma of primary interest is the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Any such number n′ which arises as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='12 below will be referred to as a valid splitting length for (the heavy state) t with respect to divisibility constant b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn and let B be a minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph GB of B with A the minimal representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose there are b ∈ N and t ∈ QB so that t is heavy for the pair (B, φA) with b a divisibility constant for t, and where ��{tφp A|p ∈ Z} �� = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In these circumstances, there is n′ ∈ N a number which divides the lengths of orbits of all edges (s, x, t) and satisfies the following conditions: i) the lowest common multiple of n′ and b is n, ii) there is m > 1 so that n′ = mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let N be the greatest common divisor of the orbit lengths of all edges (s, x, t) in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let (s, x, t) be an edge of B with orbit length k under the action of ⟨φA⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now, by the third bullet point of the definition of the state t being heavy we see that lcm(k, b) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows, as (s, x, t) is an arbitrary incoming edge for t, that lcm(N, b) = n as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Since r is the orbit length of t we see that r|k and since (s, x, t) is an arbitrary incoming edge for t we therefore have r|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By assumption, r|b, so if r = N we would have lcm(N, b) = b < n which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It then follows that N = kr for some integer k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus the set of numbers N which divide the orbit lengths of all edges (s, x, t) and satisfy points i) and ii) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now let n′ be an element of this set and determine m ∈ N so that mr = n′ (noting that 1 < m by construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn and let B be a minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph GB of B with A the minimal representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose there is b ∈ N and t ∈ QB so that t is heavy for the pair (B, φA) with b a divisibility constant for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set t0,0 = t and let t0,0, t0,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , t0,r−1 be the orbit of t under iteration by φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let n′ be a valid splitting length of t and let m > 1 be determined by n′ = mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In these circumstances we may form a new strongly synchronizing automaton B′ with QB′ = QB ⊔ {ta,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ta,r−1 | 1 ≤ a < m} such that we have i) πB′(x, s) := πB(x, s) for those pairs (x, s) ∈ Xn ×QB where πB(x, s) is not in the orbit of t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' ii) πB′(·, ta,i) := πB′(·, t0,i) for all 0 ≤ a < m, 0 ≤ i < r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' iii) The incoming transitions of B′ to the set of vertices {ta,i | 0 ≤ a < m, 0 ≤ i < r} are determined by the above rules, and by an automorphism ψA of the underlying digraph GB′ of B′ satisfying: (t0,0)ψar+i A = ta,i for 0 ≤ a < m, 0 ≤ i < r, (t0,0)ψn′ A = t0,0, and H(B′, ψA) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let n′ be a valid splitting length for the heavy state t0,0 with divisibility constant b, and let m > 1 be an integer so that n′ = mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set T := {t0,0, t0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , t0,r−1} and build a set of new objects (the extra “shadow states” arising from the splitting along the orbit of t0,0) T ′ = {ta,1, ta,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ta,r | 1 ≤ a < m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that |T ∪ T ′| = n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We will define an action ψA on QB′ := QB ∪ T ′ 34 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For s ∈ QB′, set sψA = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 sφA if s ∈ QB\\T ta,i+1 if s = ta,i and i < r − 1 ta+1,0 if s = ta,r−1 and a < m − 1 t0,0 if s = tm−1,r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It is immediate by construction that this is an action, and also that the orbit of t0,0 has size n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We will specify transitions for B′ by steadily expanding the definition of the underlying digraph GB′ of B′ through adding edges of the form (p, x, q) for p, q ∈ QB′ and x ∈ Xn (thus adding the transition (x, p)πB′ = q to B′), while simultaneously extending the function ψA on the corresponding edges of GB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Ultimately, B′ will be a strongly synchronizing automaton and ψA will be an automorphism of the digraph GB′ with H(B′, ψA) being equivalent to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Important in what follows will be a graph homomorphism ι : GB′ → GB, which we will automatically extend to the (new) edges of GB′ whenever they are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' On the set QB′, ι is defined as follows: for s ∈ QB′\\(T ∪T ′) set sι := s, and for any ta,i ∈ T ∪T ′ set ta,iι := t0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Below, whenever we extend GB′ by adding new edges, we also extend the graph homomor- phisms ι : GB′ → GB and ψA : GB′ → GB′ so as to maintain rsc, the rule of semi-conjugacy, which we define here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' rsc: (a) for all q ∈ QB′ we have qψAι = qιφA, and (b) for all edges e of GB′ we further require eψAι = eιφA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Of course we have part (a) of the rule because we have already defined ι and ψA over QB′ to satisfy this rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In the above construction of ι if eι = (r, x, s) then we will identify e as (p, x, q) where p is the source of e and q is the target of e, so after any extension we can always think of the new GB′ as an edge-labelled directed graph with edge labels “lifted” from GB by the map ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that below we will sometimes add a large collection of edges at one go, but in this case, there is always a well defined triple (p, x, q) for each new edge, as we add in edges along an orbit under ψA which always contains a well-defined edge (r, y, s), from which we can detect the correct letter labelling of all edges along the orbit by using rsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that if B′ is a strongly synchronizing automaton then H(B, φA) will represent the same element of Hn as H(B′, ψA), since the map ι never changes edge labels, and the map ψA will have to change edge labels in the corresponding fashion as φA in order to uphold rsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We now begin to specify the edges of GB′, and hence the transition function πB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Recall below that QB\\T = QB′\\(T ∪ T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 35 Partition the edges of GB into the following four sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' NT :={(p, x, q) | p, q ̸∈ T}, BT :={(p, x, q) | p, q ∈ T}, DT :={(p, x, q) | p ∈ T, q ̸∈ T}, and RT :={(p, x, q) | p ̸∈ T, q ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We observe in passing that φA acts on each of the sets NT, BT, DT, and RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For (p, x, q) an edge in NT, let (p, x, q) also be an edge of GB′ (and so (x, p)πB′ = q as well) and set (p, x, q)ψA := (p, x, q)φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Recall that for a group H acting on a set X, a traversal for the orbits is a subset Y ⊂ X so that each orbit under the group action has a unique representative in the set Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let YB be a traversal for the orbits of the edges in BT such that each edge in YB is of the form (t0,0, x, t0,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Similarly set YD to be a traversal for the orbits of the edges in DT so that each edge of YD is of the form (t0,0, x, s) for some s ∈ QB\\T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Finally set YR to be a traversal for the orbits of the edges in RT so that each edge of YR is of the form (s, x, t0,0) for some s ∈ Q\\T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Extend GB′ to include YD ∪ YR ∪ YB as edges incident on t0,0 (we will add more edges incident on t0,0 later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Furthermore, use the action of ψA on the set QB′ together with the map ι to uniquely determine new edges (of the form (p, x, q) for x ∈ Xn) that must be added to GB′ so that the resulting digraph is closed under the action of ψA, contains the transversal edges YD ∪ YR ∪ YB and satisfies rsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that this process extends the definition of ι and ψA to these new edges as well, but these extensions are inductively well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now we may use the new edges of GB′ in the obvious way to also extend the definition of the transition function πB′ so as to create a correspondingly larger automaton B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Observe that for any state s ∈ QB′\\(T ∪T ′) = QB\\T, the process above now has created a unique edge of the form (s, x, q) for each x ∈ Xn (which are the “lifts” of NT and RT to GB′ by ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For edges in NT this is simply by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For an edge (s, x, t0,i) ∈ RT, there is an edge (s′, x′, t0,0) ∈ YR so that there is a minimal non-negative integer k with (s′, x′, t0,0)φk A = (s, x, t0,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that (s′, x′, t0,0)ψk A = (s, x, ta,i) for the unique non- negative a so that k = ar + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now suppose there is an edge (s, x, tb,j) of GB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By rsc, we see that (s, x, tb,j)ι = (s, x, t0,j) but as there is a unique outgoing edge in GB from s with letter x we see that t0,j = t0,i and in particular, i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We assume without meaningful loss of generality that b ≥ a and that |b−a| is minimal amongst all such differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus by rsc the orbit length of the edge (s, x, t0,i) under φA is precisely (b − a)r or else b = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' However, (b − a)r < n′ and n′ divides the length of the orbit of (s, x, t0,i) by the definition of n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that (s, x, ta,i) is the unique pre-image of (s, x, t0,i) under ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' There remains a special concern that we must address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Specifically, there are now pairs (x, ta,i) ∈ Xn×(T ∪T ′) so that there are no edges of the form (ta,i, x, q) in GB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This happens as the orbit of t0,0 has length r under φA but length n′ = mr > r under ψA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Also, to verify the coherence of the rsc condition for edges in YB, recall that n′ divides the orbit length of these edges as they are in the orbit of an edge incident to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 36 Let us now deal with the “missing edges” issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Observe that for an edge (t0,0, x, s) ∈ YD ∪ YB, its orbit under φA may contain multiple edges of the form (t0,0, y, q) (for various y ∈ Xn and q ∈ QB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let us organise these as the sequence of pairwise distinct edges (e0, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , ek) where ei = (t0,0, x, s)φri A, and with ekφr A = e0 = (t0,0, x, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this context, the orbit of (t0,0, x, s) under φA has length (k + 1)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let us set notation ei =: (t0,0, xi, qi) so we can understand the letter xi associated to ei for each valid index i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The concern is that in our current graph GB′ we see for any index 0 < i ≤ k that there is no edge of the form (t0,0, xi, q) ∈ GB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For the letter xi observe that there is an edge of the form (ta,0, xi, qi) of GB′ for a = ir mod n′ and some state qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The rule of modification is, add the edge (t0,0, xi, qi) to GB′, for all indices 0 < i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Repeat this same procedure across all of the transversal elements (t0,0, x′, s′) ∈ YD ∪YB and as a consequence, for each letter y ∈ Xn, we see that the vertex t0,0 now has a unique outgoing edge of the form (t0,0, y, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Finally, we again use the action of ψA on vertices and the action of φA on GB along with the rsc condition to extend the definitions of ψA and ι to the necessary edges we have to add to GB′ in order to complete the orbits of our newly-added edges based at t0,0, and to discern what letters needed to be associated to these new edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now induce from GB′ the enlarged automaton B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' One observes that for any valid indices a and b and fixed index i, the states ta,i and tb,i of B′ have all the same outgoing transitions, and indeed, that the automaton B′ collapses back down to B by identifying these states for each fixed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular GB′ is strongly synchronizing as it admits a collapse sequence to the n-leafed rose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Further, the rsc condi- tion implies that ψA acts as an automorphism of the directed graph GB′ in fashion locally emulating how φA acts on GB so that H(GB′, ψA) represents A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Let A, B, t = t0,0 be as in the statement of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Assume we applied Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 to lengthen the orbit of t as in the lemma statement to create automaton B′ with automor- phism ψA so that H(B′, ψA) has minimal representative A and where the orbit of t in GB′ under the action of ψA is the set {ta,i | 0 ≤ a < m, 0 ≤ i < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Now for each state t0,i for 0 ≤ i < r (these states are in the original orbit of t in GB under the action of φA), we call the set of states {ta,i | 0 < a ≤ m − 1} ⊊ QB′ the shadow states for t0,i (in QB′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that these are precisely the states of H(B′, ψA) with local maps equivalent to the local map at t0,i for the transducer H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If we apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 inductively and perhaps repeatedly on states on the now extended orbit of t, we extend the definition of the shadow states of t0,i to be the union of the sets of states added in each round of applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 which have local maps equivalent to the local map at t0,i for the transducer H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that this process cannot go on forever as each application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 lengthens the orbit of t0,0 with n an upper bound on the length of this orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Also note that each added state will be a shadow state of one of the original states t0,j after any number of iterated applications of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn and let B be the minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph of B with A the minimal 37 representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose that all circuits in B are on orbits of length n under the action of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then there is a strongly synchronizing transducer �B such that A acts as an automorphism �ψA of �B and all edges of �B are on orbits of length n under the action of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We first observe that all states of B must be on orbits of length dividing n as the underlying digraph of B is strongly synchronizing and therefore each state is visited by some circuit which is on an orbit of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Also, we may assume that |B| > 1 otherwise all loops at the state of B will be on orbits of length n and we would be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let s ∈ QB be such that s is on an orbit of length strictly less than n under φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' (If all states of QB were on orbits of length n then all edges of B would be on orbits of length n as well and we would be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=') Inductively define states as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set QB(0, s) := {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Assume QB(i, s) is defined for some i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We now define QB(i + 1, s) ⊆ QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' An element q ∈ QB belongs to QB(i + 1, s) if the following conditions hold: (a) there are elements x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , xi+1 ∈ Xn, such that, for all 1 ≤ j ≤ i + 1, we have πB(xi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' xj, q) ∈ QB(j − 1, s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' and (b) the path (q, xi+1xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' x1, s) is on an orbit of length strictly less than n under φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We observe that for any state q in a set QB(i+1, s), the orbit of q under φA has size properly dividing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If q ∈ QB(i + 1, s) and xi+1, xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , x1 ∈ Xn satisfies points (a) and (b) then we call the path (q, xi+1xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' x1, s) conformant for QB(i + 1, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let k ∈ N be minimal so that QB(k + 1, s) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If such k did not exist then there would a long path (as in point (b) of the definition of the sets QB(i, s)) which is long enough that it must contain a circuit in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Any such circuit would be on an orbit of length strictly less than n under the action of φA, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' From the argument directly above it also follows that whenever j > 0, s /∈ QB(j, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We now apply an induction argument using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 to reduce k to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let t ∈ QB(k, s) and fix xk, xk−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , x1 ∈ Xn, such that the path (t, xkxk−1 · · ·x1, s) is conformant for QB(k, s) and where the orbit of this path under the action of φA is of length b < n (note that b|n and also that the length of the orbit of t divides b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Moreover, by choice of k, for any pair (x, p) ∈ Xn × QB with πB(x, p) = t, the lowest common multiple of the length of the orbit of the edge (p, x, t) and b is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Therefore t is heavy for the pair (B, φA) and b is a divisibility constant for t, so we may apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 with the state t and constant b to add states in the orbit of t and necessary edges to form a new strongly synchronizing automaton B′ with ψA an automorphism of GB′, and so that H(B′, ψA) still represents A (and so in particular, all circuits of GB′ are still on orbits of length n under the action of ψA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Recall that by the construction of B′, the orbit of t, which includes all of its shadow states, is now of larger size n′ t under the action of the resulting digraph automorphism ψA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' for any t′ in the orbit of t (including the shadow states we have just added),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' if there is a path from t′ to s which is conformant for QB′(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' then t′ (and therefore t) is heavy for (QB′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' ψA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' so we may again inductively increase the length of the orbit of t by adding 38 more shadow states until there are no paths from a point in the orbit of t to s which are conformant for QB′(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' s) (note this happens to be a consequence of t no longer being heavy for (B′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' ψA) which must happen eventually as the orbit of t is getting longer and is bounded above by n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that if p ∈ QB′(k, s) but p is not in the orbit of t, then p ∈ QB(k, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular, we have |QB′(k, s)| < |QB(k, s)| as this count of states for B′ no longer includes the state t nor any state in its orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We can now inductively repeat this process for s until |QB′(k, s)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that we can repeat this process for any state p with |QB′(k, p)| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus we may now proceed inductively in this fashion until finally we have constructed an automaton B′ and an automorphism ψA of GB′ so that B′ folds onto B and H(B′, ψA) represents A, and where if q is any state and j is minimal so that QB′(j, q) = ∅, then j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We set �B = B′ and �ψA = ψA in this final case, noting that the orbit of every edge of �B under the action of �ψA is of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn be an element of finite order and suppose that every point in X−N n is on an orbit of length n under the action of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='14, there is a minimal (in size) strongly synchronizing automaton B such that A acts as an automorphism φA of the underlying digraph of B and all edges of B are on orbits of length n under the action of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='4 Relabelling through shadows In this section we make use of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 to deal with situations in which we are unable to directly apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='7 or Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The lemma below says that applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 and then suitably relabelling does not move us out of the conjugacy class of the considered finite order element A ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn and let B be the minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph of B with A the minimal representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let D be a strongly synchronizing automaton which is obtained by repeated applications of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 to the automaton B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let ψA be the automorphism of the underlying digraph of D with A the minimal representative of H(D, ψA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A relabelling D′ of D is called a relabelling through shadows if, for the induced automorphism ψ′ A of the underlying digraph of D′, for any state q ∈ QB the set Sq,D of shadow states of q are all ω-equivalent in H(D′, ψ′ A) to the state q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn and let B be the minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph of B with A the minimal representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let D be a strongly synchronizing automaton which is obtained by repeated applications of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 to the automaton B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let ψA be the automorphism of underlying digraph of D with A the minimal representative of H(D, ψA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let D′ be a relabelling through shadows of D, ψA′ be the induced automorphism of the underlying digraph of D′, and A′ be the minimal representative of H(D′, ψA′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then A′ is conjugate to A and there is a strongly synchronizing automaton B′ and an automorphism φA′ of the underlying 39 digraph GB′ so that A′ is the minimal representative of H(B′, φA′), with GB′ equal to the underlying digraph of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus, the minimal strongly synchronizing automaton C on which A′ acts is carried by a digraph GC that is a graph quotient of GB and we have |GC| ≤ |GB|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As the relabelling process employed is a relabelling through shadows, it preserves the equivalence of the local maps induced by the graph automorphism across all shadow states shadowing any particular original state of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular the collapse of each state with all of its shadow states results in an automaton B′ which still admits an automorphism ψA′ of its underlying graph (which graph is isomorphic to GB) so that A′ is the minimal representative of H(B′, ψA′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The result now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn and let B be the minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph of B with A the minimal representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose that all circuits in B are on orbits of length n under the action of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let p ∈ QB, i ∈ N be less than or equal to the orbit length of p, and γ ∈ X∗ n be such that (p, γ, pφi) is a path in B from p to pφi, then (p, γ, pφi) is on an orbit of length n under φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let k be the orbit length of p and let r be the order of i in the additive group Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For 1 ≤ a < r write γa for the label of the path (p, γ, pφi)φia and set γ0 = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Write Γ = γ0γ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' γr−1, then the circuit (p, Γ, p) is on an orbit of length n by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' From this it follows that the path (p, γ, pφi) is also on an orbit of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn and suppose there is a minimal strongly synchronizing automa- ton B such that there is an automorphism φA of the underlying digraph of B with A the minimal representative of H(B, φA) and so that all circuits in B are on orbits of length n under φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let s, t, p ∈ QB be such that Letters(s, p) = Letters(t, p) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then by repeated application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 one may obtain from B a strongly synchronizing automaton D with automorphism ψA so that A is the minimal representative of H(D, ψA) and where the pair (D, ψA) satisfies the following conditions: (a) there are no shadow states of any element in the orbit of p in D (that is, each application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 creates no shadow states for elements in the orbit of p), (b) LettersD(s, p) = LettersD(t, p) = LettersB(s, p), (c) for u ∈ {s, t}, for any shadow state u′ of u we have, LettersD(u, p) = LettersD(s, p) = LettersB(s, p), (d) for any x ∈ LettersB(s, p), the length of the orbits of the edges (s, x, p), (t, x, p) under the action of ψA on D is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 40 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We proceed in a similar way to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If |B| = 1, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Therefore we assume that |B| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Inductively define subsets of QB as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set QB(0, p) = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Assume that QB(i, p) is defined for some i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Define QB(i + 1, p) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' A state q ∈ QB belongs to QB(i + 1, p), if there are elements x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , xi ∈ Xn, such that πB(xixi−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' xj, q) ∈ QB(j, p) for 0 ≤ j ≤ i, and the path (q, xixi−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' x0, p) is on an orbit of length strictly less than n under φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='14, there is a k ∈ N such that QB(k + 1, p) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set k ∈ N to be minimal such that QB(k + 1, p) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' If QB(1, p) ∩ {s, t} = ∅, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus we may assume that at least one of s, t belongs to QB(1, p) (and so k ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Observe that by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='18 for any j ∈ N, pφj is not an element of QB(i, p) for any 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We now repeatedly apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13, as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='14, until we have an automaton D such that QD(1, p)∩{s, t} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that since p is always the single element of QD(0, p), then, as for any j ∈ N, pφj is not an element of QB(i, p) for any 1 ≤ i ≤ k, we do not create a shadow state of p or elements in the orbit of p in an application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We prove the base case to illustrate how the proof goes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let q ∈ QB(k, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may find x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' , xk such that for any 0 ≤ j ≤ k, πB(q, xkxk−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' xj) ∈ QB(j, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let b be the length of the orbit of the path (q, xkxk−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' x0, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then for any state u ∈ QB for which there is an edge (u, x, q), the lowest common multiple of the length of the orbit of (u, x, q) and b is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may now apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13 to form a new transducer B′ by adding shadow states of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We may define the sets QB′(i, p) as before, noting that QB′(k + 1, p) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' This follows as for any edge (u, x, q) in B, the orbit of the edge (u′, x, q′) in B′ (for u′ and q′ either equal to u and q or shadow states of u and q respectively) is equal to the orbit of the edge (u, x, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Moreover, as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='14, the number of paths (q′, x0x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' xk, p), where q′ is either q or one of its shadow states, witnessing that q′ ∈ QB′(k) is strictly fewer than the witness paths for q in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus inductively applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13, we find an automaton, which we again denote B′, in which |QB′(k, p)| < |QB(k, p)| since neither q nor any of its shadow states belong to QB′(k, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Thus, replacing B′ with B we may repeat the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Eventually we reach an automaton D′ such that QD(1, p) ∩ {s, t} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Moreover, since, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='13, shadow states transition identically to their original counterparts on edges into states which have no shadow states added (and this transition mirrors the transition in B), the automaton D′ satisfies the requirements of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' □ In what follows, set notation olτ(⋆) to represent the orbit length of ⋆ under the action of τ a digraph automorphism of some digraph G, where ⋆ is a vertex, edge, or path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For an automaton C over alphabet Y with a, b ∈ QC, recall that EC(a, b) represents the set of edges of GC from a to b, while LettersC(a, b) := {y ∈ Y | ∃(a, y, b) ∈ EC(a, b)} represents the set of letters from Y which are the labels of these edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' 41 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let A ∈ Hn and let B be the minimal strongly synchronizing automaton such that there is an automorphism φA of the underlying digraph of B with A the minimal representative of H(B, φA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose that all circuits in B are on orbits of length n under φA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let s, t, p ∈ QB be such that s and t belong to distinct orbits but have the same orbit length under φA, and so that Letters(s, p) = Letters(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Then there is A′ ∈ Hn and an automorphism φA′ of the underlying digraph of B so that the following hold: (a) the element A is conjugate to A′ in Hn, where A′ is the minimal representative of H(B, φA′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' and, (b) there is an N ∈ N such that for all x ∈ Letters(s, p) the edges (s, x, p) and (t, x, p) have orbit length N under φA′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The strategy is to apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='19 to add shadow states to B to obtain an automa- ton D and an automorphism ψA of D, such that H(D, ψA) has minimal representative A, ψA preserves the orbit length of the state p, and, the orbit length of the edges from s and t into p is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We then apply a relabelling through shadows of D by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='17 to obtain a conjugate element A′ to A represented by a transducer H(B, φA′) for φA′ an automorphism of the underlying digraph GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The key ingredient is that the relabelling of D is chosen such that the orbits of the edges of s and t into p now all have the same length under φA′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In order to find a relabelling achieving this goal, we will need to track numerous integer constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set r = olφA(p) and determine m so that mr = lcm(olφA(s), olφA(p)) = lcm(olφA(t), olφA(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For any edge (s, x, p) ∈ EB(s, p) we have (s, x, p)φk A ∈ EB(s, p) if and only if k is an integer multiple of mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For each x ∈ LettersB(s, p) = LettersB(t, p), determine integers ux, vx so that uxmr = olφA((s, x, p)) and vxmr = olφA((t, x, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular, for any x ∈ LettersB(s, p), the number of edges in EB(s, p) which belong to the orbit of (s, x, p) under φA is precisely ux, while vx defines the analogous number for (t, x, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that there are permutations θs : LettersB(s, p) → LettersB(s, p) and θt : LettersB(t, p) → LettersB(t, p) induced from the permutations of edges from s to p (and from t to p respectively) achieved by applying φmr A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular, for x ∈ LettersB(s, p)(= LettersB(t, p)), we have the cycle of θs containing x has length ux and the cycle of θt containing x has length vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Adding shadows: Apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='19 to the quadruple (s, t, p, B) to obtain a strongly synchronizing au- tomaton D and a corresponding automorphism ψA of the digraph GD underlying D so that A is the minimal representative of H(D, ψA), and the conclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content='19 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular, for any x ∈ LettersB(s, p) we have olψA((s, x, p)) = olψA((t, x, p)) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We now determine various constants arising from the construction so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' By construction, r remains the length of the orbit of p in D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' however, the orbit lengths of s and t have possibly been padded out with shadow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Determine e, f so that emr = lcm(olψA(s), olψA(p)) and fmr = lcm(olψA(t), olψA(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As LettersB(s, p) = LettersB(t, p) = LettersD(t, p) = LettersD(s, p) we will often use the notation Letters (s||t, p) for this set, although, we might use one of the other names if we 42 specifically wish to emphasise that we are considering the action of φA or of ψA in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set as well ζ := | Letters (s||t, p)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Determining constants and orbit blocks from LettersD(s, p) and LettersD(t, p): As above, consider the permutation of LettersD(s, p) induced by applying ψemr A to GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note that this permutation is θe s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Similarly ψfmr A induces θf t on LettersD(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set qe to be the order of θe s, so that θqee s is the identity permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Analogously define qf to be the order of θf t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Note in passing that qe = n/(emr) is the number of times the orbit of an edge of the form (s, x, p) intersects ED(s, p) under the action of ⟨ψA⟩ (and that this number is independent from the choice of such edge), and similarly, qf = n/(fmr) counts the cardinality of the intersection of the orbit of an edge of the form (t, x, p) with ED(t, p) under the action of ⟨ψA⟩ (again, independent of the choice of such an edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that all cycles of θe s have length qe and that all the cycles of θf t have length qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We note that for all x ∈ LettersD(s, p) we have qee = lcm(ux, e) and qff = lcm(vx, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Therefore, for any x ∈ Letters (s||t, p), we have qe|ux and qf|vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Further, as qeemr = qffmr = n we have qee = qff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let u = gcd{ux : x ∈ Letters (s||t, p)} and v = gcd{vx : x ∈ Letters (s||t, p)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As qe|ux and qf|vx for all x ∈ Letters (s||t, p), we see that qe|u and qf|v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let u and v be such that uqe = u and vqf = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It also follows that lcm(u, e) = qee = qff = lcm(v, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let τ ∈ {s, t} and set ∼τ,p to be the equivalence relation on the set of edges from τ to p, where two edges are equivalent under ∼τ,p if they are in the same orbit under ψB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let X(τ, p) be a transversal for this equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' It follows that � (s,x,p)∈X(s,p) ux = � (t,x,p)∈X(t,p) vx = | Letters (s||t, p)| = ζ so in particular we see that both u and v divide ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Set w := lcm(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Since u and v divide ζ we have that w|ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Let α be such that αw = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Further, as u|w and v|w there are µ, ν ∈ N such that w = µu = µuqe and w = νv = νvqf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Further, qee = lcm(u, e)| lcm(w, e) and qff = lcm(v, f)| lcm(w, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Since both u and v divide qee = qff, it follows that w|qee and so lcm(w, e) = qee = qff = lcm(w, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In particular, wmr|qeemr = n and wmr|qffmr = n and we have lcm(wmr, emr) = n = lcm(wmr, fmr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Building the relabelling: Our relabelling will be a relabelling through shadows which will create a pattern of labels along an edge orbit that repeats after every wmr steps under iteration of the new automorphism ψA′ of GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' To be a relabelling through shadows we will need that for any 0 ≤ i < mr and integer k that the local map at sψi+kmr A′ agrees with the local map at sψi A′ (and similarly for the orbit of t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' We will now define a new labelling λD : Xn × QD → Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Recall that a relabelling function is always induced from a vertex fixing automorphism of the underlying digraph of an automaton in the collapse sequence of the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In our particular construction of λD, 43 the reader will see that the automorphism employed in its creation is simply a vertex fixing automorphism of the underlying digraph GD of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' Suppose (l, x′, p′) is an edge of D which does not belong to the orbit of an edge (τ, x, p) for any pair (τ, x) ∈ {s, t} × LettersD(τ, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' In this case we set (x′, l)λB = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' For the moment we focus on edges in the orbit of some edge (s, x, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' The edges in the orbit of (t, x, p) are dealt with analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFRT4oBgHgl3EQfeDf8/content/2301.13570v1.pdf'} +page_content=' As | Letters (s||t, p)| = ζ = wα we may partition Letters (s||t, p) into α blocks of size w, which partition we organise through some labelling of the set’s elements as follows: Letters (s||t, p) = {xc i | 0 ≤ i < w, 0 ≤ c < α}, where here, the cth part, denoted xc, has w elements arranged as the ordered sequence (xc i)0≤i 0. The Weierstrass σ-function +with quasi-periods 2ω, 2ω′ is defined by the following infinite product over the lattice +2ωm + 2ω′m′, m, m′ ∈ Z: +σ(x) = σ(x| ω, ω′) = x +� +s̸=0 +� +1 − x +s +� +e +x +s + x2 +2s2 , +s = 2ωm + 2ω′m′ +m, m′ ∈ Z. +(A1) +It is an odd quasiperiodic function with two linearly independent quasi-periods in the +15 + +complex plane. The expansion around x = 0 is +σ(x) = x + O(x5), +x → 0. +(A2) +The monodromy properties of the σ-function under shifts by the quasi-periods are as +follows: +σ(x + 2ω) = −e2ζ(ω)(x+ω)σ(x), +σ(x + 2ω′) = −e2ζ(ω′)(x+ω′)σ(x). +(A3) +Here ζ(x) is the Weierstrass ζ-function defined as +ζ(x) = σ′(x) +σ(x) . +(A4) +As x → 0, +ζ(x) = 1 +x + O(x3), +x → 0. +(A5) +The Weierstrass ℘-function is defined as ℘(x) = −ζ′(x). It is an even double-periodic +function with periods 2ω, 2ω′ and with second order poles at the points of the lattice +s = 2ωm + 2ω′m′ with integer m, m′. As x → 0, ℘(x) = x−2 + O(x2). +The Weierstrass functions obey many non-trivial identities. Here we present the two +which are necessary for the calculations leading to equation (3.4): +ζ(x + η) + ζ(x − η) − 2ζ(x) = +℘′(x) +℘(x) − ℘(η), +(A6) +℘(x + η) − ℘(x − η) = − +℘′(x)℘′(η) +(℘(x) − ℘(η))2. +(A7) +The proof is standard. The both sides are elliptic functions of x, and the singular terms +in the both sides coincide. Therefore, the difference between the left and right hand sides +is a constant which can be found by putting x to some special value. +Acknowledgments +This work has been supported in part within the framework of the HSE University Basic +Research Program. +References +[1] M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite-dimensional sys- +tems related to Lie algebras, Phys. Rep. 71 (1981) 313–400. +[2] F.W. Nihhoff and G.D. Pang, A time-discretized version of the Calogero-Moser +model, Phys. Lett. A 191 (1994) 101–107. +16 + +[3] F.W. Nihhoff, O. 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Kashiwara and T. Miwa, Transformation groups for soliton +equations IV. A new hierarchy of soliton equations of KP type, Physica D 4D (1982) +343–365. +[26] A. Zabrodin, Elliptic families of solutions to constrained Toda hierarchy, Theor. +Math. Phys. 213 (2022) 1362–1368, arXiv:2201.07582 +[27] N. Delice, F.W. Nijhoff and S. Yoo-Kong, On elliptic Lax systems on the lattice and +a compound theorem for hyperdeterminants, J. Phys. A: Math. Theor. 48 (2015) +035206. +18 + diff --git a/rNE4T4oBgHgl3EQfVAw3/content/tmp_files/load_file.txt b/rNE4T4oBgHgl3EQfVAw3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c49779e5d42086007fc90eed21e88e2feb391ae3 --- /dev/null +++ b/rNE4T4oBgHgl3EQfVAw3/content/tmp_files/load_file.txt @@ -0,0 +1,593 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf,len=592 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='05020v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='SI] 12 Jan 2023 Time discretization of the deformed Ruijsenaars-Schneider system A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Zabrodin∗ January 2023 ITEP-TH-04/23 Abstract We obtain B¨acklund transformations and integrable time discretization of the recently introduced deformed Ruijsenaars-Schneider many-body system which is the dynamical system for poles of elliptic solutions to the Toda lattice with con- straint of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We also show that the deformed Ruijsenaars-Schneider system in discrete time is the dynamical system for poles of elliptic solutions to the fully discrete Kadomtsev-Petviashvili equation of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Besides, we suggest a field analogue of the deformed Ruijsenaars-Schneider system on a space-time lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Contents 1 Introduction 2 2 The B¨acklund transformation 4 3 The discrete time dynamics and continuum limits 6 4 The discrete time deformed RS system from fully discrete BKP equa- tion 7 5 A field analogue of the deformed RS system on a space-time lattice 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1 Equations of motion on the space-time lattice from elliptic families for the fully discrete BKP equation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 13 ∗Skolkovo Institute of Science and Technology, Moscow 143026, Russia and National Research Univer- sity Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia and NRC “Kurchatov Institute”, Moscow, Russia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' e-mail: zabrodin@itep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='ru 1 6 Concluding remarks 15 Appendix A: The Weierstrass functions 15 Acknowledgments 16 References 16 1 Introduction Integrable many-body systems of Calogero-Moser (CM) and Ruijsenaars-Schneider (RS) type play a significant role in modern mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' They are important and meaningful from both mathematical and physical points of view (see the review [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' An interesting problem of their integrable time discretization was addressed in [2, 3], see also the book [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' This problem is closely connected with B¨acklund transformations of the CM and RS systems [5, 6, 7] and the so-called self-dual form of their equations of motion [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The idea is that the B¨acklund transformation can be regarded as evolution in discrete time by one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The equations of motion of the most general elliptic version of the N-particle RS system in discrete time obtained in [3] are as follows: N � k=1 σ(xn i − xn+1 k − η)σ(xn i − xn k + η)σ(xn i − xn−1 k ) + N � k=1 σ(xn i − xn+1 k )σ(xn i − xn k − η)σ(xn i − xn−1 k + η) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) where xn i is coordinate of the ith particle at the nth step of discrete time and η plays the role of a lattice spacing in the space lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Here and below we use the standard Weierstrass σ-, ζ- and ℘-functions σ(x), ζ(x) = σ′(x)/σ(x) and ℘(x) = −ζ′(x) (see Appendix A for their definition and properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The properly taken continuous time limit of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) yields the equations of motion of the RS model [10, 11]: ¨xi + N � j̸=i ˙xi ˙xj � ζ(xij + η) + ζ(xij − η) − 2ζ(xij) � = 0, xij ≡ xi − xj, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2) where dot means the time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Recently, a deformation of the RS model was introduced [12] as a dynamical sys- tem describing time evolution of poles of elliptic solutions to the Toda lattice with the constraint of type B [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Equations of motion of the deformed RS system are ¨xi + N � j̸=i ˙xi ˙xj � ζ(xij + η) + ζ(xij − η) − 2ζ(xij) � + σ(2η)(U− i − U+ i ) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) 2 where U± i = N � j̸=i σ(xij ± 2η)σ(xij ∓ η) σ(xij ± η)σ(xij) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4) They differ from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2) be the additional terms σ(2η)(U− i − U+ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In [12, 14] it was shown that the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) can be obtained by restriction of certain Hamiltonian flow of the 2N-particle RS system to the 2N-dimensional subspace P ⊂ F of the 4N-dimensional phase space F corresponding to the configurations in which the 2N particles stick together joining in N pairs such that the distance between particles in each pair is equal to η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The restriction gives equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3), with the xi being coordinate of the ith pair moving as a whole thing with the fixed distance between the two particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Using this observation, all integrals of motion of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) has been found in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In this paper we suggest an integrable time discretization of the deformed RS system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) obtained in the way similar to the one which leads to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The equations of motion in discrete time are: µ N � k=1 σ(xn i − xn+1 k )σ(xn i − xn k + η)σ(xn i − xn−1 k − η) + µ N � k=1 σ(xn i − xn+1 k + η)σ(xn i − xn k − η)σ(xn i − xn−1 k ) = µ−1 N � k=1 σ(xn i − xn+1 k − η)σ(xn i − xn k + η)σ(xn i − xn−1 k ) + µ−1 N � k=1 σ(xn i − xn+1 k )σ(xn i − xn k − η)σ(xn i − xn−1 k + η), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) where µ is a parameter related to the lattice spacing in the time lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The structure of each side is the same as the structure of the left hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We note that this form of equations of motion was conjectured by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='Krichever some time ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' There are two different continuous time limits of equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' One of them gives equations of motion of the deformed RS system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The other one leads to the equations obtained in [17] as equations of motion for poles of elliptic solutions to the semi-discrete Kadomtsev- Petviashvili equation of type B (BKP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Like the equations of motion for the RS system, equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) admit trigonometric (or hyperbolic) and rational degenerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In the trigonometric limit, one of the two quasi-periods of the σ-function tends to ∞ and σ(x) → sin x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In the rational limit, both quasi-periods tend to ∞ and σ(x) → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In this paper we will not discuss the specifics and details of the trigonometric and rational limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We also present the extension of the deformed RS model to a lattice field theory (a “field analogue”) in which the coordinates of particles xi become “fields” xi(x, t) depending not only on the time t but also on a space variables x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Following the method developed in [15] for the CM model and applied in [16] to the RS model, we obtain the equations of motion as equations for poles of more general elliptic solutions (called elliptic families in [15]) to the fully discrete BKP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 3 The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In Section 2 we obtain the B¨acklund transformation of the deformed RS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In Section 3 the chain of B¨acklund transformations is interpreted as a discrete time evolution and the equations of motion in discrete time are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Possible continuum limits are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In Section 4 we show that the discrete time equations of motion for the deformed RS system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) describe the dynamics of poles of elliptic solutions to the fully discrete BKP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Section 5 is devoted to obtaining the lattice field analogue of the fully discrete deformed RS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Concluding remarks are presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In Appendix A the definition and main properties of the Weierstrass functions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 2 The B¨acklund transformation Since the works [18]–[22] it became a common knowledge that the integrable many-body systems of CM and RS type are dynamical systems for poles of singular solutions to nonlinear integrable differential and difference equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The nonlinear integrable equa- tions are known to serve as compatibility conditions for linear differential or difference equations for the “wave function” ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Poles of solutions to the nonlinear equations (zeros of the tau-function) are simultaneously poles of the ψ-function, so the latter are subject to equations of motion of the CM or RS type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In fact zeros of the ψ-function are subject to the same equations, and this leads to the idea to obtain the B¨acklund transformation of the CM or RS system as passage from poles to zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' This approach works for all previously known examples, and we are going to apply it to the case of the Toda lattice with constraint of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The first linear problem for the Toda lattice with constraint of type B has the form [13] ∂tψ(x) = v(x) � ψ(x + η) − ψ(x − η) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) where v(x) is expressed through the tau-function τ(x) as v(x) = τ(x + η)τ(x − η) τ 2(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2) For elliptic solutions τ(x) = N � i=1 σ(x − xi), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) where the xj’s are assumed to be all distinct, so v(x) is an elliptic function with periods 2ω, 2ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Therefore, solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) can be sought as double-Bloch functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' functions such that ψ(x + 2ω) = Bψ(x), ψ(x + 2ω′) = B′ψ(x) with some Bloch multipliers B, B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Poles of the ψ-function are zeros of the tau-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Therefore, we can represent solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) in the form ψ(x) = µx/ηe(µ−µ−1)t ˆτ(x) τ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4) 4 where ˆτ(x) = N � i=1 σ(x − yi) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) with some yi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Then the ψ-function is indeed a double-Bloch function with the Bloch multipliers B = µ2ω/η exp � 2ζ(ω) N � j=1 (xj − yj) � , B′ = µ2ω′/η exp � 2ζ(ω′) N � j=1 (xj − yj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Below we will see that N � j=1 ( ˙xj − ˙yj) = 0, so the Bloch multipliers do not depend on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' It is proved in [12] that poles of the ψ-function xj satisfy the equations of motion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) for any µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Substituting the ansatz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4) into equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) with v(x) given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2), we obtain the equation ∂tˆτ(x) ˆτ(x) − ∂tτ(x) τ(x) + µ − µ−1 = µ ˆτ(x + η)τ(x − η) ˆτ(x) τ(x) − µ−1τ(x + η)ˆτ(x − η) τ(x) ˆτ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='6) It is evident from this equation that it is invariant under the simultaneous changes τ ↔ ˆτ, µ ↔ µ−1, so the yj’s satisfy the same equations of motion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Both sides of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='6) have simple poles at x = xj and x = yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Equating the residues, we obtain the equations ˙xi = µσ(−η) � j̸=i σ(xi−xj−η) σ(xi−xj) � k σ(xi−yk+η) σ(xi−yk) + µ−1σ(−η) � j̸=i σ(xi−xj+η) σ(xi−xj) � k σ(xi−yk−η) σ(xi−yk) , ˙yi = µσ(−η) � j̸=i σ(yi−yj+η) σ(yi−yj) � k σ(yi−xk−η) σ(yi−xk) + µ−1σ(−η) � j̸=i σ(yi−yj−η) σ(yi−yj) � k σ(yi−xk+η) σ(yi−xk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7) These are so-called self-dual equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' They are symmetric under the exchange xj ↔ yj, µ ↔ µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The passage xj → yj can be regarded as a B¨acklund transformation of the deformed RS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The equations of motion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) for xj’s and yj’s in principal can be derived from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' To this end, one should take the time derivative of these equations and use them again substituting the expressions for ˙xj, ˙yj through xj, yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) are then equivalent to a non-trivial identity for multi-variable elliptic functions which is too complicated to be proved directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' However, we do not need to prove it directly since the equations of motion for the xj’s follow from the result of [12] and those for the yj’s follow from the symmetry xj ↔ yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Note that the B¨acklund transformation for the RS 5 system differs from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7) by absence of the second terms in the right hand sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In this sense it is contained in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7) as a formal limiting case µ → ∞ (or µ → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' From equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7) it follows that N � j=1 ( ˙xj − ˙yj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='8) The proof is based on the higher degree identities for the σ-function of the Fay type (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=', [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' It is almost the same as the corresponding proof from [7] and we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 3 The discrete time dynamics and continuum limits The B¨acklund transformation xj → yj can be regarded as a time evolution by one step of the discrete time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Denoting the discrete time variable by n ∈ Z, we then write xi = xn i , yi = xn+1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Shifting n → n − 1 in the second equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7), so that the left hand sides of the two equations become equal, we conclude that the right hand sides are equal, too, which results in equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5), or N � j=1 σ(xn i − xn+1 j )σ(xn i − xn j + η)σ(xn i − xn−1 j − η) σ(xn i − xn+1 j + η)σ(xn i − xn j − η)σ(xn i − xn−1 j ) = −1 + µ−2 N � j=1 σ(xn i −xn+1 j )σ(xn i −xn−1 j +η) σ(xn i −xn+1 j +η)σ(xn i −xn−1 j ) + µ−2 N � j=1 σ(xn i −xn+1 j −η)σ(xn i −xn j +η) σ(xn i −xn+1 j +η)σ(xn i −xn j −η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) Let us discuss continuous time limit of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In fact they admit different continuum limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' For one of them, we introduce the variables Xn j = xn j − nη (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2) and assume that these variables behave smoothly when the time changes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=', Xn+1 j = Xn j + O(ε) as ε → 0, where we introduce the lattice spacing ε in the time lattice, so that the continuous time variable is t = nε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In terms of the variables Xn j , equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) acquire the form N � j=1 σ(Xn i − Xn+1 j − η)σ(Xn i − Xn j + η)σ(Xn i − Xn−1 j ) σ(Xn i − Xn+1 j )σ(Xn i − Xn j − η)σ(Xn i − Xn−1 j + η) = −1 + µ−2 N � j=1 σ(Xn i −Xn+1 j −η)σ(Xn i −Xn−1 j +2η) σ(Xn i −Xn+1 j )σ(Xn i −Xn−1 j +η) + µ−2 N � j=1 σ(Xn i −Xn+1 j −2η)σ(Xn i −Xn j +η) σ(Xn i −Xn+1 j )σ(Xn i −Xn j −η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) We should expand these equations in powers of ε taking into account that Xn±1 j = Xj ± ε ˙Xj + 1 2 ε2 ¨Xj + O(ε3) 6 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The expansion procedure is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' It is enough to expand up to the order ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' For consistency of the expansion procedure one should require that µ−1 is of order ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Putting µ−1 = ε, one obtains equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) for the Xj’s in the leading order ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Another possibility is to assume that the original variables xn j are smooth when the time changes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=', xn±1 j = xj ± ε ˙xj + 1 2 ε2¨xj + O(ε3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In this case one should expand equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' It is easy to see that in general position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' if µ−2 − 1 = O(1) as ε → 0, the leading order is ε and the expansion gives the RS equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' However, if µ−2 − 1 = O(ε), say, µ−2 = 1 + αε + O(ε2), then the first order gives the identity 0 = 0 and one should expand up to the second order in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The procedure is rather cumbersome but straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' As a result, one obtains the equations derived in [17] for dynamics of poles of elliptic solutions to the semi-discrete BKP equation: � j̸=i (¨xi ˙xj − ˙xi¨xj) � ℘′(η) ℘(xij) − ℘(η) − 2ζ(η) � + � k � j̸=i,k ˙xi ˙xj ˙xk ℘′(xij) ℘(xij) − ℘(η) � ℘′(η) ℘(xij) − ℘(η) − 2ζ(η) � − 2ζ(η) � j̸=i ˙xi ˙x2 j ℘′(xij) ℘(xij) − ℘(η) − α � ¨xi + � j̸=i ˙xi ˙xj � ζ(xij + η) + ζ(xij − η) − 2ζ(xij) �� = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4) (here we correct the misprint in [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In contrast to the previously discussed equations of motion, these equations are not resolved with respect to the ¨xj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 4 The discrete time deformed RS system from fully discrete BKP equation In this section we show that the discrete time elliptic deformed RS system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) is a dynamical system for poles of elliptic solutions to the fully discrete BKP equation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We begin with the continuous BKP hierarchy [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Let t = {t1, t3, t5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='} be an infinite set of continuous “times” indexed by odd numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' They are independent variables of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The dependent variable is the tau-function τ = τ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The BKP hierarchy is encoded in the generating basic bilinear relation for the tau-function [25]: � C∞ dz 2πiz eξ(t,z)−ξ(t′,z)τ � t − 2[z−1] � τ � t′ + 2[z−1] � = τ(t)τ(t′) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) valid for all t, t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Here we use the standard notation ξ(t, z) = � k=1,3,5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' tkzk, t ± 2[z−1] = � t1 ± 2 z, t3 ± 2 z3 , t5 ± 2 z5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 7 The contour C∞ is a big circle around infinity such that the singularities coming from the factors with the tau-functions are inside it and those coming from the exponential factor are outside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The discrete BKP equation is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Put τ(ℓ, m, n) := τ � t − 2ℓ[a−1] − 2m[b−1] − 2n[c−1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2) Setting t′ k = tk − 2a−k/k − 2b−k/k − 2c−k/k in the bilinear relation, one can calculate the integral in the left hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) with the help of the residue calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (In doing so one should take into account that the poles at the points a, b, c which arise from the exponential factor are outside the contour, and the contour should be shrunk to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=') As a result, one obtains from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) the fully discrete BKP equation for τ(ℓ, m, n) first appeared in [23]: (a + b)(a + c)(b − c)τ(ℓ + 1)τ(m + 1, n + 1) − (a + b)(b + c)(a − c)τ(m + 1)τ(ℓ + 1, n + 1) + (a + c)(b + c)(a − b)τ(n + 1)τ(ℓ + 1, m + 1) = (a − b)(a − c)(b − c)τ τ(ℓ + 1, m + 1, n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) For simplicity of the notation, we explicitly indicate only the variables that are subject to shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Let us introduce the wave function ψ(ℓ, m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' z) by the formula ψ(ℓ, m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' z) = eξ(t,z) �a − z a + z �ℓ �b − z b + z �m τ � t − 2ℓ[a−1] − 2m[b−1] − 2[z−1] � τ � t − 2l[a−1] − 2m[b−1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4) A direct calculation shows that the fully discrete BKP equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) is equivalent to the following linear equation for the wave function: ψ(ℓ, m + 1) − ψ(ℓ + 1, m) = a − b a + b u(ℓ, m) � ψ(ℓ + 1, m + 1) − ψ(ℓ, m) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) where u(ℓ, m) = τ τ(ℓ + 1, m + 1) τ(ℓ + 1)τ(m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='6) So far all the discrete variables entered symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Now we are going to break the symmetry by distinguishing one of the variables, say ℓ, treating it as a space variable on the space lattice with lattice spacing η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Accordingly, we introduce the continuous variable x = ℓη with ℓ ± 1 corresponding to x ± η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The discrete time variable is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' It is convenient to change the notation as τ(ℓ, m, n) → τ m(x), ψ(ℓ, m) → ψm(x) (the variable n is supposed to be fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' With this notation, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) acquires the form ψm+1(x) − ψm(x + η) = κ um(x) � ψm+1(x + η) − ψm(x) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7) 8 um(x) = τ m(x) τ m+1(x + η) τ m(x + η) τ m+1(x), κ = a − b a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='8) Let us point out how the two continuous time limits discussed in the previous section look like in these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' For the first limit (which leads to the Toda lattice with constraint of type B) one should tend b → a and m → ∞ in such a way that the continuous time variable t = m(b−1 − a−1) remains finite and nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' For the second limit (which leads to the semi-discrete BKP equation) one should tend b → ∞ and m → ∞ in such a way that the continuous time variable t = mb−1 remains finite and nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Now we are going to obtain equations of motion for poles in x of the wave function ψm(x) (which are zeros of the τ m(x)) as functions of the discrete time m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' According to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4), we represent the wave function in the form ψm(x) = λx/ηνm ˆτ m(x) τ m(x) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='9) with λ = a − z a + z , ν = b − z b + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Note that in terms of λ and ν we have κ = λ − ν 1 − λν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='10) For elliptic solutions we set τ m(x) = N � j=1 σ(x − xm j ), ˆτ m(x) = N � j=1 σ(x − ym j ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='11) and assume that the xm j ’s are all distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='9) into the linear equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7), we obtain the following equation connecting τ m(x) and ˆτ m(x): ν ˆτ m+1(x) τ m+1(x) − λ ˆτ m(x + η) τ m(x + η) = λνκ τ m(x)ˆτ m+1(x + η) τ m(x + η)τ m+1(x) − κ ˆτ m(x)τ m+1(x + η) τ m(x + η)τ m+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='12) The both sides have simple poles at x = xm+1 i and x = xm i − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Equating the residues at these poles and making the shift m → m − 1 when necessary, we get the equations ντ m−1(xm i +η)ˆτ m(xm i ) = λνκτ m−1(xm i )ˆτ m(xm i +η) − κτ m(xm i +η)ˆτ m−1(xm i ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='13) λτ m+1(xm i −η)ˆτ m(xm i ) = κτ m+1(xm i )ˆτ m(xm i −η) − λνκτ m(xm i −η)ˆτ m+1(xm i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='14) Substituting x = xm i and x = xm+1 i −η into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='12), we see that one term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Making the shift m → m − 1 when necessary, we get the equations κτ m+1(xm i +η)ˆτ m(xm i ) = λτ m+1(xm i )ˆτ m(xm i +η) − ντ m(xm i +η)ˆτ m+1(xm i ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='15) λνκτ m−1(xm i −η)ˆτ m(xm i ) = ντ m−1(xm i )ˆτ m(xm i −η) − λτ m(xm i −η)ˆτ m−1(xm i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='16) Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='15) can be regarded as a system of linear equations for ˆτ m(xm i ) and ˆτ m(xm i +η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' According to the Cramer’s rule, the solution for ˆτ m(xm i ) is ˆτ m(xm i ) = −κτ m(xm i + η) ������� ˆτ m−1(xm i ) λντ m−1(xm i ) νˆτ m+1(xm i ) λτ m+1(xm i ) ������� ������� ντ m−1(xm i +η) −λνκτ m−1(xm i ) −κτ m+1(xm i +η) λτ m+1(xm i ) ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='17) 9 In their turn, equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='16) can be regarded as a system of linear equations for ˆτ m(xm i ) and ˆτ m(xm i −η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The solution for ˆτ m(xm i ) is ˆτ m(xm i ) = −κτ m(xm i − η) ������� λνˆτ m+1(xm i ) τ m+1(xm i ) λˆτ m−1(xm i ) ντ m−1(xm i ) ������� ������� −λτ m+1(xm i −η) κτ m+1(xm i ) λνκτ m−1(xm i −η) −ντ m−1(xm i ) ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='18) Equating the right hand sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='18), we obtain the equations for the discrete time dynamics of poles xm i : τ m+1(xm i )τ m(xm i − η)τ m−1(xm i + η) + τ m+1(xm i − η)τ m(xm i + η)τ m−1(xm i ) = κ2τ m+1(xm i +η)τ m(xm i −η)τ m−1(xm i ) + κ2τ m+1(xm i )τ m(xm i +η)τ m−1(xm i −η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='19) These are equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 5 A field analogue of the deformed RS system on a space-time lattice It is known that integrable models of the CM and RS type admit extensions to field the- ories (“field analogues”) in which the coordinates of particles xi become “fields” xi(x, t) depending not only on the time t but also on a space variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Equations of motion of these more general models can be obtained as equations for poles of more general elliptic solutions (called elliptic families in [15]) to nonlinear integrable equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In this case, one considers solutions which are elliptic functions of a linear combination λ of higher times tk of the hierarchy, their poles λi(x, t) being functions of the space and time vari- ables x, t (in the CM/KP case x = t1, t = t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' They obey a system of partial differential or difference equations which are equations of motion of the field analogue of the CM or RS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' They were obtained by this method in [15] and [16] respectively (see also [26], where elliptic families of solutions to the constrained Toda lattice were discussed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In this section we apply this method to the fully discrete BKP equation and obtain the field extension of the deformed RS model on a space-time lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' To wit, we will con- sider elliptic families of solutions to the fully discrete BKP equation and find dynamical equations for their poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1 Equations of motion on the space-time lattice from elliptic families for the fully discrete BKP equation We begin with the BKP hierarchy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Let λ = � j odd βjtj be an arbitrary linear com- bination of the times of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' According to [15], the tau-function τ(λ, t) corre- 10 sponding to a solution which is an elliptic function of λ is of the general form τ(λ, t) = ρ(t)ec1λ+c2λ2 N � j=1 σ(λ − λj(t)), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) where ρ(t) does not depend on λ and c1, c2 are some constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Note that the shift of λ by any period should give an equivalent tau-function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' the tau-function which differs from the initial one by multiplying by exponential function of a linear form in the times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Therefore, the zeros λi of the tau-function (poles of the solution) should satisfy the condition � i λi(t) = linear form in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2) This means that the “center of masses” of the set of the λi’s moves uniformly in all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' For the fully discrete BKP equation we can consider elliptic solutions of the form τ m(λ, x) = ρm(x)ec1λ+c2λ2 N � j=1 σ(λ − λm j (x)), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3) where x is a space variable and m is the discrete time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We assume that all λj’s are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Set σm(λ, x) := N � j=1 σ(λ − λm j (x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4) We can find solutions to equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7) in the form ψm(x) = ˆτ m(λ, x) σm(λ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) Plugging (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7), we get the equation ˆτ m+1(λ, x) σm+1(λ, x) − ˆτ m(λ, x + η) σm(λ, x + η) = κm(x)σm(λ, x)ˆτ m+1(λ, x + η) σm(λ, x + η)σm+1(λ, x) − κm(x)σm+1(λ, x + η)ˆτ m(λ, x) σm(λ, x + η)σm+1(λ, x), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='6) where κm(x) = κ ρm(x)ρm+1(x + η) ρm(x + η)ρm+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='7) The further calculation is similar to the one performed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='6) are elliptic functions of λ with simple poles at λ = λm+1 i (x) and λ = λm i (x − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Equating the residues at these poles and making the shifts m → m − 1, x → x − η when necessary, we get the equations σm−1(λm i (x), x + η)ˆτ m(λm i (x), x) = κm−1(x)σm−1(λm i (x), x)ˆτ m(λm i (x), x + η) − κm−1(x)σm(λm i (x), x + η)ˆτ m−1(λm i (x), x), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='8) σm+1(λm i (x), x − η)ˆτ m(λm i (x), x) = κm(x − η)σm+1(λm i (x), x)ˆτ m(λm i (x), x − η) − κm(x − η)σm(λm i (x), x − η)ˆτ m+1(λm i (x), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='9) 11 Substituting λ = λm i (x) and λ = λm+1 i (x−η) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='6), we see that one term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Making the shifts m → m − 1, x → x − η when necessary, we get the equations κm(x)σm+1(λm i (x), x + η)ˆτ m(λm i (x), x) = σm+1(λm i (x), x)ˆτ m(λm i (x), x + η) − σm(λm i (x), x + η)ˆτ m+1(λm i (x), x), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='10) κm−1(x − η)σm−1(λm i (x), x − η)ˆτ m(λm i (x), x) = σm−1(λm i (x), x)ˆτ m(λm i (x), x − η) − σm(λm i (x), x − η)ˆτ m−1(λm i (x), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='11) Consider equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='8) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' They can be regarded as a system of linear equations for ˆτ m(λm i (x), x) and ˆτ m(λm i (x), x+η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In their turn, equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='11) can be regarded as a system of linear equations for ˆτ m(λm i (x), x) and ˆτ m(λm i (x), x−η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Solving them for ˆτ m(λm i (x), x) with the help of the Cramer’s rule, as in the previous section, and equating the results, we get the equations (κm−1(x))−1σm+1(λm i (x), x)σm(λm i (x), x − η)σm−1(λm i (x), x + η) +(κm(x − η))−1σm+1(λm i (x), x − η)σm(λm i (x), x + η)σm−1(λm i (x), x) = κm−1(x − η)σm+1(λm i (x), x)σm(λm i (x), x + η)σm−1(λm i (x), x − η) +κm(x)σm+1(λm i (x), x + η)σm(λm i (x), x − η)σm−1(λm i (x), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='12) These are equations of motion for the field analogue of the deformed RS model on the space-time lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' They resemble equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) and reduce to them if one sets λm i (x) = xm i + x and κm(x) = κ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In the limit κ → 0, when the right hand side vanishes, equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='12) reduce to the fully discrete version of the field extension of the RS model [16] (see also [27], where similar equations were obtained from the elliptic Lax pair of general form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Note also that equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='12) can be written entirely in terms of the function τ m(λ, x) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3): τ m+1(λm i (x), x)τ m(λm i (x), x − η)τ m−1(λm i (x), x + η) +τ m+1(λm i (x), x − η)τ m(λm i (x), x + η)τ m−1(λm i (x), x) = τ m+1(λm i (x), x)τ m(λm i (x), x + η)τ m−1(λm i (x), x − η) +τ m+1(λm i (x), x + η)τ m(λm i (x), x − η)τ m−1(λm i (x), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='13) In the next subsection we will discuss continuous time limits of these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2 Continuous time limits For the continuous time limit, we rewrite equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='12) in the form κm(x−η) κm−1(x) � j σ(λm i (x) − λm+1 j (x))σ(λm i (x) − λm j (x−η))σ(λm i (x) − λm−1 j (x+η)) σ(λm i (x) − λm−1 j (x))σ(λm i (x) − λm+1 j (x−η))σ(λm i (x) − λm j (x+η)) = −1 + κm(x)κm(x−η) � j σ(λm i (x) − λm+1 j (x+η))σ(λm i (x) − λm j (x−η)) σ(λm i (x) − λm j (x+η))σ(λm i (x) − λm+1 j (x−η)) + κm−1(x−η)κm(x−η) � j σ(λm i (x) − λm+1 j (x))σ(λm i (x) − λm−1 j (x−η)) σ(λm i (x) − λm−1 j (x))σ(λm i (x) − λm+1 j (x−η)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='14) and assume that the λm j ’s and ρm behave smoothly when the time changes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' λm±1 i (x) = λi(x) ± ε ˙λi(x) + 1 2 ε2¨λi(x) + O(ε3), ρm±1(x) = ρ(x) ± ε ˙ρ(x) + O(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The limit is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We should expand the equation in powers of ε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' If κ2 ̸= 1, then the first non-vanishing order is ε, and we get the equations ¨λi(x) + � j �˙λi(x) ˙λj(x−η)ζ � λi(x)−λj(x−η) � + ˙λi(x) ˙λj(x+η)ζ � λi(x)−λj(x+η) �� − 2 � j̸=i ˙λi(x) ˙λj(x)ζ � λi(x) − λj(x) � + � c(x − η) − c(x) �˙λi(x) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='15) where c(x) = ˙ρ(x + η) ρ(x + η) − ˙ρ(x) ρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Summing the equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='15) over all i and taking into account the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='2), we find that c(x) is expressed in terms of the λi’s as follows: c(x) = �� k ˙λk(x) �−1 � i,j ˙λi(x) ˙λj(x+η)ζ � λi(x)−λj(x+η) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='16) Equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='15), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='16) were obtained in [16] as equations of motion for the field analogue of the RS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' If κ2 = 1, then the order ε gives the identity 0 = 0, and one has to expand up to the order ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In this way it is possible to obtain the field analogue of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Like in the previous section, another continuum limit is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We introduce the lattice fields ϕm i (x), ωm(x) by setting λm i (x) = ϕm i (x + mη), ρm(x) = ωm(x + mη) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='17) and assume that these new fields have a smooth continuous time limit in the sense that ϕm±1 i (x) = ϕi(x) ± ε ˙ϕi(x) + 1 2 ε2 ¨ϕi(x) + O(ε3), ωm±1(x) = ω(x) ± ε ˙ω(x) + O(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 13 In terms of the new fields, equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='14) read: κm(x−η) κm−1(x) � j σ(ϕm i (x) − ϕm+1 j (x+η))σ(ϕm i (x) − ϕm j (x−η))σ(ϕm i (x) − ϕm−1 j (x)) σ(ϕm i (x) − ϕm j (x+η))σ(ϕm i (x) − ϕm−1 j (x−η))σ(ϕm i (x) − ϕm+1 j (x)) = −1 + κm(x)κm(x−η) � j σ(ϕm i (x) − ϕm+1 j (x+2η))σ(ϕm i (x) − ϕm j (x−η)) σ(ϕm i (x) − ϕm j (x+η))σ(ϕm i (x) − ϕm+1 j (x)) + κm−1(x−η)κm(x−η) � j σ(ϕm i (x) − ϕm−1 j (x−2η))σ(ϕm i (x) − ϕm+1 j (x+η)) σ(ϕm i (x) − ϕm−1 j (x−η))σ(ϕm i (x) − ϕm+1 j (x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='18) The limit ε → 0 is consistent if κ = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Then in the order ε we obtain the equations ¨ϕi(x) + � j � ˙ϕi(x) ˙ϕj(x−η)ζ � ϕi(x)−ϕj(x−η) � + ˙ϕi(x) ˙ϕj(x+η)ζ � ϕi(x)−ϕj(x+η) �� − 2 � j̸=i ˙ϕi(x) ˙ϕj(x)ζ � ϕi(x) − ϕj(x) � − ∂t log w(x) ˙ϕi(x) + w(x)w(x + η)G+ i + w(x)w(x − η)G− i = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='19) where w(x) = ω(x + η)ω(x − η) ω2(x) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='20) G+ i = σ(ϕi(x)−ϕi(x+2η))σ(ϕi(x)−ϕi(x−η)) σ(ϕi(x)−ϕi(x+η)) × � j̸=i σ(ϕi(x)−ϕj(x+2η))σ(ϕi(x)−ϕj(x−η)) σ(ϕi(x)−ϕj(x+η))σ(ϕi(x)−ϕj(x)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='21) and G− i differs from G+ i by the change η → −η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Summing equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='19) over all i, we obtain the equation ∂t log w(x) � i ˙ϕi(x) = F(x) + w(x)w(x + η) � i G+ i + w(x)w(x − η) � i G− i , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='22) where F(x) = � i,j � ˙ϕi(x) ˙ϕj(x+η)ζ � ϕi(x)−ϕj(x+η) � + ˙ϕi(x) ˙ϕj(x−η)ζ � ϕi(x)−ϕj(x−η) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='23) Equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='19) together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='22) form a system of N + 1 differential equations for the N + 1 fields ϕj(x) = ϕj(x, t) (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' , N), w(x) = w(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In contrast to equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='15), where the extra field was not dynamical and could be excluded from the equations of motion, in the present case the extra field w is dynamical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='19), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='22) provide the field extension of the deformed RS system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' 14 6 Concluding remarks In this paper we have obtained integrable time discretization of the deformed RS sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The method is based on the explicit form of the B¨acklund transformation which is obtained as equations connecting the dynamics of poles and zeros of double-Bloch solu- tions ψ to the linear problem for the Toda lattice with constraint of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' As in other systems of the Calogero-Moser and Ruijsenaars-Schneider type, the B¨acklund transfor- mation is the passage from poles to zeros of the ψ-function, and this is interpreted as one step forward in the discrete time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Possible continuum limits of the discrete time equations obtained were also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' One of them gives equations of motion of the deformed RS system while another one gives equations of motion for poles of elliptic solutions to the semi-discrete BKP equation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We have also shown that the discrete time equations of motion for the deformed RS system describe evolution of poles of elliptic solutions to the fully discrete BKP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Besides, by considering more general elliptic solutions of the latter (the so-called elliptic families), we have obtained a field extension of the deformed RS model on a space-time lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' It is an important open question whether the obtained equations admit any commuta- tion representation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' whether they can be represented as a relation between matrices whose matrix elements depend on the dynamical variables, like (discrete versions of) Lax and Manakov triple representations or representation of the Zakharov-Shabat type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' We hope to address this problem elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Finally, we would like to mention that the discrete time equations of motion for the RS system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='1) mysteriously coincide with the nested Bethe ansatz equations arising in the theory of quantum integrable systems with elliptic R-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' It is still not clear whether this is just a coincidence or this fact has some profound reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' In this connection it is natural to ask whether the discrete time equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5) for the deformed RS system have any relation to quantum integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' To wit, the question is whether there exists any quantum integrable system solved by Bethe ansatz or any other method whose Bethe-like equations would be of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Appendix A: The Weierstrass functions In this appendix we present the definition and main properties of the Weierstrass func- tions: the σ-function, the ζ-function and the ℘-function which are used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Let ω, ω′ be complex numbers such that Im(ω′/ω) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The Weierstrass σ-function with quasi-periods 2ω, 2ω′ is defined by the following infinite product over the lattice 2ωm + 2ω′m′, m, m′ ∈ Z: σ(x) = σ(x| ω, ω′) = x � s̸=0 � 1 − x s � e x s + x2 2s2 , s = 2ωm + 2ω′m′ m, m′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (A1) It is an odd quasiperiodic function with two linearly independent quasi-periods in the 15 complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The expansion around x = 0 is σ(x) = x + O(x5), x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (A2) The monodromy properties of the σ-function under shifts by the quasi-periods are as follows: σ(x + 2ω) = −e2ζ(ω)(x+ω)σ(x), σ(x + 2ω′) = −e2ζ(ω′)(x+ω′)σ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (A3) Here ζ(x) is the Weierstrass ζ-function defined as ζ(x) = σ′(x) σ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (A4) As x → 0, ζ(x) = 1 x + O(x3), x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (A5) The Weierstrass ℘-function is defined as ℘(x) = −ζ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' It is an even double-periodic function with periods 2ω, 2ω′ and with second order poles at the points of the lattice s = 2ωm + 2ω′m′ with integer m, m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' As x → 0, ℘(x) = x−2 + O(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The Weierstrass functions obey many non-trivial identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' Here we present the two which are necessary for the calculations leading to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content='4): ζ(x + η) + ζ(x − η) − 2ζ(x) = ℘′(x) ℘(x) − ℘(η), (A6) ℘(x + η) − ℘(x − η) = − ℘′(x)℘′(η) (℘(x) − ℘(η))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' (A7) The proof is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} +page_content=' The both sides are elliptic functions of x, and the singular terms in the both sides coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfVAw3/content/2301.05020v1.pdf'} 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McLaughlin,1 Ryan O. Milligan,1, 2 Graham S. Kerr,2, 3 Aaron J. Monson,1 Paulo J. A. Sim˜oes,4, 5 and +Mihalis Mathioudakis1 +1Queen’s University Belfast, University Rd, Belfast BT7 1NN, Northern Ireland +2Department of Physics Catholic University of America, 620 Michigan Avenue, Northeast, Washington, DC 20064, USA +3Solar Physics Laboratory (Code 671), Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA +4Center for Radio Astronomy and Astrophysics Mackenzie, School of Engineering, Mackenzie Presbyterian University, S˜ao Paulo, Brazil +5SUPA, School of Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ, UK +ABSTRACT +The Lyman Continuum (LyC; < 911.12˚A) forms at the top of the chromosphere in the quiet-Sun, +making LyC a powerful tool for probing the chromospheric plasma during solar flares. To understand +the effects of non-thermal energy deposition in the chromosphere during flares, we analysed LyC profiles +from a grid of field-aligned radiative hydrodynamic models generated using the RADYN code as part +of the F-CHROMA project. The spectral response of LyC, the temporal evolution of the departure +coefficient of hydrogen, b1, and the color temperature, Tc, in response to a range of non-thermal +electron distribution functions, were investigated. The LyC intensity was seen to increase by 4-5.5 +orders of magnitude during solar flares, responding most strongly to the non-thermal electron flux of +the beam. Generally, b1 decreased from 102–103 to closer to unity during solar flares, indicating a +stronger coupling to local conditions, while Tc increased from 8–9 kK to 10–16 kK. Tc was found to +be approximately equal to the electron temperature of the plasma when b1 was at a minimum. Both +optically thick and optically thin components of LyC were found in agreement with the interpretation of +recent observations. The optically thick layer forms deeper in the chromosphere during a flare compared +to quiescent periods, whereas the optically thin layers form at higher altitudes due to chromospheric +evaporation, in low-temperature, high-density regions propagating upwards. We put these results in +the context of current and future missions. +1. INTRODUCTION +Solar flares are the most energetic events to occur +in the solar system, releasing up to 1032 erg of energy +over the course of tens of minutes (Fletcher et al. 2011). +This energy is typically thought to be released via mag- +netic reconnection in the solar corona. Following release, +this energy manifests itself in several ways, including in +the acceleration of large quantities of charged particles +that propagate along magnetic field lines, deeper into +the Sun’s atmosphere. These charged particles undergo +Coulomb collisions with the ambient plasma depositing +energy in the solar chromosphere, where intense heat- +ing, ionization, and mass flows result. The subsequent +increase in radiation allows us to shed light on the physi- +cal mechanisms that release and transport energy. Mod- +elling by Allred et al. (2005) suggests that solar flare +emission is energetically dominated by the Balmer con- +tinuum. However, Balmer continuum observations have +been severely lacking. Lyman Continuum flare obser- +vations have been more readily available in recent years +from the Extreme Ultraviolet Experiment (EVE; Woods +et al. 2012) on board the Solar Dynamic Observatory +(SDO; Pesnell et al. 2012), with the potential of upcom- +ing flare observations from the Spectral Imaging of the +Coronal Environment (SPICE; Spice Consortium et al. +2020) instrument on board the Solar Orbiter (M¨uller +et al. 2020), and the EUV High-Throughput Spectro- +scopic Telescope (EUVST; Shimizu et al. 2019) on board +Solar-C (Watanabe 2014). +The Lyman Continuum (LyC; <911.12˚A) results from +the free-bound transition of a free electron to the ground +state of an ambient hydrogen nuclei. In the quiet-Sun, +LyC forms at the top of the chromosphere/base of the +transition region where the temperature rises from the +cooler, denser chromosphere (T ≤ 104K) to the corona, +(T ≥ 106K; Vernazza et al. 1973, 1976, 1981; Avrett +& Loeser 2008; Machado et al. 2018). Therefore, the +LyC is sensitive to energy perturbations in the chromo- +sphere induced during solar flares, and since thermaliza- +tion occurs very rapidly at the higher densities located +here, its spectrum may reflect the local plasma tempera- +ture (Noyes & Kalkofen 1970). The LyC is a potentially +powerful diagnostic tool of the chromospheric response +to flare energy injection, but this potential is presently +largely untapped. +One of the earliest investigations into LyC formation +during solar flares was performed by Machado & Noyes +(1978), who observed nine flares using the extreme- +ultraviolet (EUV) spectrograph on the Apollo Telescope +arXiv:2301.01648v1 [astro-ph.SR] 4 Jan 2023 + +2 +McLaughlin et al. +Mount onboard Skylab (Ise & Cagle 1974). They found +that the non-local thermodynamic equilibrium (NLTE) +departure coefficient of the first level of hydrogen (b1) +tended towards unity during flares, more so than in ac- +tive regions or the quiet-Sun. The departure coefficient +is given by n1/n∗ +1, where n∗ +1 is the ground state LTE pop- +ulation and n1 is the NLTE ground state populations, +meaning the LyC formation region was driven closer to +LTE conditions during the solar flares (Menzel & Cilli´e +1937). They also found that the color temperature (Tc) +derived from the head of the continuum was around +Tc ∼ 7.9–9 kK, comparable to that of the surrounding +quiet-Sun, but hotter than active regions. +They con- +cluded that in flares LyC is optically thick and forms +close to LTE in a region of higher density deeper in the +chromosphere, whereas in active regions or the quiet- +Sun b1 ≈ 102 (Noyes & Kalkofen 1970). However, they +also showed that Tc measured at shorter wavelengths +(700–790 ˚A) revealed higher temperatures (10–15 kK, +although with large uncertainties), suggesting the pres- +ence of an optically thin LyC layer that formed higher in +the atmosphere. One limitation of this study was that +several of the observations were carried out in spectral +scanning mode; spectra were obtained in a 5 arcsec2 +area over 300–1335 ˚A, over 3.8 minutes, meaning that +different parts of the spectrum may have been sampled +at different times and may not reflect true changes due +to the flare. +More recently, Machado et al. (2018) performed a +study using Sun-as-a-star observations from EVE on +board SDO which allowed the temporal evolution of LyC +emission during six major solar flares to be investigated. +After converting the EVE data from spectral irradiance +to specific intensity, by assuming flaring areas, they re- +ported that the LyC intensities at the head of the con- +tinuum were enhanced by 3–4 orders of magnitude rela- +tive to preflare values. Their results supported the ear- +lier conclusions of Machado & Noyes (1978). However, +Machado et al. (2018) found a larger color temperature, +noting that Tc determined between 870–912 ˚A increased +from Tc = 8–9 kK to Tc ≈ 9–12 kK, perhaps due to the +study of more energetic events. +Milligan et al. (2012) reported an enhancement in LyC +emission during an X-class flare that occurred on 2011 +February 15 also using data from SDO/EVE. A follow- +up study showed that the total energy radiated by LyC +was a few percent of the total non-thermal electron en- +ergy, amounting to 1.8×1029 erg over the course of the +flare, comparable to the radiated soft X-ray energy (Mil- +ligan et al. 2014). LyC irradiance increased by approxi- +mately a factor of 10 during the impulsive phase. +Lemaire et al. (2004) presented observations of an +X5.3 solar flare from the head of the LyC obtained +from scattered light detected by the Solar Ultravio- +let Measurements of Emitted Radiation spectrometer +(SUMER; Wilhelm et al. 1995) on board the Solar & +Heliospheric Observatory (SOHO; Domingo et al. 1995) +mission. +They determined the local increase of LyC +radiance to be a factor of several thousand, in agree- +ment with Machado & Noyes (1978). Using the ratio of +the spectral radiance at 910˚A and 890˚A they estimated +Tc = 12.15 kK. +There have been several attempts at modelling LyC +during solar flares. Ding & Schleicher (1997) calculated +LyC intensities in response to a precipitating beam of +non-thermal particles using a non-LTE, radiative trans- +fer code. +They found that LyC is very sensitive to +the incoming flux of electrons, and they predicted that +the temperature at the formation height of LyC will +increase in conjunction with a downward shift of the +transition region. In particular, they noted that non- +thermal collisions were responsible for driving down b1. +In their experiments, Ding & Schleicher (1997) manu- +ally modified semi-empirical model atmospheres to in- +vestigate the impact of the chromospheric temperature +gradients, and transition region depth on the LyC for- +mation, in a manner that was not self-consistent (i.e. +they did not simulate the time-dependent chromospheric +radiation-hydrodynamic response to energy injection by +a beam of non-thermal electrons). +For the purposes +of including non-thermal collisional ionization of hy- +drogen, they assumed non-thermal electron fluxes of +F = [1010, 1011, 1012] erg cm−2 s−1, with spectral indices +δ = [3, 4, 5], and a fixed low-energy cutoff Ec = 20 keV. +They found that LyC intensities increased by 1–3 or- +ders of magnitude compared to quiet-Sun values, and +that the LyC intensities were highly sensitive to the +flux of the non-thermal electron beam. +The LyC in- +tensities were less sensitive to the spectral index of the +non-thermal electron distribution. +More recent modelling of LyC emission was performed +by Druett & Zharkova (2019) using the radiation trans- +port and hydrodynamic code, HYDRO2GEN. They in- +vestigated the spectral response of LyC to various elec- +tron beam fluxes during the impulsive and gradual +phases of solar flares. They found that the LyC intensi- +ties increased by 2 orders of magnitude from their F10 +model to their F12 model, highlighting the sensitivity +of the LyC intensities to the flux of the non-thermal +electron beam. From continuum contribution functions, +they determined that LyC forms under optically thick +conditions at the top of the chromosphere. During the +impulsive phase, they noted a flattening of the LyC spec- +tra but found that the gradient steepened further from +the head of the continuum, in agreement with Machado +& Noyes (1978) and Machado et al. (2018). During the +gradual phase, however, they observed a flattening of the +LyC spectra at shorter wavelengths compared to the im- +pulsive phase spectra. They stated that the flattening +below 700˚A is due to emission from high-altitude regions +that cool sufficiently to allow recombinations to occur, +resulting in an enhanced intensity longward of ∼ 700 ˚A. +In this paper, we build upon these prior modelling ef- +forts, by investigating the plasma properties (namely Tc + +Formation of The Lyman Continuum During Solar Flares +3 +and b1) in the region of LyC formation, and the diagnos- +tic potential of the LyC in flares. We present a detailed +analysis of LyC emission from a large parameter space +of solar flare models using the Flare CHRomospheres: +Observations, Models and Archives (F-CHROMA1) grid +of models generated by the 1D field-aligned radiative- +hydrodynamic code RADYN. +2. NUMERICAL SIMULATIONS +RADYN is a 1D field-aligned radiative hydrodynamic +code that solves the coupled time-dependent equa- +tions of hydrodynamics, NLTE radiative transfer, non- +equilibrium atomic level populations, and charge conser- +vation (Carlsson & Stein 1992, 1995, 1997; Allred et al. +2005; Allred et al. 2015). The code’s adaptive grid (Dorfi +& Drury 1987) allows RADYN to resolve steep gradients +or shocks that easily form in flares. RADYN solves the +NLTE radiation transfer for a 6-level hydrogen atom, +9-level helium atom, and a 6-level Ca II ion. +Important for modelling the LyC, RADYN models the +non-equilibrium populations, and for hydrogen and he- +lium includes non-thermal collisional rates between the +injected non-thermal electron distribution and ambient +plasma. For helium, we use the expressions from Arnaud +& Rothenflug (1985) to model non-thermal collisional +ionisation of He I→He II, and from He II→He III. For +hydrogen, we include non-thermal collisional ionisation +from the ground state, and excitation from the ground +state to n = [2, 3, 4], following the approach of Fang +et al. (1993). In this approach, we are able to include +the important effects of secondary collisions. +As dis- +cussed in several prior studies (e.g. Ricchiazzi & Canfield +1983; Fang et al. 1993; Karlick´y et al. 2004; Kaˇsparov´a +et al. 2009), non-thermal collisional ionisation from the +excited states are of lesser importance, in part because +the population of the excited states are orders of magni- +tude smaller than the ground state. Thus the ionisation +stratification is unlikely to be drastically altered by the +omission of non-thermal collisions from the ground state. +Note, however, that some models have included non- +thermal effects from the excited states (see e.g. Zharkova +& Kobylinskii 1993; Druett & Zharkova 2018) though +it is unclear just how much of a role those collisions +played compared to those from the ground state. Fig- +ure 3 of Zharkova & Kobylinskii (1993) does show that +non-thermal collisions from n = 2 became important +relative to the thermal collisional rates, but only deep +in the atmosphere. For the reasons that we discussed +above, and since we are focused on the LyC which forms +in the upper chromosphere/lower transition region, we +believe that we are justified in using only ground state +non-thermal collisional rates including secondary effects. +RADYN is a well-established resource that has been +extensively used to model the response of the solar at- +1 https://star.pst.qub.ac.uk/wiki/public/solarmodels/start.html +mosphere to flare energy injection (e.g. Abbett & Haw- +ley 1999; Allred et al. 2005; Allred et al. 2015; Kowal- +ski et al. 2015, 2017, 2022; Kowalski & Allred 2018; +Kennedy et al. 2015; Rubio da Costa et al. 2016; Kerr +et al. 2016, 2019, 2020, 2021; Sim˜oes et al. 2017; Brown +et al. 2018; Graham et al. 2020). +When simulating +flares driven by electron beams (the ‘standard model’) +the non-thermal electron distribution is modelled as a +power law characterised by the spectral index, δ, the +low-energy cutoff, Ec, and energy flux density, F. The +transport and thermalization of these non-thermal elec- +trons are modelled by solving the Fokker-Planck equa- +tions, including various diffusion terms (Allred et al. +2015). Note that we solve the Coulomb operator using +general forms of the Rosenbluth potentials and so no as- +sumption is made as to the target temperature; that is, +there is no need to make a cold or warm target approx- +imation (see also Allred et al. 2020). Beam parameters +can be obtained through backward modelling of hard X- +ray observations taken with space-borne satellites such +as the Ramaty High Energy Solar Spectroscopic Im- +ager (RHESSI; Lin et al. 2002) or the Fermi Gamma-ray +Space Telescope (Meegan et al. 2009), usually under the +assumption of thick target collisions (Brown 1971; Hol- +man et al. 2003; Krucker et al. 2007, 2008; Kontar et al. +2008). In our study, we do not model a specific flare, +but instead survey a large parameter space of δ, Ec & +F values, with the values of those parameters guided by +typical observed ranges (e.g. Kontar et al. 2011; Holman +et al. 2011). +The F-CHROMA grid of RADYN flare models, hosted +by Queen’s University Belfast, consists of 72 RADYN +simulations (Carlsson et al. 2022). This grid includes +δ = [3, 4, 5, 6, 7, 8], and Ec = [10, 15, 20, 25] keV, and +models various magnitudes of injected non-thermal elec- +tron flux as a 20 s triangular profile in time, peaking at +t = 10 s. The time-integrated energy flux ranges Ftot = +[3×1010, 1×1011, 3×1011, 1×1012] erg cm−2. It is typical +in the flare community to refer to the energy flux den- +sity, and so for the remainder of this paper we quote the +peak injected flux (at t = 10 s in each simulation), which +are F = [3×109, 1×1010, 3×1010, 1×1011] erg cm−2 s−1. +For brevity, we refer to these simulations as, e.g., 1F11, +equating to 1 × 1011 erg cm−2 s−1. +In this study, we have selected 25 of the 72 models +(Table 1). +Not every combination of δ and Ec were +modelled for inclusion in this grid for the highest en- +ergy flares. +The models omitted from our subset are +the 1F11, Ec=20keV, δ = [6, 7], models and the 1F11, +δ = 5, Ec=15keV model. +3. LYC RESPONSE TO FLARE ENERGY +INJECTION +3.1. Synthetic LyC spectra +Figure 1 shows continuum spectra at the time of the +peak LyC emission in each simulation, between 9.7 s +and 13.6 s. The He I edge at 503.98˚A is also visible. + +4 +McLaughlin et al. +400 +600 +800 +1000 +Wavelength (Å) +0 +2 +4 +6 +8 +log Intensity (erg cm−2 s−1 sr−1 Å−1) +3F9, Ec=20keV +pre−flare +δ= 3 +δ= 4 +δ= 5 +δ= 6 +δ= 7 +400 +600 +800 +1000 +Wavelength (Å) +1F10, Ec=20keV +400 +600 +800 +1000 +Wavelength (Å) +3F10, Ec=20keV +400 +600 +800 +1000 +Wavelength (Å) +1F11, Ec=20keV +3F9, δ= 5 +400 +600 +800 +1000 +Wavelength (Å) +0 +2 +4 +6 +8 +log Intensity (erg cm−2 s−1 sr−1 Å−1) +pre−flare +Ec= 15keV +Ec= 20keV +Ec= 25keV +1F10, δ= 5 +400 +600 +800 +1000 +Wavelength (Å) +3F10, δ= 5 +400 +600 +800 +1000 +Wavelength (Å) +1F11, δ= 5 +400 +600 +800 +1000 +Wavelength (Å) +Figure 1. Synthetic RADYN continua spectra from 400–1000˚A showing the head of the He I continuum (<503.98˚A), LyC +(<911.12˚A), and the tail of the Ca II continuum (<1044.00˚A). The top row shows spectra for a fixed Ec = 20 keV, with peak +beam fluxes of 3F9, 1F10, 3F10, and 1F11, all shown at the peak in the LyC spectrum (between 9.7 s–13.6 s for all models). +The black curve denotes the preflare spectra, while the colored curves show the spectra for spectral indices of δ=3–7. The +bottom row shows the spectra but for a fixed δ=5, varying Ec=15, 20, and 25 keV. Note the 3F10, δ=5, Ec = 20 keV and the +1F10, δ=5, Ec = 20 keV models have transient negative intensities in the tail of the LyC continuum at the time of the peak +LyC emission. This is due to numerical noise in the simulation at these times, which only lasts a very short amount of time. +Therefore, the spectra shown for these models have been shifted by 1 s. +F (erg cm−2 s−1) +δ +Ec(kev) +3 × 109 +3, 4, 5, 6, 7 +15, 20, 25 +1 × 1010 +3, 4, 5, 6, 7 +15, 20, 25 +3 × 1010 +3, 4, 5, 6, 7 +15, 20, 25 +1 × 1011 +3, 4, 5 +20, 25 +Table 1. Non-thermal electron beam parameters used for +generating the RADYN models analysed in this work. +The top row shows spectra generated for a fixed low- +energy cutoff of 20 keV, while increasing the beam flux +from 3F9 to 1F11 (left to right), and varying the spec- +tral indices from δ = 3–7 (colored lines). The pre-flare +spectra are shown in black. There were some snapshots +(< 1 s) in a few simulations in which the EUV con- +tinua tails had negative intensities, likely due to numer- +ical noise. The simulations where the LyC was affected +are: the 3F10, δ=5, Ec = 20 keV model and the 1F10, +δ=5, Ec = 20 keV model. Therefore, the spectra shown +for these models have been shifted by 1 s. The spec- +tra for the models where only the Ca II continuum was +affected were not shifted. As the electron beam intensi- +fies, so too does the response of LyC, peaking roughly +co-temporally with the peak of the heating rate. The +1F11, δ = 4, 20keV model is an exception, peaking at +a time of 13.6s as will be discussed in Section 3.2. For +the 3F9 models, the head of the continuum is around +four orders of magnitude greater than the pre-flare value. +This increases to around 5.5 orders of magnitude for the +1F11 models. The magnitude of the LyC enhancement +during solar flares is strongly dependent on the flux of +non-thermal electrons. +The dependence on the spec- +tral index of the beam is weaker; softer beams result in +slightly higher intensities. This is because softer beams +(larger δ) deposit their energy over a narrower region +in the upper chromosphere since they thermalize more +easily, resulting in more localized heating and electron +density enhancements. +The bottom row of Figure 1 shows synthetic LyC +spectra for a fixed spectral index of δ = 5 and Ec = +[15, 20, 25] keV (colored lines). +The LyC spectra are +weakly dependent on the low-energy cutoff, similar +to the behaviour seen with varying the spectral in- +dex; Beams with larger Ec values are composed of a +greater proportion of high-energy electrons that pene- +trate deeper into the atmosphere, resulting in the beam +heating region shifting downwards, affecting the LyC re- +sponse. The increase in intensity from varying the spec- +tral index and low-energy cutoff is much weaker com- + +Formation of The Lyman Continuum During Solar Flares +5 +pared with that caused by the increasing flux of the +non-thermal electrons. +Like Ding & Schleicher (1997), we found a strong +dependence of the LyC spectral intensity upon the +flux of the non-thermal electron beam, as seen in Fig- +ures 1. +However, RADYN predicts larger impulsive +phase LyC intensities than the modelling of Ding & +Schleicher (1997). Ding & Schleicher (1997) also pre- +dicted a stronger dependence on the spectral index than +RADYN. These differences may arise for a number of +reasons. Chiefly, Ding & Schleicher (1997) used a static +code with semi-empirical atmospheres and manually +varied the temperature and transition region location. +The electron beam was only used to add non-thermal +collisions, and did not drive the chromospheric response. +In our dynamic simulations, while these non-thermal +collisions certainly play an important role, their impact +on driving the overall dynamics is taken into account, +so that there is a feedback between the atmosphere and +the beam propagation/thermalisation. +Recent modelling of LyC emission during solar flares +was also performed by Druett & Zharkova (2019) using +the radiative hydrodynamic code, HYDRO2GEN. They +present LyC spectra for their 1F10, 1F11, and 1F12 solar +flare models with Ec = 10 keV. The magnitude of our +impulsive phase LyC intensities are in agreement with +their spectra. However, the F-CHROMA grid does not +include any 1F12 simulations. +3.2. Synthetic LyC lightcurves +The flare excess integrated LyC lightcurves are shown +in Figure 2 for the 3F9, 1F10, 3F10, and 1F11 models, +with Ec = 20 +keV, and spectral indices ranging from +δ = 3–7 (colored lines). Note the saw-toothing seen for +some of the models is due to numerical noise in the spec- +tra between time steps. The synthetic lightcurves again +highlight that the magnitude of the LyC enhancement is +primarily driven by the magnitude of the non-thermal +electron flux, whereas variations in the spectral index +are less significant. Softer beams result in slightly higher +LyC fluxes. This is because softer beams (larger δ) de- +posit their energy over a narrower region in the upper +chromosphere, resulting in more localized heating close +to the LyC formation region. There is a period of time +near the beam onset where the LyC emission is barely +enhanced (recall that the flux of the non-thermal elec- +tron distribution ramps up in a triangular profile). The +duration of this period is dependent upon the flux of the +non-thermal electron beam, with larger fluxes having a +shorter delay in the enhancement. All models consis- +tently showed that notable LyC enhancement occurred +after approximately 3 × 109–4 × 109 erg cm−2 of energy +was deposited by the beam. +Flares that inject larger +non-thermal electron fluxes more rapidly produce con- +ditions conducive to LyC ionizations. +Despite this variable onset time, each LyC flare +reaches a maximum around t ∼ 10 s and falls thereafter +when the energy flux of the electrons decreases. +The +LyC response largely tracks the beam heating profile, +in agreement with Druett & Zharkova (2019). However, +the 1F11, δ = 4, Ec = 20 keV model shows a secondary +peak in the LyC lightcurve at t = 13.6 s. This second +maximum occurs as a chromospheric ‘bubble’ (that is, +a region of dense, low-temperature material, appearing +at high altitude) dissipates at that time, resulting in +a decrease in the optical depth (Reid et al. 2020). At +the same time, a condensation front producing optically +thin LyC emission merges with the heated chromosphere +producing the optically thick LyC layer, resulting in a +sudden, secondary increase in the LyC intensity. The +optically thin LyC layers are discussed in more detail in +Section 3.3. +The 1F11, δ = 3, Ec = 20 keV and 3F10, Ec = +20 keV, δ = 6, 7 models also show a secondary increase +in LyC emission after the peak beam heating. +This +is caused by the merging of optically thin condensa- +tion fronts with the optically thick LyC layer. +How- +ever, for these models, the chromospheric bubbles do +not dissipate before this merging. For the 1F11, δ = 5, +Ec = 20 keV model the merging of the two layers occurs +before the peak beam heating, resulting in the peak seen +between 9–10 s in Figure 2 (the formation layers for this +model are discussed in further detail in Section 3.3). +All this is to say that features in the lightcurves, while +largely following the profiles of energy injection, are in- +fluenced by the complex dynamics of the flaring chro- +mosphere. +3.3. LyC Formation Properties +3.3.1. Formation Heights and Optical Depth +The contribution function to the emergent intensity ef- +fectively shows where in the solar atmosphere the emis- +sion originates (for brevity we refer to this simply as +the ‘contribution function’ going forward). The formal +solution of the emergent intensity can be expressed as: +Iλ = 1 +µ +� +z +CI(z)dz = +� +z +jλ exp (τλ/µ)dz +(1) +where the integrand is the contribution function, CI(z) +(Magain 1986; Carlsson & Stein 1994). +Following +Kowalski et al. (2017) we express CI(z) in terms of emis- +sivity, jλ, attenuated by some optical depth τλ. +The +emissivity and opacities are the sum of various sources, +including hydrogen bound-free, hydrogen free-free, H−, +scattering by metals, Rayleigh scattering, and Thom- +son scattering. Of these, the bound-free transitions are +treated fully in NLTE, whereas the others are treated +in LTE, but using the NLTE, non-equilibrium electron +or proton densities as appropriate (See Kowalski et al. +(2017) for more details). +Generally, if the emission originates close to the τν = 1 +surface then the emission forms under optically thick +conditions, and if it originates in a region where τν << 1 + +6 +McLaughlin et al. +0 +5 +10 +15 +20 +Time (s) +0.0 +0.1 +0.2 +0.3 +0.4 +Flux (x109 erg cm−2 s−1) +3F9, Ec=20keV +δ= 3 +δ= 4 +δ= 5 +δ= 6 +δ= 7 +0 +5 +10 +15 +20 +Time (s) +0.0 +0.1 +0.2 +0.3 +0.4 +1F10, Ec=20keV +0 +5 +10 +15 +20 +Time (s) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +3F10, Ec=20keV +0 +5 +10 +15 +20 +Time (s) +0 +2 +4 +6 +8 +1F11, Ec=20keV +Figure 2. Flare excess lightcurves of spectrally integrated LyC emission from RADYN (505.0˚A to 911.7˚A) for the 3F9, 1F10, +3F10, and 1F11 models with δ=3–7 and Ec=20keV. +1000 +1500 +2000 +Height (km) +1000 +1500 +2000 +Height (km) +τ=0.01 +τ=0.1 +τ=1 +τ=0.01 +τ=0.1 +τ=1 +−10.00 +−8.00 +−6.00 +−4.00 +−2.00 +log Contribution function (erg cm−3 s−1 sr−1 Å−1) +t=0.0s +3 +4 +5 +6 +7 +log Temp (K) +t=8.0s +3 +4 +5 +6 +7 +log Temp (K) +t=10.0s +3 +4 +5 +6 +7 +log Temp (K) +500 +600 +700 +800 +900 +Wavelength (Å) +1000 +1500 +2000 +Height (km) +500 +600 +700 +800 +900 +Wavelength (Å) +1000 +1500 +2000 +Height (km) +t=14.0s +500 +600 +700 +800 +900 +Wavelength (Å) +500 +600 +700 +800 +900 +Wavelength (Å) +t=18.0s +500 +600 +700 +800 +900 +Wavelength (Å) +500 +600 +700 +800 +900 +Wavelength (Å) +t=20.0s +Figure 3. The log of the contribution function for the 3F9, δ = 5, Ec = 20 keV model at times t = [0, 8, 10, 14, 18, 20] s. The +heights at which τ=1 (grey solid line), τ=0.1 (grey dashed line), and τ=0.01 (grey dot-dashed line) are shown. The temperature +profile is shown in red at each time step. +then it can be considered to be optically thin. It is possi- +ble that some flares may exhibit both optically thick and +thin components with significant emission coming from +multiple layers, e.g. from an optically thin layer overly- +ing the chromosphere. Knowledge of the plasma proper- +ties in the LyC forming region(s), in response to differ- +ent heating conditions, will broaden our understanding +of energy transport processes during flares, and where +this energy is deposited. +To illustrate the general formation properties we dis- +cuss detailed examples from two simulations: the 3F9, +δ = 5, Ec = 20 keV and the 1F11, δ = 5, Ec = 20 keV + +Formation of The Lyman Continuum During Solar Flares +7 +800 +1000 +1200 +1400 +1600 +1800 +Height (km) +800 +1000 +1200 +1400 +1600 +1800 +Height (km) +τ=0.01 +τ=0.1 +τ=1 +τ=0.01 +τ=0.1 +τ=1 +−10.00 +−8.00 +−6.00 +−4.00 +−2.00 +0.00 +log Contribution function (erg cm−3 s−1 sr−1 Å−1) +t=0.0s +3 +4 +5 +6 +7 +log Temp (K) +t=7.0s +3 +4 +5 +6 +7 +log Temp (K) +t=8.0s +3 +4 +5 +6 +7 +log Temp (K) +500 +600 +700 +800 +900 +Wavelength (Å) +800 +1000 +1200 +1400 +1600 +1800 +Height (km) +500 +600 +700 +800 +900 +Wavelength (Å) +800 +1000 +1200 +1400 +1600 +1800 +Height (km) +t=8.8s +500 +600 +700 +800 +900 +Wavelength (Å) +500 +600 +700 +800 +900 +Wavelength (Å) +t=11.0s +500 +600 +700 +800 +900 +Wavelength (Å) +500 +600 +700 +800 +900 +Wavelength (Å) +t=15.0s +Figure 4. Same as Figure 3, but for the 1F11, δ = 5, Ec = 20 keV model. Note the color bar scale has changed. +models. +These showcase a weaker and stronger flare, +respectively. Figure 3 shows the temporal evolution of +the log of the contribution function for the 3F9, δ = 5, +Ec = 20 keV model during the heating phase. The tem- +perature stratification (solid red line) and the heights +at which τλ = [0.01, 0.1, 1] are also indicated (grey dot- +dashed, dashed, and solid lines, respectively). Initially, +at t = 0 s, there is a peak in the contribution function at +the base of the transition region (z ∼ 1750 km), form- +ing between 0.1 ≲ τλ ≲ 1, corresponding to an optically +thick LyC formation, in agreement with the literature +(Machado & Noyes 1978; Machado et al. 2018; Druett +& Zharkova 2019). By t = 8 s, the formation region +has widened, and shifted deeper into the chromosphere +(z ≈ 1200 − 1400 km) as the beam heating strength- +ens. +The height difference between τλ = 1 for short +and long wavelengths has also narrowed. At this time, +opacity effects are still significant, with the bulk of the +emission forming between 0.1 ≲ τλ ≲ 1. A secondary +peak in the contribution function is present higher in +altitude (z = 1700 km), co-spatial with a chromospheric +‘bubble’, seen as a narrow dip in the temperature pro- +file forming at the base of the transition region at t = 8, +10, 14, 18, and 20 s in Figure 3. Emission from this +very narrow layer forms under optically thin conditions +τλ < 0.01. Similar regions of cooler, yet dense plasma, +are present in most of our simulations (see also Allred +et al. 2005; Reid et al. 2020). Druett & Zharkova (2019) +also presented evidence for an overlying optically thin +layer in their 1F11, Ec=10keV model. +As time pro- +gresses, this chromospheric ‘bubble’ propagates upwards +towards the corona along a chromospheric ablation (also +referred to as ‘evaporation’) front. The bubble collapses +as it propagates, narrowing until all of the cool material +has been heated to coronal temperatures (or higher). In +the declining phase of the heating (t > 10 s) the τλ = 1 +layer begins to gradually return to the pre-flare height +as the atmosphere cools and the number of recombina- +tions in low opacity regions increases whilst ionization +decreases, in agreement with (Druett & Zharkova 2019). +The τλ = 0.1 layer begins shifting to the location of the +bubble as the density there increases, raising the LyC +opacity at those altitudes. +Figure 4 shows the temporal evolution of the log of +the LyC contribution function for the 1F11, δ = 5, +Ec = 20 keV model during the heating phase. In this +more energetic simulation, the LyC forms over much +more narrow regions (z ≈ 1000–1100 km) due to the +more dramatic response of the chromosphere that be- +comes very compressed. By t = 8 s, two optically thin + +8 +McLaughlin et al. +layers have formed due to bubbles flanking a rapidly ex- +panding high-temperature region (one upflowing bubble +due to explosive evaporation at z ≈ 1450 km, and one +downflowing bubble at z ≈ 1250 km, commonly referred +to as a chromospheric condensation). When the very +dense downflowing bubble effectively merges with the +bulk of the chromosphere the density at temperatures +favorable to produce LyC can rapidly increase. Since +opacity is a strong function of wavelength there can be +times at which there is a marked optical depth strati- +fication, resulting in variations to the spectral shape of +the LyC (e.g. flattening at certain wavelengths). Ex- +amples are shown at t = [8.8, 11] s, where the τλ = 1 +layer forms at a higher altitude for the head of the con- +tinuum compared to the tail. In this stronger flare, the +chromosphere becomes so compressed that the LyC ul- +timately forms from a vanishingly narrow downflowing +layer in the latter stages of the flare, shown at t = 15 s +of Figure 4 at z ≈ 1000 km. +These formation properties are generally true of the +other simulations. Most simulations contain some com- +bination of optically thick and overlying optically thin +contributions to the emergent spectra (though the bulk +of the emission does tend to originate between 0.1 ≲ +τλ ≲ 1). Stronger flares produce more dramatic atmo- +spheric responses, leading to the bubbles and shocks, +and associated phenomenon. +These dynamics also +emerge and develop faster with increasing flare strength, +and for softer non-thermal electron distributions. This +is because these electrons heat the uppermost chromo- +sphere very efficiently and do not penetrate as deeply, +driving flows more easily. Weaker flares and harder spec- +tra do not tend to produce as many shocks, do not signif- +icantly compress the chromosphere, and exhibit mainly +upflowing features. +3.3.2. Emissivity and Opacity +To understand some of the features described we can +briefly look at the components of opacity and emissivity +as functions of space, time, and wavelength. We illus- +trate the components of emissivity and opacity at two +times from one flare for a long (λ = 900 ˚A) and short +(700 ˚A) wavelengths in Figures 5. The dominant source +of opacity at all wavelengths, throughout the chromo- +sphere and through the lower transition region was hy- +drogen free-bound opacity, χλ,Hbf (red dashed line). To- +wards the upper transition region and corona, where +hydrogen is ionized, Thomson scattering begins to dom- +inate but the overall effect on the emerging radiation is +small, compared to the free-bound opacity (pink dashed +line). +Once the flare starts in earnest, we saw that the for- +mation height of the LyC dropped in altitude. This is +because the upper chromosphere was strongly heated, +ionizing hydrogen and decreasing χλ,Hbf , meaning that +the τλ = 1 forms much deeper in the atmosphere, shown +in the top panels of Figure 5. +Later, the merging of +the dense bubbles with the bulk of the chromosphere +means that χλ,Hbf increases, but since this is a function +of wavelength, this happens first for longer wavelengths +during this process. +The emissivity of the LyC is also dominated by hy- +drogen recombinations, jλ,Hbf at longer wavelengths +(roughly speaking λ > 700 ˚A), but at shorter wave- +lengths Thomson scattering can be important. Thom- +son scattering can also compete with jλ,Hbf at longer +wavelengths at certain times, but once the electron den- +sity is high (e.g. in the narrow chromospheric bubbles) +then jλ,Hbf once again dominates. The more extended +region of flaring LyC formation compared to the quiet +Sun can be understood from the stratification of emis- +sivity and opacity also. +During a flare, a greater ex- +tent of the chromosphere is at an elevated temperature, +and hence electron density, so the free-bound emissivity +throughout the greater geometric height range is raised. +Integrating over height yields a higher emergent inten- +sity. +In Figures 5, note the initial decrease in opacity (top +row t=2s panels for 700˚A and 900˚A), pushing LyC for- +mation deeper, and then the influence of the bubbles. +The emissivity is largest at the optically thick layer (be- +tween τ = 0.1–1 heights) as the density of emitting par- +ticles is greatest here. The optically thin bubbles also +have an increased emissivity compared to the ambient +plasma. +At shorter wavelengths the emissivity of the +optically thicker layer is comparable to the emissivity of +the optically thin bubbles, meaning the optically thin +layers will reinforce the lower wavelength LyC intensi- +ties to a greater extent than the continuum head, in +agreement with the literature (Machado & Noyes 1978; +Machado et al. 2018; Druett & Zharkova 2019). +4. LYC COLOR TEMPERATURE AND +DEPARTURE FROM LTE +4.1. Fitting the LyC spectra +Following the method used by Machado & Noyes +(1978) and Machado et al. (2018), the synthetic LyC +spectra were fit using the Eddington-Barbier (EB) rela- +tion to determine values for b1 and Tc. This approxima- +tion equates the emergent intensity to the source func- +tion at optical depth unity, and is given by: +Iλ(µ) ≈ Sλ(τλ = µ) = Bλ(Tc) +b1 +, +(2) +where Bλ(Tc) is the Planck function, +Bλ(Tc) = 2hc2 +λ5 +1 +exp +� +hc +λkBTc +� +− 1 +. +(3) +Iλ is the continuum intensity and Sλ is the source func- +tion. The λ subscripts indicate that these variables are +functions of wavelength. Tc is the color temperature, b1 +is the non-local thermodynamic equilibrium departure + +Formation of The Lyman Continuum During Solar Flares +9 +Figure 5. The components of λ = [700, 900] ˚A opacity (top row) and emissivity (bottom row) for two snapshots in the 1F11, +δ = 5, Ec = 20 keV model. The left-hand panels show 700 ˚A and the right-hand panels show 900 ˚A. The light grey dot-dashed +line in each panel shows the total opacity or emissivity at t = 0 s. The colored lines correspond to various contributions to the +total emissivity or opacity (the sums of these contributions are shown as thick black lines). The vertical lines correspond to the +heights at which τ = 1 (dashed), τ = 0.1 (dotted), and τ = 0.01 (dot-dashed). +coefficient of the first level of hydrogen, and all other +constants have their usual meanings. +An illustration of this spectral fitting is shown in the +left-hand panel of Figure 6, for various times during the +3F9, δ = 5, Ec = 20 keV model. The right-hand panel +shows the temporal evolution of the log of the goodness- +of-fit (ϵ) with vertical dashed lines denoting the time of +the spectra in the left panel. The goodness-of-fit values +are given by: +ϵ = 1 +N +N +� +i=1 +|yi − Oi| +yi +, +(4) +where i is the element index, N is the number of data +points, yi is the data value and Oi is the fitted value. As +the goodness-of-fit is weighted by the data for each wave- +length and time, smaller values correspond to a smaller +relative difference between the data and the fit, meaning +the fits are better. +In the right-hand panel of Figure 6, at t ∼ 5 s the +beam has mostly heated a region in the atmosphere be- +low the formation height of the LyC (i.e below τλ = 1, +where photons cannot readily escape). As discussed in +Section 3.3, the height at which τλ = 1 varies with wave- +length, typically forming lower in the atmosphere with +decreasing wavelength. The difference is a few tens of +km at t = 5 s. The flare does produce a small increase in +emissivity above the height at which τλ = 1 for shorter +wavelengths, increasing the emergent intensity towards +the tail of the LyC. At that height, however, τλ > 1 +for the head of the continuum and so there is no mean- +ingful change in the emergent intensity for longer wave- +lengths. During these times, the LyC spectrum cannot +be approximated by the EB assumption, resulting in the +large spike in ϵ seen between t = 5–6 s in the right-hand +panel of Figure 6, indicating a poor fit. Further discus- +sion on this behaviour can be found in Section 5.2. + +^ = 700 A +入 = 900 A +t = 8.80 s +t = 2.00 s +t = 8.80 s +t = 2.00 s +10-3 +10-3 +10-6 +10-6 +[cm' +10-9 +10- +10-12 +10-12 +10-15/ +10-15L +1000 +2000 +3000 +1000 +2000 +3000 +1000 +2000 +3000 +1000 +2000 +3000 +t = 8.80 s +t = 2.00 s +t = 8.80 s +t = 2.00 s +102 +102 +A-1] +100 +100 +sr-1 +Emissivity +10 +10- +10 +10-6 +10° +1000 +3000 +1000 +1000 +2000 +1000 +2000 +3000 +2000 +3000 +2000 +3000 +Height [km] +Height [km] +Height [km] +Height [km] +Total, t= Os +Hff +TScatt + = 5, Ec= 20 keV +Total +Metals +RScatt +Fpeak = le+1l erg cm-2 s-1 +Hell bf +Hbf +.H10 +McLaughlin et al. +Figure 6. Left-hand panel shows synthetic RADYN spectra from 400–911˚A for the 3F9, δ=5, Ec=20keV model, at times t=0s, +5.3s, 5.7s, 10s, and 20s (colored dots). Fits to LyC using Equation 2 are shown by solid colored lines. The right-hand panel +shows the log of the goodness-of-fit (ϵ) as a function of time for the fits shown in the left panel (solid black curve), while the +blue curve illustrates the ϵ values when the data was fit between 800.0˚A and 911˚A. The dashed colored lines denote the times +of the spectra in the left panel. +In general, the tail of the continuum does not conform +to the EB assumption as well as longer wavelengths, +particularly in the decay phase following the flare peak. +This is due to the presence of the overlying optically +thin LyC layers discussed in Section 3.3. When Equa- +tion 2 is applied only between λ = 800−911˚A we obtain +better fits (the blue line on the goodness-of-fit panel in +Figure 6) compared to fitting the full range λ = 505– +911˚A (black line). This suggests two distinct gradients +of the LyC, in agreement with the literature (Machado +& Noyes 1978; Machado et al. 2018; Druett & Zharkova +2019). +For times in the simulation where the fit was +reasonable, we obtain Tc values consistent with previ- +ous studies, increasing from Tc ∼ 9 kK to Tc ∼ 13.5 kK, +with b1 decreasing, though not reaching unity. Where +we see a poor fit (e.g. t = 5.7 s) we obtain an exception- +ally large b1 and Tc > 34 kK. +Values of Tc and b1 were obtained from fitting the LyC +spectra by applying the EB assumption (Equations 2 +and 3) at each time step from every flare in our study. +Figure 7 shows the temporal evolution of b1 (top row) +and Tc (bottom row) for the 3F9, 1F10, 3F10, and 1F11 +models, with δ = [3–7] (colored lines), and Ec = 20 keV +where we fit between λ = 700–911 ˚A. This range was +selected as Figure 6 shows that the Eddington-Barbier +approximation generally fits the data well at those wave- +lengths, but shorter wavelengths can deviate at certain +times. +During the beam onset, the initial response of b1 and +Tc is somewhat dependent on the spectral index. For +beams with δ ≥ 5, b1 initially decreases steadily during +the beam onset followed by a rapid and sudden increase +whereas, beams with δ ≤ 4 show an initial gradual in- +crease in b1, peaking around the time of the spikes in +the other models. We limit the range of the figures to +the scales shown, but the b1 and Tc values at the time of +these spikes can be several orders of magnitude larger. +The spikes in b1 and Tc are caused by the flattening in +the continuum head, at which times the EB approxima- +tion is no longer a valid assumption; the b1 and Tc values +at these times are unreliable. However, there are three +key times when the behaviour of b1 is consistent across +all models. At t = 0 s, b1 ∼ 102–103, decreasing to a +minimum around t ∼ 10 s (flare peak), then increasing +again during the decay phase. +The magnitude of the +b1 minima depends upon the flux of the non-thermal +electron beam: b1 values for the 3F9 models decrease +from b1 ∼ 103 to b1 ∼ 102, whereas the 3F10 and 1F11 + +t = 0.0 s +6 = 5 +(Tc=8922K;b1=1.06e+03) +E. = 20 keV +t = 5.3 s +t= 10.0s +(Tc=8887K; b1=7.29e+02) +(Tc=13763K;b1=6.49e+01) +t = 5.7 s +t= 15.0s +(Tc=34143K;b1=3.02e+07) +(Tc=13480K;b1=9.62e+01) +GoF[505-911A] +GoF.[800-911A] +105 +10 +103 +Intensity +cm +101 +[erg +10- +10 +0 +5 +10 +15 +20 +600 +700 +800 +900 +Time [s] +Wavelength [A]Formation of The Lyman Continuum During Solar Flares +11 +0 +2 +4 +6 +log Departure coefficent +3F9, Ec=20keV +1F10, Ec=20keV +3F10, Ec=20keV +1F11, Ec=20keV +δ= 3 +δ= 4 +δ= 5 +δ= 6 +δ= 7 +0 +5 +10 +15 +20 +Time (s) +6.0×103 +8.0×103 +1.0×104 +1.2×104 +1.4×104 +1.6×104 +1.8×104 +Colour Temperature (K) +3F9, Ec=20keV +0 +5 +10 +15 +20 +Time (s) +1F10, Ec=20keV +0 +5 +10 +15 +20 +Time (s) +3F10, Ec=20keV +0 +5 +10 +15 +20 +Time (s) +1F11, Ec=20keV +Figure 7. The temporal evolution of b1 and Tc for the 3F9, 1F10, 3F10, 1F11 models with Ec=20keV and δ=3–7. +models decrease closer 10−1 < b1 < 101. +The infer- +ence there being that more energetic simulations drive +LyC formation closer to LTE due to the increased elec- +tron densities. Both thermal and non-thermal collisions +will significantly increase in those simulations. Ding & +Schleicher (1997) presented a similar finding whereby +beams with larger non-thermal electron fluxes caused b1 +to decrease to a greater extent. +As discussed earlier, Machado et al. (2018) determined +b1 and Tc values from pre-flare and flaring spectra for +the six solar flares they analysed. The pre-flare fit re- +sults from our modelling are consistent with their qui- +escent Tc and b1 results. Further, the range of Tc values +we measured in our flare simulations are also generally +consistent with the observations. During the observed +flares, Machado et al. (2018) reported b1 ≈ 1, whereas +our model results show a wide range of values (between +0.1–102) depending upon the flux of the non-thermal +electron beam. One possible reason for this discrepancy +may be due to the dependency of the observed b1 values +scaling with the assumed flaring area. Further details +can be found in Machado et al. (2018). +To visualise the spread of Tc and b1 values we +produced a 2D histogram that collates the informa- +tion from all simulations for all times with parame- +ters: +δ = [3, 4, 5], Ec = [15, 20, 25] keV, Fpeak = +[3F9, 1F10, 3F10, 1F11] erg s−1 cm−2 (recall that the +Fpeak = 1F11 simulations did not include the Ec = +15 keV scenario). This is shown in Figure 8, where it is +clear that Tc increases from quiet Sun values, Tc ∼ 8– +9 kK, to values roughly in the range Tc ∼ 10 − 16 kK, +in agreement with the literature (Machado & Noyes +Figure 8. Histogram of Tc and b1 obtained from fitting the +EB approximation to our simulation grid, weighted by the +number of counts. +1978; Ding & Schleicher 1997; Lemaire et al. 2004; +Machado et al. 2018). There are two distinct clusters +within the histogram; the cooler cluster (Tc ∼ 6–11 kK, +b1 ∼ 101 − 105) and the hotter cluster (Tc ∼ 10–20 kK, +b1 ∼ 101 −103). The cooler cluster corresponds to times +before the brief breakdown in the EB approximation, +whilst the hotter cluster corresponds to times after this. +4.2. Comparing Spectral Fitting to Derived Plasma +Properties + +25 +Fit:入=[700.00-911.001A +102 +20 +Telfit [kK] +Counts +15 +101 +10 +100 +101 +bi fit12 +McLaughlin et al. +We can assess how consistent the properties derived +from fitting the spectra are with the actual plasma con- +ditions in the models by comparing the b1 values directly +from RADYN, that we refer to as b1,rad. The atomic +level populations are functions of height and wavelength, +and so to obtain b1,rad averaged over the LyC formation +region we calculate the normalised cumulative distribu- +tion function (NCDF) of the contribution function, Ccdf +(see also e.g. Kowalski et al. 2017). The heights corre- +sponding to where the bulk of the emission originates +are selected. The weighted average of b1,rad in that for- +mation region was then obtained, weighted by the con- +tribution function: +< b1,rad >= +� z(Ccdf =upp) +z(Ccdf =low) CI(z) b1,rad dz +� z(Ccdf =upp) +z(Ccdf =low) CI(z) dz +, +(5) +where z(Ccdf = low) refers to the height at which the +Ccdf reaches the lower bound, and z(Ccdf = upp) the +height of the upper bound. For example, the heights +corresponding to 10% & 90% of the Ccdf, respectively. +Figure 9. Temporal evolution of < b1,rad > values for the +3F10, δ = 6, Ec = 20 keV model. The < b1,rad > values are +shown for where λ = 900.3 ˚A emission originates, considering +Ccdf = [10–90%, 50–90%, 10–40%] (red, blue, and purple, +respectively). Also shown is the result of fitting the spectra +between λ = 700–911 ˚A using the EB approximation (solid +black line). The b1=1 line is shown by the dashed black line +for reference +Figure 9 shows < b1,rad > compared with b1 obtained +from our spectral fitting (solid black line). Formation +height ranges corresponding to [10–90%, 50–90%, 10– +40%] (red, blue, and purple lines, respectively) were con- +sidered. These ranges were selected by integrating the +contribution function as a function of height and deter- +mining the ranges where the emission became optically +thick or thin. The emission range 10–90% considers both +optically thick and thin components, 50–90% considers +emission between τλ ≈ 0.1–1 (optically thick), and 10– +40% considers emission above the τλ < 0.1 layer (opti- +cally thin) at 900.3˚A. The < b1,rad > values from the +optically thick layer are more consistent with those ob- +tained from the spectra but deviate during the declining +phase of the beam heating. +The < b1,rad > values determined over 10–90% and +10–40% of the NCDF are significantly larger than the +values determined from the spectra. +The former rep- +resents an assessment of what the < b1,rad > value +when including both thick and thin LyC emitting re- +gions. This suggests that the thin components (originat- +ing from the bubbles) have such large < b1,rad > values +that they drag the overall average up considerably from +the values in the chromosphere. The fact that < b1,rad > +derived from the optically thick region is more consis- +tent with b1 derived from spectral fitting could be due +to the optically thick emission dominating the emergent +intensity. +Figure 10. +The ratio of Tc to the electron temperature +(Te), determined when b1 is at a minimum, for the 3F9, +1F10, 3F10, and 1F11 models, with δ=3–7, and Ec=20 keV +where available. Te was determined from RADYN for emis- +sion between 50–90% of the normalised contribution function +at 900.3 ˚A. Tc and b1 values were determined from fitting the +spectra between 700.0–911.0 ˚A. +In a similar manner, the average electron temperature, +< Te > from the LyC forming regions was calculated (re- +placing b1,rad for Te in Equation 5).As b1 tends towards +unity, Tc is expected to tend towards the electron tem- +perature, Te (Machado et al. 2018). Figure 10 shows +the ratio of Tc to < Te >, determined when b1 is at a +minimum for the 3F9, 1F10, 3F10, and 1F11 models, +with δ = 5–7, and Ec = 20 keV outside of the times of +the anomalous b1 values. < Te > was determined over +the optically thick layer (50–90% of the Ccdf) as the +< b1,rad > values over this emission range are in general +agreement with the b1 values obtained from the fits at +these times. From Figure 10, the ratio of Tc to < Te > + +E =20keV +c +1.3 +3F9 +83 +1F10 +3F10 +85 +1.2 +1F11 +86 米 +87 +1.1 +X +e +△ ++ +1.0 +米 +米 +—— +米 +0.9 +X +0.8 E +1.5 +-0.5 +2.0 +0.5 +0.0 +1.0 +log min b10 + from Eddington-Barbier +b. from CDF 10-90% +8 +b: from CDF 50-90% +b' from CDF 10-40% +b,=1 +b +2 +-2 +15 +20 +10 +0 +5 +Time (s)Formation of The Lyman Continuum During Solar Flares +13 +is clustered around 1 when b1 reaches a minimum. The +minimum b1 has a rather large range, extending from +b1 < 1 to b1 ∼ 100. That is, the ratio Tc :< Te > ap- +proaches unity at the minimum value of b1 but b1 itself +does not necessarily have a value of unity, somewhat con- +trary to our expectations. Machado et al. (2018) present +a similar finding where the 6 different X-class flares they +observed had varying b1 values (b1 < 1 to b1 ∼ 68). +However, as they used EVE Sun-as-a-star observations, +they converted EVE spectral irradiances to specific in- +tensities by assuming the flaring area. Machado et al. +(2018) assumed a fixed flaring area of 1018 cm−2, rep- +resenting the middle of a rather large range of reported +areas of X-class flares (see Section 4 of Machado et al. +2018). Varying this area would in effect shift the LyC +specific intensity up or down, while keeping the spectral +slope fixed. This would vary the b1 value but not Tc. +Figure 11. The ratio of Tc to electron temperature (Te) +as a function of b1 for all models, weighted by time. +Te +was determined from RADYN for emission between 50–90% +of the normalised contribution function at 850.0˚A. Tc and +b1 values were determined from fitting the spectra between +750.0–911.0˚A. +Outside of the times when b1 is at a minimum, the +ratio of Tc to Te was found to have a large range of val- +ues. Figure 11 shows the ratio of Tc to Te as a function +of b1 for all models, weighted by time. As seen in Fig- +ure 11, even as b1 approaches unity, the range of Tc/Te +extends from 0.1–1.5. This may be because < Te > is +determined using the NCDF of the contribution func- +tion. As < Te > is determined over a given emission +range, the heights considered will vary from model to +model, particularly for the more energetic beams where +the dynamics of the formation layers occur on shorter +time scales (see Figures 3 and 4). Therefore, the value +of < Te > is dependent on the height range considered. +Figure 12. Histogram of < Te > and < ne > in the forma- +tion region of λ = 850 ˚A, weighted by the average intensity +of the emission. +Finally, we determined < Te > and < ne > for +λ = 850 ˚A to illustrate the general range of plasma +properties where LyC forms. +Those are presented +as a 2D histogram in Figure 12 in which properties +from δ = [3, 4, 5], Ec = [15, 20, 25] keV, Fpeak += +[3F9, 1F10, 3F10, 1F11] erg s−1 cm−2 were collated. +Figure 12 shows that while there is some spread to high +temperatures (due mostly to contributions from dense, +optically thin bubbles) the bulk of the flare emission +forms in the range < Te >= 10–30 kK, with electron +densities spanning < ne >= 1011−14 cm−3. The data +has been weighted by the average intensity of emission, +and indicates that higher intensity is generally associ- +ated with higher < ne >, though temperature also plays +a role. +5. DISCUSSION +5.1. Percentage of energy radiated away by the LyC +To determine the percentage of energy radiated by +the LyC compared to the total energy injected via +non-thermal electrons, the LyC lightcurves in Figure 2 +were integrated over time and divided by the total +non-thermal electron energy. We found the LyC radi- +ated away between 1–3% of the total non-thermal elec- +tron energy injected. +The 3F9 models radiated away +around 1% of the total injected energy through the +LyC, whereas the 3F10 models radiated around 3%. +The overall distribution from a number of simualtions +is shown in Figure 13 +Milligan et al. (2014) provided a study of the global +energy budget in a strong flare, comparing the energy +radiated in the lower solar atmosphere at optical, UV, +and EUV wavelengths to the energy injected via non- +thermal electrons. They found the LyC radiated away + +Fit:入=[750.00-911.001A +Plasma Props: [0.50-0.90] of ^ref = 850.00A +17.5 +1.5 +15.0 +12.5 +1.0 +} +7.5 +Av +0.5 +5.0 +0.0 +2.5 +0 +2 +b1|fitPlasma Props: [0.05-0.95] of ^ref = 850.00A +107 +60 +2 sr-1 A-1J +106 +[kK] +105 +Intensity +40 +104 +103 +20 +[erg +102 +1011 +1012 +1013 +nelRADYN [cm-3]14 +McLaughlin et al. +Figure 13. +The ratio of energy radiated by LyC to the +energy injected by non-thermal electrons (NTE), as a func- +tion of the non-thermal electron energy. +Circles represent +δ = 3, plus symbol δ = 4, and diamonds δ = 5. The symbols +increase in size with increasing Ec. +approximately 1% of the total non-thermal electron en- +ergy. This is in agreement with our findings that predict +LyC to radiate away a few percent of the total non- +thermal electron energy. +5.2. Spikes in the Departure Coefficient and color +Temperature values +We mentioned several times previously the appearance +of strong spikes in b1 and Tc, caused by a flattening +towards the head of the LyC, as seen, for example, at t = +5.7 s in the left-hand panel of Figure 6. The Eddington- +Barbier approximation is clearly no longer valid during +these times. +This phenomenon can be understood from Figure 14, +which shows the temporal evolution of the ground state +of hydrogen level population for the 3F9 models, with +Ec = 20 keV, and δ = [3, 5]. The height of the peak +of the LyC contribution function is shown by the ver- +tical dashed lines at each time step. +At t = 0 s, n1 +is large in the chromosphere and decreases across the +transition region for both models. As the beam heat- +ing begins, n1 decreases within the chromosphere due +to excitation and ionization following the temperature +increase and non-thermal collisions. However, there is a +small region of plasma between the beam heating region +and the transition region that is only minimally heated +by the non-thermal electrons, resulting in n1 remaining +large compared to the adjacent plasma. At later times, +the optically thick layer shifts much deeper into the chro- +mosphere. The photons emitted from the optically thick +layer subsequently get absorbed by this region of plasma, +1000 +1200 +1400 +1600 +1800 +Height (Km) +6 +8 +10 +12 +14 +log n1 (cm−3) +3F9, δ=3, Ec=20keV +t=8s +t=7s +t=6s +t=5s +t=4s +t=3s +t=0s +6 +8 +10 +12 +14 +log n1 (cm−3) +3F9, δ=5, Ec=20keV +Figure 14. The temporal evolution of the ground state of +hydrogen level population for the 3F9 models, with Ec = +20 keV, and δ = [3, 5]. The height of the peak of the LyC +contribution function is shown by the vertical dashed lines +at each time step. +resulting in the flattening of the head of the LyC, and +the delayed enhancement of the LyC lightcurves. For the +δ = 3 model, the optically thick layer of the LyC forms +below the region of minimal heating for a few seconds +until the region dissipates, resulting in the extended in- +crease in b1 and Tc observed for harder beams (see δ=3– +4 curves in Figure 7). Whereas for softer beams (see +δ ≥5 curves in Figure 7), the optically thick layer only +forms below the region of minimal heating for a shorter +duration, resulting in the sudden steep spikes in b1 and +Tc. +In Figure 14, the n1 values for the harder beam (δ=3) +are reduced more uniformly and at higher altitudes +within the chromosphere despite harder beams being +composed of a greater number of high energy, deeply +penetrating, electrons. +This can be understood from +Figure 15, which shows the temperature and beam heat- +ing profile for the 3F9 models, with Ec=20keV, and + +10 +[%] +LyC Flux / NTE Flux [ +Ec[keV] +15 keV +20 keV +25 keV +6 +6=3 +10 +6=4 +6=5 +NTE Flux [erg cm-2jFormation of The Lyman Continuum During Solar Flares +15 +δ=3 (harder beam; blue lines) and 5 (softer beam; red +lines). The region of minimal heating is thicker for softer +beams (z ≈1720–1770 km). The δ=3 beam is harder +and therefore has a greater number of high-energy elec- +trons, which results in a faster atmospheric response and +evaporation of chromospheric plasma. +As the evapo- +ration front propagates the mass density of the upper +atmosphere is increased by approximately a few orders +of magnitude. The increased column depth means that +the higher energy electrons in the δ=3 beam are ther- +malised higher up in the atmosphere at later times in +the simulation despite the δ=5 beam being composed of +more low energy electrons, as can be seen by the blue +and red dashed lines in Figure 15. This results in the +narrower region of minimal heating (z ≈1790 km) seen +for the harder δ=3 case. The thickness of this region +also contributes to the varying profiles during the sud- +den increases (spikes) in b1 and Tc. +1700 +1720 +1740 +1760 +1780 +1800 +3.8 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +1700 +1720 +1740 +1760 +1780 +1800 +Height (km) +3.8 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +log Temp (K) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Normalised Beam heating +3F9,δ=3, Ec20keV @7s +3F9,δ=5, Ec20keV @7s +Beam heating 3F9, δ=3, 20keV @7s +Beam heating 3F9, δ=5, 20keV @7s +Figure 15. Temperature profiles for the 3F9 models, with +Ec = 20 keV, and δ = 3 and 5 at t=7 s. The normalised +beam heating is shown by the dashed lines for both models. +5.3. Wavelength dependency of b1 and Tc +Machado & Noyes (1978) hypothesised from observa- +tional evidence that a higher-lying region contributes +optically thin component to the LyC during flares along- +side the bulk of the chromospheric emission. Evidence +for this optically thin layer comes from a steepening +in the LyC spectrum’s gradient away from the contin- +uum head, resulting in increased Tc values determined at +shorter wavelengths. In the right-hand panel of Figure 6, +there is a local peak in ϵ at t=10s. Shorter wavelengths, +between λ ∼ 505.0 − 700.0˚A, are poorly fit compared to +longer wavelengths. This spectrum has two distinct gra- +dients, one at shorter wavelengths, λ ≤ 700 ˚A, and the +other at longer wavelengths, λ ≥ 700 ˚A, in agreement +with Machado & Noyes (1978), Machado et al. (2018), +and Druett & Zharkova (2019). +Figure 16. +Evolution of b1 and Tc for the 3F9, δ = 5, +Ec = 20 keV model. Equations 2 and 3 have been applied +between λ = [505−700] ˚A and λ = [700−911] ˚A. The values +are only shown between t = [7 − 30] s to omit times where +we know that spectral fitting produced poor results. +Figure 16 shows the b1 and Tc values for the 3F9, +δ = 5, Ec = 20 keV model, where the EB approxima- +tion has been applied between λ = 505 − 700 ˚A and +λ = 700−911 ˚A. Generally, the Tc values determined at +shorter wavelengths were a few thousand Kelvin hotter +than at longer wavelengths, in agreement with the litera- +ture (Machado & Noyes 1978; Machado et al. 2018). The +b1 values are also generally larger at shorter wavelengths. +This is due to the optically thin components of the LyC +that enhance the spectrum at shorter wavelengths. The +number of optically thin components that form depends +on the type of evaporation observed. Gentle evapora- +tion resulted in one upwardly propagating optically thin +layer forming. Whereas, explosive evaporation resulted +in two or three optically thin components correspond- +ing to evaporation and condensation fronts. +The up- +ward propagating optically thin components of the LyC +in Figure 4 are due to bubbles of chromospheric material +travelling immediately ahead of the evaporation front. +Reid et al. (2020) state that these bubbles are a source +of optically thin Ca II 8542˚A line emission and do not +always emit strongly in the Hα line, while Brown et al. +(2018) found solar bubbles to be among the dominant +sources of Lyman alpha line emission. Such propagating +high-density features could be indirectly detected via +LyC observations due to the effect they have on the +spectral shape, manifesting as an increase in both Tc + +250 +200 +b- 150 +100 +50 +1.6日 +1.5 +K +(x104 +T +二 +1.2 +505.0-700.0A +700.0-911.3A +12 +20 +10 +14 +16 +18 +8 +Time (s)16 +McLaughlin et al. +and b1 at shorter wavelengths. Machado et al. (2018) +only observed an increase in Tc at shorter wavelengths +for one of the six events that they analysed. This may be +due to the fact that the other five flares did not provide +the correct conditions for a solar bubble to form, or at +the time of observation the bubbles had dissipated. +6. CONCLUSIONS +Using the F-CHROMA grid of RADYN models, we +have shown that the LyC is greatly enhanced during +solar flares. The LyC spectral response is highly sensi- +tive to the flux of the non-thermal electron beam, but +is less dependent upon the spectral index or low-energy +cutoff. LyC was found to radiate away between 1–3% +of the total non-thermal electron energy injected. In- +creases in solar irradiance associated with solar flares +are known to drive dynamic and compositional changes +in Earth’s ionosphere, which can have adverse implica- +tions for modern technology on which society has be- +come dependent. The 850–1027˚A range, in particular, +is absorbed at an altitude of around 105–120km in the +ionosphere (E-layer), where it drives the partial dissoci- +ation of molecular oxygen (Robinson 1959). This part of +the spectrum is dominated by LyC, along with higher- +order Lyman emission lines. +Both optically thin and thick layers of the LyC were +found to form during solar flares, in agreement with the +literature (Machado & Noyes 1978; Machado et al. 2018; +Druett & Zharkova 2019). The optically thick layer is +formed in NLTE in the QS and forms at the top of +the chromosphere. During solar flares, this layer shifts +deeper into the solar chromosphere due to the evapora- +tion of the upper chromosphere, forming near the peak +beam heating region, with < Te >≈ 10–40 kK. It also +forms closer to LTE conditions. The optically thin com- +ponents of the LyC formed due to chromospheric evap- +oration, with the number of optically thin components +forming being dependent on the type of evaporation ob- +served, gentle or explosive. These optically thin com- +ponents cause an enhancement in intensities away from +the LyC head, resulting in increased b1 and Tc values +determined at shorter wavelengths. Fitting with the EB +relation we find that Tc ≈ 10–16 kK, with b1 at times +dropping to b1 ≈ few × 10, but which have a large scat- +ter. +Our results suggest that the LyC spectral response is +indicative of a chromospheric temperature and density +enhancement largely probing the chromosphere. +Gra- +dients in the derived Tc as a function of wavelength +can also indicate the presence of regions of propagat- +ing dense, cool material in the upper atmosphere. The +number of optically thin layers formed was found to be +greater for stronger solar flares, and the LyC contribu- +tion functions presented show that the optically thin lay- +ers form and dissipate over a shorter period for stronger +flares. Model-data discrepancies (e.g. the large scatter +of b1 values that don’t always approach unity, contrary +to Machado et al. 2018), could result from an exagger- +ated optically thin component to the LyC forming in +the dense bubbles, if those bubbles are denser than in +actual flares. For example, Graham et al. (2020) mod- +elled the ratio of the intensity of redshifted ‘satellite’ +components of the Fe II line to the intensity of the sta- +tionary component, noting that it was larger than the +observed flare they were simulating. This could be due +to the density in the modelled condensation (from which +the satellite component originated) being larger than the +condensation produced during the observed flare. Our +chromospheric bubbles may be similarly over-dense. +EVE currently provides LyC observations with the +greatest coverage.However, these are Sun-as-a-star ob- +servations with a cadence of 60 s (observations prior to +2014 had a 10 s cadence, but instrument degradation +has forced a longer exposure time). +Therefore, EVE +LyC flare observations will observe a range of flaring +loops at various heating or cooling stages. Thus, some +of the dynamic features may become smeared tempo- +rally, especially for stronger flares. As the breakdown in +the EB approximation and delayed enhancement in the +LyC lightcurves also occur over second to sub-second +timescales, it is unlikely that EVE can observe these +phenomena. +Additionally, as EVE provides Sun-as-a- +star observations, EVE spectral irradiances can be con- +verted to specific intensities by assuming the flaring area +as discussed in Section 4.2. Varying this area shifts the +LyC specific intensity up or down, while keeping the +spectral slope fixed. This would vary the b1 value but +not Tc. +SPICE on board the Solar Orbiter mission that was +launched in 2020, provides EUV coverage in the wave- +length ranges of λ = 704–790 ˚A and λ = 973–1049 ˚A. +This provides partial coverage of LyC. It remains to be +seen to what extent it is possible to extract continuum +intensities, which will depend in part on how well we can +resolve spectral lines with that passband, but SPICE +observations may be used to determine b1 and Tc val- +ues below the head the LyC. Unlike SDO/EVE, these +observations would have spatial resolution and obtain +higher cadences. However, as SPICE only provides par- +tial coverage of LyC, the derived b1 and Tc values may be +elevated due to the presence of the optically thin LyC +layers, enhancing the LyC spectrum at shorter wave- +lengths. To determine if this is the case, b1 and Tc val- +ues should be derived and compared from SPICE and +EVE spectra over multiple wavelength ranges for any +flares both instruments capture. Our analysis paves the +way for an interpretation of solar flare LyC observations +taken by current and future missions. + +Formation of The Lyman Continuum During Solar Flares +17 +We thank the anonymous referee for a careful review +of our manuscript, and for useful comments and sug- +gestions. +SAM would like to thank the Science and +Technology Facilities Council (UK) for the award of +a PGR studentship. AJM acknowledges funding from +the Science Technology Funding Council (STFC) Grant +Code ST/T506369/1. +ROM and GSK would like to +acknowledge support from NASA Heliophysics Sup- +porting Research grant NNH19ZDA001N and ROM +thanks the Science and Technologies Facilities Council +(UK) for the award of an Ernest Rutherford Fellowship +(ST/N004981/2). 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A. 1993, SoPh, 143, +259, doi: 10.1007/BF00646487 + diff --git a/sdAzT4oBgHgl3EQfrv1r/content/tmp_files/load_file.txt b/sdAzT4oBgHgl3EQfrv1r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54a68ebb95c6b595679f9ae688267582c58de4fc --- /dev/null +++ b/sdAzT4oBgHgl3EQfrv1r/content/tmp_files/load_file.txt @@ -0,0 +1,1218 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf,len=1217 +page_content='Draft version January 5, 2023 Typeset using LATEX twocolumn style in AASTeX631 Formation Of The Lyman Continuum During Solar Flares Shaun A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' McLaughlin,1 Ryan O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Milligan,1, 2 Graham S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kerr,2, 3 Aaron J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Monson,1 Paulo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Sim˜oes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 5 and Mihalis Mathioudakis1 1Queen’s University Belfast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' University Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Belfast BT7 1NN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Northern Ireland 2Department of Physics Catholic University of America,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 620 Michigan Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Northeast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' DC 20064,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' USA 3Solar Physics Laboratory (Code 671),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Heliophysics Science Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' NASA Goddard Space Flight Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Greenbelt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' MD 20771,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' USA 4Center for Radio Astronomy and Astrophysics Mackenzie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' School of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Mackenzie Presbyterian University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' S˜ao Paulo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Brazil 5SUPA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' School of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' University of Glasgow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Glasgow G12 8QQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' UK ABSTRACT The Lyman Continuum (LyC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' < 911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='12˚A) forms at the top of the chromosphere in the quiet-Sun, making LyC a powerful tool for probing the chromospheric plasma during solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' To understand the effects of non-thermal energy deposition in the chromosphere during flares, we analysed LyC profiles from a grid of field-aligned radiative hydrodynamic models generated using the RADYN code as part of the F-CHROMA project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The spectral response of LyC, the temporal evolution of the departure coefficient of hydrogen, b1, and the color temperature, Tc, in response to a range of non-thermal electron distribution functions, were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The LyC intensity was seen to increase by 4-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 orders of magnitude during solar flares, responding most strongly to the non-thermal electron flux of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Generally, b1 decreased from 102–103 to closer to unity during solar flares, indicating a stronger coupling to local conditions, while Tc increased from 8–9 kK to 10–16 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Tc was found to be approximately equal to the electron temperature of the plasma when b1 was at a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Both optically thick and optically thin components of LyC were found in agreement with the interpretation of recent observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The optically thick layer forms deeper in the chromosphere during a flare compared to quiescent periods, whereas the optically thin layers form at higher altitudes due to chromospheric evaporation, in low-temperature, high-density regions propagating upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' We put these results in the context of current and future missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' INTRODUCTION Solar flares are the most energetic events to occur in the solar system, releasing up to 1032 erg of energy over the course of tens of minutes (Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This energy is typically thought to be released via mag- netic reconnection in the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Following release, this energy manifests itself in several ways, including in the acceleration of large quantities of charged particles that propagate along magnetic field lines, deeper into the Sun’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' These charged particles undergo Coulomb collisions with the ambient plasma depositing energy in the solar chromosphere, where intense heat- ing, ionization, and mass flows result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The subsequent increase in radiation allows us to shed light on the physi- cal mechanisms that release and transport energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Mod- elling by Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2005) suggests that solar flare emission is energetically dominated by the Balmer con- tinuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, Balmer continuum observations have been severely lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Lyman Continuum flare obser- vations have been more readily available in recent years from the Extreme Ultraviolet Experiment (EVE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Woods et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2012) on board the Solar Dynamic Observatory (SDO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Pesnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2012), with the potential of upcom- ing flare observations from the Spectral Imaging of the Coronal Environment (SPICE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Spice Consortium et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2020) instrument on board the Solar Orbiter (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2020), and the EUV High-Throughput Spectro- scopic Telescope (EUVST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Shimizu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2019) on board Solar-C (Watanabe 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The Lyman Continuum (LyC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' <911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='12˚A) results from the free-bound transition of a free electron to the ground state of an ambient hydrogen nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In the quiet-Sun, LyC forms at the top of the chromosphere/base of the transition region where the temperature rises from the cooler, denser chromosphere (T ≤ 104K) to the corona, (T ≥ 106K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Vernazza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 1973, 1976, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Avrett & Loeser 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Therefore, the LyC is sensitive to energy perturbations in the chromo- sphere induced during solar flares, and since thermaliza- tion occurs very rapidly at the higher densities located here, its spectrum may reflect the local plasma tempera- ture (Noyes & Kalkofen 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The LyC is a potentially powerful diagnostic tool of the chromospheric response to flare energy injection, but this potential is presently largely untapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' One of the earliest investigations into LyC formation during solar flares was performed by Machado & Noyes (1978), who observed nine flares using the extreme- ultraviolet (EUV) spectrograph on the Apollo Telescope arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01648v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='SR] 4 Jan 2023 2 McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Mount onboard Skylab (Ise & Cagle 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They found that the non-local thermodynamic equilibrium (NLTE) departure coefficient of the first level of hydrogen (b1) tended towards unity during flares, more so than in ac- tive regions or the quiet-Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The departure coefficient is given by n1/n∗ 1, where n∗ 1 is the ground state LTE pop- ulation and n1 is the NLTE ground state populations, meaning the LyC formation region was driven closer to LTE conditions during the solar flares (Menzel & Cilli´e 1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They also found that the color temperature (Tc) derived from the head of the continuum was around Tc ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='9–9 kK, comparable to that of the surrounding quiet-Sun, but hotter than active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They con- cluded that in flares LyC is optically thick and forms close to LTE in a region of higher density deeper in the chromosphere, whereas in active regions or the quiet- Sun b1 ≈ 102 (Noyes & Kalkofen 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, they also showed that Tc measured at shorter wavelengths (700–790 ˚A) revealed higher temperatures (10–15 kK, although with large uncertainties), suggesting the pres- ence of an optically thin LyC layer that formed higher in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' One limitation of this study was that several of the observations were carried out in spectral scanning mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' spectra were obtained in a 5 arcsec2 area over 300–1335 ˚A, over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8 minutes, meaning that different parts of the spectrum may have been sampled at different times and may not reflect true changes due to the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' More recently, Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018) performed a study using Sun-as-a-star observations from EVE on board SDO which allowed the temporal evolution of LyC emission during six major solar flares to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' After converting the EVE data from spectral irradiance to specific intensity, by assuming flaring areas, they re- ported that the LyC intensities at the head of the con- tinuum were enhanced by 3–4 orders of magnitude rela- tive to preflare values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Their results supported the ear- lier conclusions of Machado & Noyes (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018) found a larger color temperature, noting that Tc determined between 870–912 ˚A increased from Tc = 8–9 kK to Tc ≈ 9–12 kK, perhaps due to the study of more energetic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Milligan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2012) reported an enhancement in LyC emission during an X-class flare that occurred on 2011 February 15 also using data from SDO/EVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' A follow- up study showed that the total energy radiated by LyC was a few percent of the total non-thermal electron en- ergy, amounting to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8×1029 erg over the course of the flare, comparable to the radiated soft X-ray energy (Mil- ligan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' LyC irradiance increased by approxi- mately a factor of 10 during the impulsive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Lemaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2004) presented observations of an X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3 solar flare from the head of the LyC obtained from scattered light detected by the Solar Ultravio- let Measurements of Emitted Radiation spectrometer (SUMER;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Wilhelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 1995) on board the Solar & Heliospheric Observatory (SOHO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Domingo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 1995) mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They determined the local increase of LyC radiance to be a factor of several thousand, in agree- ment with Machado & Noyes (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Using the ratio of the spectral radiance at 910˚A and 890˚A they estimated Tc = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='15 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' There have been several attempts at modelling LyC during solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ding & Schleicher (1997) calculated LyC intensities in response to a precipitating beam of non-thermal particles using a non-LTE, radiative trans- fer code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They found that LyC is very sensitive to the incoming flux of electrons, and they predicted that the temperature at the formation height of LyC will increase in conjunction with a downward shift of the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In particular, they noted that non- thermal collisions were responsible for driving down b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In their experiments, Ding & Schleicher (1997) manu- ally modified semi-empirical model atmospheres to in- vestigate the impact of the chromospheric temperature gradients, and transition region depth on the LyC for- mation, in a manner that was not self-consistent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' they did not simulate the time-dependent chromospheric radiation-hydrodynamic response to energy injection by a beam of non-thermal electrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For the purposes of including non-thermal collisional ionization of hy- drogen, they assumed non-thermal electron fluxes of F = [1010, 1011, 1012] erg cm−2 s−1, with spectral indices δ = [3, 4, 5], and a fixed low-energy cutoff Ec = 20 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They found that LyC intensities increased by 1–3 or- ders of magnitude compared to quiet-Sun values, and that the LyC intensities were highly sensitive to the flux of the non-thermal electron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The LyC in- tensities were less sensitive to the spectral index of the non-thermal electron distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' More recent modelling of LyC emission was performed by Druett & Zharkova (2019) using the radiation trans- port and hydrodynamic code, HYDRO2GEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They in- vestigated the spectral response of LyC to various elec- tron beam fluxes during the impulsive and gradual phases of solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They found that the LyC intensi- ties increased by 2 orders of magnitude from their F10 model to their F12 model, highlighting the sensitivity of the LyC intensities to the flux of the non-thermal electron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' From continuum contribution functions, they determined that LyC forms under optically thick conditions at the top of the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' During the impulsive phase, they noted a flattening of the LyC spec- tra but found that the gradient steepened further from the head of the continuum, in agreement with Machado & Noyes (1978) and Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' During the gradual phase, however, they observed a flattening of the LyC spectra at shorter wavelengths compared to the im- pulsive phase spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They stated that the flattening below 700˚A is due to emission from high-altitude regions that cool sufficiently to allow recombinations to occur, resulting in an enhanced intensity longward of ∼ 700 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In this paper, we build upon these prior modelling ef- forts, by investigating the plasma properties (namely Tc Formation of The Lyman Continuum During Solar Flares 3 and b1) in the region of LyC formation, and the diagnos- tic potential of the LyC in flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' We present a detailed analysis of LyC emission from a large parameter space of solar flare models using the Flare CHRomospheres: Observations, Models and Archives (F-CHROMA1) grid of models generated by the 1D field-aligned radiative- hydrodynamic code RADYN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' NUMERICAL SIMULATIONS RADYN is a 1D field-aligned radiative hydrodynamic code that solves the coupled time-dependent equa- tions of hydrodynamics, NLTE radiative transfer, non- equilibrium atomic level populations, and charge conser- vation (Carlsson & Stein 1992, 1995, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The code’s adaptive grid (Dorfi & Drury 1987) allows RADYN to resolve steep gradients or shocks that easily form in flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' RADYN solves the NLTE radiation transfer for a 6-level hydrogen atom, 9-level helium atom, and a 6-level Ca II ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Important for modelling the LyC, RADYN models the non-equilibrium populations, and for hydrogen and he- lium includes non-thermal collisional rates between the injected non-thermal electron distribution and ambient plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For helium, we use the expressions from Arnaud & Rothenflug (1985) to model non-thermal collisional ionisation of He I→He II, and from He II→He III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For hydrogen, we include non-thermal collisional ionisation from the ground state, and excitation from the ground state to n = [2, 3, 4], following the approach of Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In this approach, we are able to include the important effects of secondary collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As dis- cussed in several prior studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ricchiazzi & Canfield 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Karlick´y et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kaˇsparov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2009), non-thermal collisional ionisation from the excited states are of lesser importance, in part because the population of the excited states are orders of magni- tude smaller than the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Thus the ionisation stratification is unlikely to be drastically altered by the omission of non-thermal collisions from the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Note, however, that some models have included non- thermal effects from the excited states (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Zharkova & Kobylinskii 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Druett & Zharkova 2018) though it is unclear just how much of a role those collisions played compared to those from the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Fig- ure 3 of Zharkova & Kobylinskii (1993) does show that non-thermal collisions from n = 2 became important relative to the thermal collisional rates, but only deep in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For the reasons that we discussed above, and since we are focused on the LyC which forms in the upper chromosphere/lower transition region, we believe that we are justified in using only ground state non-thermal collisional rates including secondary effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' RADYN is a well-established resource that has been extensively used to model the response of the solar at- 1 https://star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='pst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='qub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='uk/wiki/public/solarmodels/start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='html mosphere to flare energy injection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Abbett & Haw- ley 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kowal- ski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2015, 2017, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kowalski & Allred 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Rubio da Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2016, 2019, 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Sim˜oes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' When simulating flares driven by electron beams (the ‘standard model’) the non-thermal electron distribution is modelled as a power law characterised by the spectral index, δ, the low-energy cutoff, Ec, and energy flux density, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The transport and thermalization of these non-thermal elec- trons are modelled by solving the Fokker-Planck equa- tions, including various diffusion terms (Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Note that we solve the Coulomb operator using general forms of the Rosenbluth potentials and so no as- sumption is made as to the target temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' that is, there is no need to make a cold or warm target approx- imation (see also Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Beam parameters can be obtained through backward modelling of hard X- ray observations taken with space-borne satellites such as the Ramaty High Energy Solar Spectroscopic Im- ager (RHESSI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2002) or the Fermi Gamma-ray Space Telescope (Meegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2009), usually under the assumption of thick target collisions (Brown 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Hol- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Krucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2007, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In our study, we do not model a specific flare, but instead survey a large parameter space of δ, Ec & F values, with the values of those parameters guided by typical observed ranges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Holman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The F-CHROMA grid of RADYN flare models, hosted by Queen’s University Belfast, consists of 72 RADYN simulations (Carlsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This grid includes δ = [3, 4, 5, 6, 7, 8], and Ec = [10, 15, 20, 25] keV, and models various magnitudes of injected non-thermal elec- tron flux as a 20 s triangular profile in time, peaking at t = 10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The time-integrated energy flux ranges Ftot = [3×1010, 1×1011, 3×1011, 1×1012] erg cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' It is typical in the flare community to refer to the energy flux den- sity, and so for the remainder of this paper we quote the peak injected flux (at t = 10 s in each simulation), which are F = [3×109, 1×1010, 3×1010, 1×1011] erg cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For brevity, we refer to these simulations as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=', 1F11, equating to 1 × 1011 erg cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In this study, we have selected 25 of the 72 models (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Not every combination of δ and Ec were modelled for inclusion in this grid for the highest en- ergy flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The models omitted from our subset are the 1F11, Ec=20keV, δ = [6, 7], models and the 1F11, δ = 5, Ec=15keV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' LYC RESPONSE TO FLARE ENERGY INJECTION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Synthetic LyC spectra Figure 1 shows continuum spectra at the time of the peak LyC emission in each simulation, between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='7 s and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The He I edge at 503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='98˚A is also visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 4 McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 400 600 800 1000 Wavelength (Å) 0 2 4 6 8 log Intensity (erg cm−2 s−1 sr−1 Å−1) 3F9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ec=20keV pre−flare δ= 3 δ= 4 δ= 5 δ= 6 δ= 7 400 600 800 1000 Wavelength (Å) 1F10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ec=20keV 400 600 800 1000 Wavelength (Å) 3F10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ec=20keV 400 600 800 1000 Wavelength (Å) 1F11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ec=20keV 3F9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' δ= 5 400 600 800 1000 Wavelength (Å) 0 2 4 6 8 log Intensity (erg cm−2 s−1 sr−1 Å−1) pre−flare Ec= 15keV Ec= 20keV Ec= 25keV 1F10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' δ= 5 400 600 800 1000 Wavelength (Å) 3F10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' δ= 5 400 600 800 1000 Wavelength (Å) 1F11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' δ= 5 400 600 800 1000 Wavelength (Å) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Synthetic RADYN continua spectra from 400–1000˚A showing the head of the He I continuum (<503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='98˚A), LyC (<911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='12˚A), and the tail of the Ca II continuum (<1044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The top row shows spectra for a fixed Ec = 20 keV, with peak beam fluxes of 3F9, 1F10, 3F10, and 1F11, all shown at the peak in the LyC spectrum (between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='7 s–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6 s for all models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The black curve denotes the preflare spectra, while the colored curves show the spectra for spectral indices of δ=3–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The bottom row shows the spectra but for a fixed δ=5, varying Ec=15, 20, and 25 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Note the 3F10, δ=5, Ec = 20 keV and the 1F10, δ=5, Ec = 20 keV models have transient negative intensities in the tail of the LyC continuum at the time of the peak LyC emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is due to numerical noise in the simulation at these times, which only lasts a very short amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Therefore, the spectra shown for these models have been shifted by 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' F (erg cm−2 s−1) δ Ec(kev) 3 × 109 3, 4, 5, 6, 7 15, 20, 25 1 × 1010 3, 4, 5, 6, 7 15, 20, 25 3 × 1010 3, 4, 5, 6, 7 15, 20, 25 1 × 1011 3, 4, 5 20, 25 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Non-thermal electron beam parameters used for generating the RADYN models analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The top row shows spectra generated for a fixed low- energy cutoff of 20 keV, while increasing the beam flux from 3F9 to 1F11 (left to right), and varying the spec- tral indices from δ = 3–7 (colored lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The pre-flare spectra are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' There were some snapshots (< 1 s) in a few simulations in which the EUV con- tinua tails had negative intensities, likely due to numer- ical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The simulations where the LyC was affected are: the 3F10, δ=5, Ec = 20 keV model and the 1F10, δ=5, Ec = 20 keV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Therefore, the spectra shown for these models have been shifted by 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The spec- tra for the models where only the Ca II continuum was affected were not shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As the electron beam intensi- fies, so too does the response of LyC, peaking roughly co-temporally with the peak of the heating rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The 1F11, δ = 4, 20keV model is an exception, peaking at a time of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6s as will be discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For the 3F9 models, the head of the continuum is around four orders of magnitude greater than the pre-flare value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This increases to around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 orders of magnitude for the 1F11 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The magnitude of the LyC enhancement during solar flares is strongly dependent on the flux of non-thermal electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The dependence on the spec- tral index of the beam is weaker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' softer beams result in slightly higher intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is because softer beams (larger δ) deposit their energy over a narrower region in the upper chromosphere since they thermalize more easily, resulting in more localized heating and electron density enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The bottom row of Figure 1 shows synthetic LyC spectra for a fixed spectral index of δ = 5 and Ec = [15, 20, 25] keV (colored lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The LyC spectra are weakly dependent on the low-energy cutoff, similar to the behaviour seen with varying the spectral in- dex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Beams with larger Ec values are composed of a greater proportion of high-energy electrons that pene- trate deeper into the atmosphere, resulting in the beam heating region shifting downwards, affecting the LyC re- sponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The increase in intensity from varying the spec- tral index and low-energy cutoff is much weaker com- Formation of The Lyman Continuum During Solar Flares 5 pared with that caused by the increasing flux of the non-thermal electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Like Ding & Schleicher (1997), we found a strong dependence of the LyC spectral intensity upon the flux of the non-thermal electron beam, as seen in Fig- ures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, RADYN predicts larger impulsive phase LyC intensities than the modelling of Ding & Schleicher (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ding & Schleicher (1997) also pre- dicted a stronger dependence on the spectral index than RADYN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' These differences may arise for a number of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Chiefly, Ding & Schleicher (1997) used a static code with semi-empirical atmospheres and manually varied the temperature and transition region location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The electron beam was only used to add non-thermal collisions, and did not drive the chromospheric response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In our dynamic simulations, while these non-thermal collisions certainly play an important role, their impact on driving the overall dynamics is taken into account, so that there is a feedback between the atmosphere and the beam propagation/thermalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Recent modelling of LyC emission during solar flares was also performed by Druett & Zharkova (2019) using the radiative hydrodynamic code, HYDRO2GEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They present LyC spectra for their 1F10, 1F11, and 1F12 solar flare models with Ec = 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The magnitude of our impulsive phase LyC intensities are in agreement with their spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, the F-CHROMA grid does not include any 1F12 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Synthetic LyC lightcurves The flare excess integrated LyC lightcurves are shown in Figure 2 for the 3F9, 1F10, 3F10, and 1F11 models, with Ec = 20 keV, and spectral indices ranging from δ = 3–7 (colored lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Note the saw-toothing seen for some of the models is due to numerical noise in the spec- tra between time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The synthetic lightcurves again highlight that the magnitude of the LyC enhancement is primarily driven by the magnitude of the non-thermal electron flux, whereas variations in the spectral index are less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Softer beams result in slightly higher LyC fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is because softer beams (larger δ) de- posit their energy over a narrower region in the upper chromosphere, resulting in more localized heating close to the LyC formation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' There is a period of time near the beam onset where the LyC emission is barely enhanced (recall that the flux of the non-thermal elec- tron distribution ramps up in a triangular profile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The duration of this period is dependent upon the flux of the non-thermal electron beam, with larger fluxes having a shorter delay in the enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' All models consis- tently showed that notable LyC enhancement occurred after approximately 3 × 109–4 × 109 erg cm−2 of energy was deposited by the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Flares that inject larger non-thermal electron fluxes more rapidly produce con- ditions conducive to LyC ionizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Despite this variable onset time, each LyC flare reaches a maximum around t ∼ 10 s and falls thereafter when the energy flux of the electrons decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The LyC response largely tracks the beam heating profile, in agreement with Druett & Zharkova (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, the 1F11, δ = 4, Ec = 20 keV model shows a secondary peak in the LyC lightcurve at t = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This second maximum occurs as a chromospheric ‘bubble’ (that is, a region of dense, low-temperature material, appearing at high altitude) dissipates at that time, resulting in a decrease in the optical depth (Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' At the same time, a condensation front producing optically thin LyC emission merges with the heated chromosphere producing the optically thick LyC layer, resulting in a sudden, secondary increase in the LyC intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The optically thin LyC layers are discussed in more detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The 1F11, δ = 3, Ec = 20 keV and 3F10, Ec = 20 keV, δ = 6, 7 models also show a secondary increase in LyC emission after the peak beam heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is caused by the merging of optically thin condensa- tion fronts with the optically thick LyC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' How- ever, for these models, the chromospheric bubbles do not dissipate before this merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For the 1F11, δ = 5, Ec = 20 keV model the merging of the two layers occurs before the peak beam heating, resulting in the peak seen between 9–10 s in Figure 2 (the formation layers for this model are discussed in further detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' All this is to say that features in the lightcurves, while largely following the profiles of energy injection, are in- fluenced by the complex dynamics of the flaring chro- mosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' LyC Formation Properties 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Formation Heights and Optical Depth The contribution function to the emergent intensity ef- fectively shows where in the solar atmosphere the emis- sion originates (for brevity we refer to this simply as the ‘contribution function’ going forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The formal solution of the emergent intensity can be expressed as: Iλ = 1 µ � z CI(z)dz = � z jλ exp (τλ/µ)dz (1) where the integrand is the contribution function, CI(z) (Magain 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Carlsson & Stein 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Following Kowalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2017) we express CI(z) in terms of emis- sivity, jλ, attenuated by some optical depth τλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The emissivity and opacities are the sum of various sources, including hydrogen bound-free, hydrogen free-free, H−, scattering by metals, Rayleigh scattering, and Thom- son scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Of these, the bound-free transitions are treated fully in NLTE, whereas the others are treated in LTE, but using the NLTE, non-equilibrium electron or proton densities as appropriate (See Kowalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2017) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Generally, if the emission originates close to the τν = 1 surface then the emission forms under optically thick conditions, and if it originates in a region where τν << 1 6 McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 0 5 10 15 20 Time (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='4 Flux (x109 erg cm−2 s−1) 3F9, Ec=20keV δ= 3 δ= 4 δ= 5 δ= 6 δ= 7 0 5 10 15 20 Time (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='4 1F10, Ec=20keV 0 5 10 15 20 Time (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 3F10, Ec=20keV 0 5 10 15 20 Time (s) 0 2 4 6 8 1F11, Ec=20keV Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Flare excess lightcurves of spectrally integrated LyC emission from RADYN (505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0˚A to 911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='7˚A) for the 3F9, 1F10, 3F10, and 1F11 models with δ=3–7 and Ec=20keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 1000 1500 2000 Height (km) 1000 1500 2000 Height (km) τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01 τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 τ=1 τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01 τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 τ=1 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 log Contribution function (erg cm−3 s−1 sr−1 Å−1) t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 3 4 5 6 7 log Temp (K) t=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 3 4 5 6 7 log Temp (K) t=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 3 4 5 6 7 log Temp (K) 500 600 700 800 900 Wavelength (Å) 1000 1500 2000 Height (km) 500 600 700 800 900 Wavelength (Å) 1000 1500 2000 Height (km) t=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 500 600 700 800 900 Wavelength (Å) 500 600 700 800 900 Wavelength (Å) t=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 500 600 700 800 900 Wavelength (Å) 500 600 700 800 900 Wavelength (Å) t=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The log of the contribution function for the 3F9, δ = 5, Ec = 20 keV model at times t = [0, 8, 10, 14, 18, 20] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The heights at which τ=1 (grey solid line), τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 (grey dashed line), and τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01 (grey dot-dashed line) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The temperature profile is shown in red at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' then it can be considered to be optically thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' It is possi- ble that some flares may exhibit both optically thick and thin components with significant emission coming from multiple layers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' from an optically thin layer overly- ing the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Knowledge of the plasma proper- ties in the LyC forming region(s), in response to differ- ent heating conditions, will broaden our understanding of energy transport processes during flares, and where this energy is deposited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' To illustrate the general formation properties we dis- cuss detailed examples from two simulations: the 3F9, δ = 5, Ec = 20 keV and the 1F11, δ = 5, Ec = 20 keV Formation of The Lyman Continuum During Solar Flares 7 800 1000 1200 1400 1600 1800 Height (km) 800 1000 1200 1400 1600 1800 Height (km) τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01 τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 τ=1 τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01 τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 τ=1 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 log Contribution function (erg cm−3 s−1 sr−1 Å−1) t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 3 4 5 6 7 log Temp (K) t=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 3 4 5 6 7 log Temp (K) t=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 3 4 5 6 7 log Temp (K) 500 600 700 800 900 Wavelength (Å) 800 1000 1200 1400 1600 1800 Height (km) 500 600 700 800 900 Wavelength (Å) 800 1000 1200 1400 1600 1800 Height (km) t=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8s 500 600 700 800 900 Wavelength (Å) 500 600 700 800 900 Wavelength (Å) t=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s 500 600 700 800 900 Wavelength (Å) 500 600 700 800 900 Wavelength (Å) t=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Same as Figure 3, but for the 1F11, δ = 5, Ec = 20 keV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Note the color bar scale has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' These showcase a weaker and stronger flare, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 3 shows the temporal evolution of the log of the contribution function for the 3F9, δ = 5, Ec = 20 keV model during the heating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The tem- perature stratification (solid red line) and the heights at which τλ = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1, 1] are also indicated (grey dot- dashed, dashed, and solid lines, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Initially, at t = 0 s, there is a peak in the contribution function at the base of the transition region (z ∼ 1750 km), form- ing between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 ≲ τλ ≲ 1, corresponding to an optically thick LyC formation, in agreement with the literature (Machado & Noyes 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Druett & Zharkova 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' By t = 8 s, the formation region has widened, and shifted deeper into the chromosphere (z ≈ 1200 − 1400 km) as the beam heating strength- ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The height difference between τλ = 1 for short and long wavelengths has also narrowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' At this time, opacity effects are still significant, with the bulk of the emission forming between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 ≲ τλ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' A secondary peak in the contribution function is present higher in altitude (z = 1700 km), co-spatial with a chromospheric ‘bubble’, seen as a narrow dip in the temperature pro- file forming at the base of the transition region at t = 8, 10, 14, 18, and 20 s in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Emission from this very narrow layer forms under optically thin conditions τλ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Similar regions of cooler, yet dense plasma, are present in most of our simulations (see also Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Druett & Zharkova (2019) also presented evidence for an overlying optically thin layer in their 1F11, Ec=10keV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As time pro- gresses, this chromospheric ‘bubble’ propagates upwards towards the corona along a chromospheric ablation (also referred to as ‘evaporation’) front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The bubble collapses as it propagates, narrowing until all of the cool material has been heated to coronal temperatures (or higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In the declining phase of the heating (t > 10 s) the τλ = 1 layer begins to gradually return to the pre-flare height as the atmosphere cools and the number of recombina- tions in low opacity regions increases whilst ionization decreases, in agreement with (Druett & Zharkova 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The τλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 layer begins shifting to the location of the bubble as the density there increases, raising the LyC opacity at those altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 4 shows the temporal evolution of the log of the LyC contribution function for the 1F11, δ = 5, Ec = 20 keV model during the heating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In this more energetic simulation, the LyC forms over much more narrow regions (z ≈ 1000–1100 km) due to the more dramatic response of the chromosphere that be- comes very compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' By t = 8 s, two optically thin 8 McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' layers have formed due to bubbles flanking a rapidly ex- panding high-temperature region (one upflowing bubble due to explosive evaporation at z ≈ 1450 km, and one downflowing bubble at z ≈ 1250 km, commonly referred to as a chromospheric condensation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' When the very dense downflowing bubble effectively merges with the bulk of the chromosphere the density at temperatures favorable to produce LyC can rapidly increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Since opacity is a strong function of wavelength there can be times at which there is a marked optical depth strati- fication, resulting in variations to the spectral shape of the LyC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' flattening at certain wavelengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ex- amples are shown at t = [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8, 11] s, where the τλ = 1 layer forms at a higher altitude for the head of the con- tinuum compared to the tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In this stronger flare, the chromosphere becomes so compressed that the LyC ul- timately forms from a vanishingly narrow downflowing layer in the latter stages of the flare, shown at t = 15 s of Figure 4 at z ≈ 1000 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' These formation properties are generally true of the other simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Most simulations contain some com- bination of optically thick and overlying optically thin contributions to the emergent spectra (though the bulk of the emission does tend to originate between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 ≲ τλ ≲ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Stronger flares produce more dramatic atmo- spheric responses, leading to the bubbles and shocks, and associated phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' These dynamics also emerge and develop faster with increasing flare strength, and for softer non-thermal electron distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is because these electrons heat the uppermost chromo- sphere very efficiently and do not penetrate as deeply, driving flows more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Weaker flares and harder spec- tra do not tend to produce as many shocks, do not signif- icantly compress the chromosphere, and exhibit mainly upflowing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Emissivity and Opacity To understand some of the features described we can briefly look at the components of opacity and emissivity as functions of space, time, and wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' We illus- trate the components of emissivity and opacity at two times from one flare for a long (λ = 900 ˚A) and short (700 ˚A) wavelengths in Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The dominant source of opacity at all wavelengths, throughout the chromo- sphere and through the lower transition region was hy- drogen free-bound opacity, χλ,Hbf (red dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' To- wards the upper transition region and corona, where hydrogen is ionized, Thomson scattering begins to dom- inate but the overall effect on the emerging radiation is small, compared to the free-bound opacity (pink dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Once the flare starts in earnest, we saw that the for- mation height of the LyC dropped in altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is because the upper chromosphere was strongly heated, ionizing hydrogen and decreasing χλ,Hbf , meaning that the τλ = 1 forms much deeper in the atmosphere, shown in the top panels of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Later, the merging of the dense bubbles with the bulk of the chromosphere means that χλ,Hbf increases, but since this is a function of wavelength, this happens first for longer wavelengths during this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The emissivity of the LyC is also dominated by hy- drogen recombinations, jλ,Hbf at longer wavelengths (roughly speaking λ > 700 ˚A), but at shorter wave- lengths Thomson scattering can be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Thom- son scattering can also compete with jλ,Hbf at longer wavelengths at certain times, but once the electron den- sity is high (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' in the narrow chromospheric bubbles) then jλ,Hbf once again dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The more extended region of flaring LyC formation compared to the quiet Sun can be understood from the stratification of emis- sivity and opacity also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' During a flare, a greater ex- tent of the chromosphere is at an elevated temperature, and hence electron density, so the free-bound emissivity throughout the greater geometric height range is raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Integrating over height yields a higher emergent inten- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In Figures 5, note the initial decrease in opacity (top row t=2s panels for 700˚A and 900˚A), pushing LyC for- mation deeper, and then the influence of the bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The emissivity is largest at the optically thick layer (be- tween τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1–1 heights) as the density of emitting par- ticles is greatest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The optically thin bubbles also have an increased emissivity compared to the ambient plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' At shorter wavelengths the emissivity of the optically thicker layer is comparable to the emissivity of the optically thin bubbles, meaning the optically thin layers will reinforce the lower wavelength LyC intensi- ties to a greater extent than the continuum head, in agreement with the literature (Machado & Noyes 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Druett & Zharkova 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' LYC COLOR TEMPERATURE AND DEPARTURE FROM LTE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Fitting the LyC spectra Following the method used by Machado & Noyes (1978) and Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018), the synthetic LyC spectra were fit using the Eddington-Barbier (EB) rela- tion to determine values for b1 and Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This approxima- tion equates the emergent intensity to the source func- tion at optical depth unity, and is given by: Iλ(µ) ≈ Sλ(τλ = µ) = Bλ(Tc) b1 , (2) where Bλ(Tc) is the Planck function, Bλ(Tc) = 2hc2 λ5 1 exp � hc λkBTc � − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (3) Iλ is the continuum intensity and Sλ is the source func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The λ subscripts indicate that these variables are functions of wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Tc is the color temperature, b1 is the non-local thermodynamic equilibrium departure Formation of The Lyman Continuum During Solar Flares 9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The components of λ = [700, 900] ˚A opacity (top row) and emissivity (bottom row) for two snapshots in the 1F11, δ = 5, Ec = 20 keV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The left-hand panels show 700 ˚A and the right-hand panels show 900 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The light grey dot-dashed line in each panel shows the total opacity or emissivity at t = 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The colored lines correspond to various contributions to the total emissivity or opacity (the sums of these contributions are shown as thick black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The vertical lines correspond to the heights at which τ = 1 (dashed), τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 (dotted), and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='01 (dot-dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' coefficient of the first level of hydrogen, and all other constants have their usual meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' An illustration of this spectral fitting is shown in the left-hand panel of Figure 6, for various times during the 3F9, δ = 5, Ec = 20 keV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The right-hand panel shows the temporal evolution of the log of the goodness- of-fit (ϵ) with vertical dashed lines denoting the time of the spectra in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The goodness-of-fit values are given by: ϵ = 1 N N � i=1 |yi − Oi| yi , (4) where i is the element index, N is the number of data points, yi is the data value and Oi is the fitted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As the goodness-of-fit is weighted by the data for each wave- length and time, smaller values correspond to a smaller relative difference between the data and the fit, meaning the fits are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In the right-hand panel of Figure 6, at t ∼ 5 s the beam has mostly heated a region in the atmosphere be- low the formation height of the LyC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='e below τλ = 1, where photons cannot readily escape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3, the height at which τλ = 1 varies with wave- length, typically forming lower in the atmosphere with decreasing wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The difference is a few tens of km at t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The flare does produce a small increase in emissivity above the height at which τλ = 1 for shorter wavelengths, increasing the emergent intensity towards the tail of the LyC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' At that height, however, τλ > 1 for the head of the continuum and so there is no mean- ingful change in the emergent intensity for longer wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' During these times, the LyC spectrum cannot be approximated by the EB assumption, resulting in the large spike in ϵ seen between t = 5–6 s in the right-hand panel of Figure 6, indicating a poor fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Further discus- sion on this behaviour can be found in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' ^ = 700 A 入 = 900 A t = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='80 s t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 s t = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='80 s t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content="00 s 10-3 10-3 10-6 10-6 [cm' 10-9 10- 10-12 10-12 10-15/ 10-15L 1000 2000 3000 1000 2000 3000 1000 2000 3000 1000 2000 3000 t = 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='80 s t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 s t = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='80 s t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 s 102 102 A-1] 100 100 sr-1 Emissivity 10 10- 10 10-6 10° 1000 3000 1000 1000 2000 1000 2000 3000 2000 3000 2000 3000 Height [km] Height [km] Height [km] Height [km] Total, t= Os Hff TScatt = 5, Ec= 20 keV Total Metals RScatt Fpeak = le+1l erg cm-2 s-1 Hell bf Hbf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='H10 McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Left-hand panel shows synthetic RADYN spectra from 400–911˚A for the 3F9, δ=5, Ec=20keV model, at times t=0s, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3s, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='7s, 10s, and 20s (colored dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Fits to LyC using Equation 2 are shown by solid colored lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The right-hand panel shows the log of the goodness-of-fit (ϵ) as a function of time for the fits shown in the left panel (solid black curve), while the blue curve illustrates the ϵ values when the data was fit between 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0˚A and 911˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The dashed colored lines denote the times of the spectra in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In general, the tail of the continuum does not conform to the EB assumption as well as longer wavelengths, particularly in the decay phase following the flare peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is due to the presence of the overlying optically thin LyC layers discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' When Equa- tion 2 is applied only between λ = 800−911˚A we obtain better fits (the blue line on the goodness-of-fit panel in Figure 6) compared to fitting the full range λ = 505– 911˚A (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This suggests two distinct gradients of the LyC, in agreement with the literature (Machado & Noyes 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Druett & Zharkova 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For times in the simulation where the fit was reasonable, we obtain Tc values consistent with previ- ous studies, increasing from Tc ∼ 9 kK to Tc ∼ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 kK, with b1 decreasing, though not reaching unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Where we see a poor fit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='7 s) we obtain an exception- ally large b1 and Tc > 34 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Values of Tc and b1 were obtained from fitting the LyC spectra by applying the EB assumption (Equations 2 and 3) at each time step from every flare in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 7 shows the temporal evolution of b1 (top row) and Tc (bottom row) for the 3F9, 1F10, 3F10, and 1F11 models, with δ = [3–7] (colored lines), and Ec = 20 keV where we fit between λ = 700–911 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This range was selected as Figure 6 shows that the Eddington-Barbier approximation generally fits the data well at those wave- lengths, but shorter wavelengths can deviate at certain times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' During the beam onset, the initial response of b1 and Tc is somewhat dependent on the spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For beams with δ ≥ 5, b1 initially decreases steadily during the beam onset followed by a rapid and sudden increase whereas, beams with δ ≤ 4 show an initial gradual in- crease in b1, peaking around the time of the spikes in the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' We limit the range of the figures to the scales shown, but the b1 and Tc values at the time of these spikes can be several orders of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The spikes in b1 and Tc are caused by the flattening in the continuum head, at which times the EB approxima- tion is no longer a valid assumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' the b1 and Tc values at these times are unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, there are three key times when the behaviour of b1 is consistent across all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' At t = 0 s, b1 ∼ 102–103, decreasing to a minimum around t ∼ 10 s (flare peak), then increasing again during the decay phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The magnitude of the b1 minima depends upon the flux of the non-thermal electron beam: b1 values for the 3F9 models decrease from b1 ∼ 103 to b1 ∼ 102, whereas the 3F10 and 1F11 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 s 6 = 5 (Tc=8922K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='b1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='06e+03) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' = 20 keV t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3 s t= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s (Tc=8887K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' b1=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='29e+02) (Tc=13763K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='b1=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='49e+01) t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='7 s t= 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0s (Tc=34143K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='b1=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='02e+07) (Tc=13480K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='b1=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='62e+01) GoF[505-911A] GoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' [800-911A] 105 10 103 Intensity cm 101 [erg 10- 10 0 5 10 15 20 600 700 800 900 Time [s] Wavelength [A]Formation of The Lyman Continuum During Solar Flares 11 0 2 4 6 log Departure coefficent 3F9, Ec=20keV 1F10, Ec=20keV 3F10, Ec=20keV 1F11, Ec=20keV δ= 3 δ= 4 δ= 5 δ= 6 δ= 7 0 5 10 15 20 Time (s) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0×103 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0×103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0×104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2×104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='4×104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6×104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8×104 Colour Temperature (K) 3F9, Ec=20keV 0 5 10 15 20 Time (s) 1F10, Ec=20keV 0 5 10 15 20 Time (s) 3F10, Ec=20keV 0 5 10 15 20 Time (s) 1F11, Ec=20keV Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The temporal evolution of b1 and Tc for the 3F9, 1F10, 3F10, 1F11 models with Ec=20keV and δ=3–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' models decrease closer 10−1 < b1 < 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The infer- ence there being that more energetic simulations drive LyC formation closer to LTE due to the increased elec- tron densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Both thermal and non-thermal collisions will significantly increase in those simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ding & Schleicher (1997) presented a similar finding whereby beams with larger non-thermal electron fluxes caused b1 to decrease to a greater extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As discussed earlier, Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018) determined b1 and Tc values from pre-flare and flaring spectra for the six solar flares they analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The pre-flare fit re- sults from our modelling are consistent with their qui- escent Tc and b1 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Further, the range of Tc values we measured in our flare simulations are also generally consistent with the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' During the observed flares, Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018) reported b1 ≈ 1, whereas our model results show a wide range of values (between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1–102) depending upon the flux of the non-thermal electron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' One possible reason for this discrepancy may be due to the dependency of the observed b1 values scaling with the assumed flaring area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Further details can be found in Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' To visualise the spread of Tc and b1 values we produced a 2D histogram that collates the informa- tion from all simulations for all times with parame- ters: δ = [3, 4, 5], Ec = [15, 20, 25] keV, Fpeak = [3F9, 1F10, 3F10, 1F11] erg s−1 cm−2 (recall that the Fpeak = 1F11 simulations did not include the Ec = 15 keV scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is shown in Figure 8, where it is clear that Tc increases from quiet Sun values, Tc ∼ 8– 9 kK, to values roughly in the range Tc ∼ 10 − 16 kK, in agreement with the literature (Machado & Noyes Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Histogram of Tc and b1 obtained from fitting the EB approximation to our simulation grid, weighted by the number of counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Ding & Schleicher 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Lemaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' There are two distinct clusters within the histogram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' the cooler cluster (Tc ∼ 6–11 kK, b1 ∼ 101 − 105) and the hotter cluster (Tc ∼ 10–20 kK, b1 ∼ 101 −103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The cooler cluster corresponds to times before the brief breakdown in the EB approximation, whilst the hotter cluster corresponds to times after this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Comparing Spectral Fitting to Derived Plasma Properties 25 Fit:入=[700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00-911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='001A 102 20 Telfit [kK] Counts 15 101 10 100 101 bi fit12 McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' We can assess how consistent the properties derived from fitting the spectra are with the actual plasma con- ditions in the models by comparing the b1 values directly from RADYN, that we refer to as b1,rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The atomic level populations are functions of height and wavelength, and so to obtain b1,rad averaged over the LyC formation region we calculate the normalised cumulative distribu- tion function (NCDF) of the contribution function, Ccdf (see also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Kowalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The heights corre- sponding to where the bulk of the emission originates are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The weighted average of b1,rad in that for- mation region was then obtained, weighted by the con- tribution function: < b1,rad >= � z(Ccdf =upp) z(Ccdf =low) CI(z) b1,rad dz � z(Ccdf =upp) z(Ccdf =low) CI(z) dz , (5) where z(Ccdf = low) refers to the height at which the Ccdf reaches the lower bound, and z(Ccdf = upp) the height of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For example, the heights corresponding to 10% & 90% of the Ccdf, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Temporal evolution of < b1,rad > values for the 3F10, δ = 6, Ec = 20 keV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The < b1,rad > values are shown for where λ = 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3 ˚A emission originates, considering Ccdf = [10–90%, 50–90%, 10–40%] (red, blue, and purple, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Also shown is the result of fitting the spectra between λ = 700–911 ˚A using the EB approximation (solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The b1=1 line is shown by the dashed black line for reference Figure 9 shows < b1,rad > compared with b1 obtained from our spectral fitting (solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Formation height ranges corresponding to [10–90%, 50–90%, 10– 40%] (red, blue, and purple lines, respectively) were con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' These ranges were selected by integrating the contribution function as a function of height and deter- mining the ranges where the emission became optically thick or thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The emission range 10–90% considers both optically thick and thin components, 50–90% considers emission between τλ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1–1 (optically thick), and 10– 40% considers emission above the τλ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 layer (opti- cally thin) at 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The < b1,rad > values from the optically thick layer are more consistent with those ob- tained from the spectra but deviate during the declining phase of the beam heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The < b1,rad > values determined over 10–90% and 10–40% of the NCDF are significantly larger than the values determined from the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The former rep- resents an assessment of what the < b1,rad > value when including both thick and thin LyC emitting re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This suggests that the thin components (originat- ing from the bubbles) have such large < b1,rad > values that they drag the overall average up considerably from the values in the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The fact that < b1,rad > derived from the optically thick region is more consis- tent with b1 derived from spectral fitting could be due to the optically thick emission dominating the emergent intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The ratio of Tc to the electron temperature (Te), determined when b1 is at a minimum, for the 3F9, 1F10, 3F10, and 1F11 models, with δ=3–7, and Ec=20 keV where available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Te was determined from RADYN for emis- sion between 50–90% of the normalised contribution function at 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Tc and b1 values were determined from fitting the spectra between 700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0–911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In a similar manner, the average electron temperature, < Te > from the LyC forming regions was calculated (re- placing b1,rad for Te in Equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='As b1 tends towards unity, Tc is expected to tend towards the electron tem- perature, Te (Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 10 shows the ratio of Tc to < Te >, determined when b1 is at a minimum for the 3F9, 1F10, 3F10, and 1F11 models, with δ = 5–7, and Ec = 20 keV outside of the times of the anomalous b1 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' < Te > was determined over the optically thick layer (50–90% of the Ccdf) as the < b1,rad > values over this emission range are in general agreement with the b1 values obtained from the fits at these times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' From Figure 10, the ratio of Tc to < Te > E =20keV c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3 3F9 83 1F10 3F10 85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2 1F11 86 米 87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1 X e △ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 米 米 —— 米 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='9 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 log min b10 from Eddington-Barbier b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=" from CDF 10-90% 8 b: from CDF 50-90% b' from CDF 10-40% b,=1 b 2 2 15 20 10 0 5 Time (s)Formation of The Lyman Continuum During Solar Flares 13 is clustered around 1 when b1 reaches a minimum." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The minimum b1 has a rather large range, extending from b1 < 1 to b1 ∼ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' That is, the ratio Tc :< Te > ap- proaches unity at the minimum value of b1 but b1 itself does not necessarily have a value of unity, somewhat con- trary to our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018) present a similar finding where the 6 different X-class flares they observed had varying b1 values (b1 < 1 to b1 ∼ 68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, as they used EVE Sun-as-a-star observations, they converted EVE spectral irradiances to specific in- tensities by assuming the flaring area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018) assumed a fixed flaring area of 1018 cm−2, rep- resenting the middle of a rather large range of reported areas of X-class flares (see Section 4 of Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Varying this area would in effect shift the LyC specific intensity up or down, while keeping the spectral slope fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This would vary the b1 value but not Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The ratio of Tc to electron temperature (Te) as a function of b1 for all models, weighted by time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Te was determined from RADYN for emission between 50–90% of the normalised contribution function at 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Tc and b1 values were determined from fitting the spectra between 750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0–911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Outside of the times when b1 is at a minimum, the ratio of Tc to Te was found to have a large range of val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 11 shows the ratio of Tc to Te as a function of b1 for all models, weighted by time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As seen in Fig- ure 11, even as b1 approaches unity, the range of Tc/Te extends from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This may be because < Te > is determined using the NCDF of the contribution func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As < Te > is determined over a given emission range, the heights considered will vary from model to model, particularly for the more energetic beams where the dynamics of the formation layers occur on shorter time scales (see Figures 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Therefore, the value of < Te > is dependent on the height range considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Histogram of < Te > and < ne > in the forma- tion region of λ = 850 ˚A, weighted by the average intensity of the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Finally, we determined < Te > and < ne > for λ = 850 ˚A to illustrate the general range of plasma properties where LyC forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Those are presented as a 2D histogram in Figure 12 in which properties from δ = [3, 4, 5], Ec = [15, 20, 25] keV, Fpeak = [3F9, 1F10, 3F10, 1F11] erg s−1 cm−2 were collated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 12 shows that while there is some spread to high temperatures (due mostly to contributions from dense, optically thin bubbles) the bulk of the flare emission forms in the range < Te >= 10–30 kK, with electron densities spanning < ne >= 1011−14 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The data has been weighted by the average intensity of emission, and indicates that higher intensity is generally associ- ated with higher < ne >, though temperature also plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Percentage of energy radiated away by the LyC To determine the percentage of energy radiated by the LyC compared to the total energy injected via non-thermal electrons, the LyC lightcurves in Figure 2 were integrated over time and divided by the total non-thermal electron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' We found the LyC radi- ated away between 1–3% of the total non-thermal elec- tron energy injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The 3F9 models radiated away around 1% of the total injected energy through the LyC, whereas the 3F10 models radiated around 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The overall distribution from a number of simualtions is shown in Figure 13 Milligan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2014) provided a study of the global energy budget in a strong flare, comparing the energy radiated in the lower solar atmosphere at optical, UV, and EUV wavelengths to the energy injected via non- thermal electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' They found the LyC radiated away Fit:入=[750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00-911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='001A Plasma Props: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='50-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='90] of ^ref = 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00A 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 } 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 Av 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 0 2 b1|fitPlasma Props: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='95] of ^ref = 850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00A 107 60 2 sr-1 A-1J 106 [kK] 105 Intensity 40 104 103 20 [erg 102 1011 1012 1013 nelRADYN [cm-3]14 McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The ratio of energy radiated by LyC to the energy injected by non-thermal electrons (NTE), as a func- tion of the non-thermal electron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Circles represent δ = 3, plus symbol δ = 4, and diamonds δ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The symbols increase in size with increasing Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' approximately 1% of the total non-thermal electron en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is in agreement with our findings that predict LyC to radiate away a few percent of the total non- thermal electron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Spikes in the Departure Coefficient and color Temperature values We mentioned several times previously the appearance of strong spikes in b1 and Tc, caused by a flattening towards the head of the LyC, as seen, for example, at t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='7 s in the left-hand panel of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The Eddington- Barbier approximation is clearly no longer valid during these times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This phenomenon can be understood from Figure 14, which shows the temporal evolution of the ground state of hydrogen level population for the 3F9 models, with Ec = 20 keV, and δ = [3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The height of the peak of the LyC contribution function is shown by the ver- tical dashed lines at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' At t = 0 s, n1 is large in the chromosphere and decreases across the transition region for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As the beam heat- ing begins, n1 decreases within the chromosphere due to excitation and ionization following the temperature increase and non-thermal collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, there is a small region of plasma between the beam heating region and the transition region that is only minimally heated by the non-thermal electrons, resulting in n1 remaining large compared to the adjacent plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' At later times, the optically thick layer shifts much deeper into the chro- mosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The photons emitted from the optically thick layer subsequently get absorbed by this region of plasma, 1000 1200 1400 1600 1800 Height (Km) 6 8 10 12 14 log n1 (cm−3) 3F9, δ=3, Ec=20keV t=8s t=7s t=6s t=5s t=4s t=3s t=0s 6 8 10 12 14 log n1 (cm−3) 3F9, δ=5, Ec=20keV Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The temporal evolution of the ground state of hydrogen level population for the 3F9 models, with Ec = 20 keV, and δ = [3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The height of the peak of the LyC contribution function is shown by the vertical dashed lines at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' resulting in the flattening of the head of the LyC, and the delayed enhancement of the LyC lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For the δ = 3 model, the optically thick layer of the LyC forms below the region of minimal heating for a few seconds until the region dissipates, resulting in the extended in- crease in b1 and Tc observed for harder beams (see δ=3– 4 curves in Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Whereas for softer beams (see δ ≥5 curves in Figure 7), the optically thick layer only forms below the region of minimal heating for a shorter duration, resulting in the sudden steep spikes in b1 and Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In Figure 14, the n1 values for the harder beam (δ=3) are reduced more uniformly and at higher altitudes within the chromosphere despite harder beams being composed of a greater number of high energy, deeply penetrating, electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This can be understood from Figure 15, which shows the temperature and beam heat- ing profile for the 3F9 models, with Ec=20keV, and 10 [%] LyC Flux / NTE Flux [ Ec[keV] 15 keV 20 keV 25 keV 6 6=3 10 6=4 6=5 NTE Flux [erg cm-2jFormation of The Lyman Continuum During Solar Flares 15 δ=3 (harder beam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' blue lines) and 5 (softer beam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The region of minimal heating is thicker for softer beams (z ≈1720–1770 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The δ=3 beam is harder and therefore has a greater number of high-energy elec- trons, which results in a faster atmospheric response and evaporation of chromospheric plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As the evapo- ration front propagates the mass density of the upper atmosphere is increased by approximately a few orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The increased column depth means that the higher energy electrons in the δ=3 beam are ther- malised higher up in the atmosphere at later times in the simulation despite the δ=5 beam being composed of more low energy electrons, as can be seen by the blue and red dashed lines in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This results in the narrower region of minimal heating (z ≈1790 km) seen for the harder δ=3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The thickness of this region also contributes to the varying profiles during the sud- den increases (spikes) in b1 and Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 1700 1720 1740 1760 1780 1800 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 1700 1720 1740 1760 1780 1800 Height (km) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 log Temp (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='10 Normalised Beam heating 3F9,δ=3, Ec20keV @7s 3F9,δ=5, Ec20keV @7s Beam heating 3F9, δ=3, 20keV @7s Beam heating 3F9, δ=5, 20keV @7s Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Temperature profiles for the 3F9 models, with Ec = 20 keV, and δ = 3 and 5 at t=7 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The normalised beam heating is shown by the dashed lines for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Wavelength dependency of b1 and Tc Machado & Noyes (1978) hypothesised from observa- tional evidence that a higher-lying region contributes optically thin component to the LyC during flares along- side the bulk of the chromospheric emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Evidence for this optically thin layer comes from a steepening in the LyC spectrum’s gradient away from the contin- uum head, resulting in increased Tc values determined at shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In the right-hand panel of Figure 6, there is a local peak in ϵ at t=10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Shorter wavelengths, between λ ∼ 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0 − 700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0˚A, are poorly fit compared to longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This spectrum has two distinct gra- dients, one at shorter wavelengths, λ ≤ 700 ˚A, and the other at longer wavelengths, λ ≥ 700 ˚A, in agreement with Machado & Noyes (1978), Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018), and Druett & Zharkova (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Evolution of b1 and Tc for the 3F9, δ = 5, Ec = 20 keV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Equations 2 and 3 have been applied between λ = [505−700] ˚A and λ = [700−911] ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The values are only shown between t = [7 − 30] s to omit times where we know that spectral fitting produced poor results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Figure 16 shows the b1 and Tc values for the 3F9, δ = 5, Ec = 20 keV model, where the EB approxima- tion has been applied between λ = 505 − 700 ˚A and λ = 700−911 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Generally, the Tc values determined at shorter wavelengths were a few thousand Kelvin hotter than at longer wavelengths, in agreement with the litera- ture (Machado & Noyes 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The b1 values are also generally larger at shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This is due to the optically thin components of the LyC that enhance the spectrum at shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The number of optically thin components that form depends on the type of evaporation observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Gentle evapora- tion resulted in one upwardly propagating optically thin layer forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Whereas, explosive evaporation resulted in two or three optically thin components correspond- ing to evaporation and condensation fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The up- ward propagating optically thin components of the LyC in Figure 4 are due to bubbles of chromospheric material travelling immediately ahead of the evaporation front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2020) state that these bubbles are a source of optically thin Ca II 8542˚A line emission and do not always emit strongly in the Hα line, while Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018) found solar bubbles to be among the dominant sources of Lyman alpha line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Such propagating high-density features could be indirectly detected via LyC observations due to the effect they have on the spectral shape, manifesting as an increase in both Tc 250 200 b- 150 100 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='6日 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='5 K (x104 T 二 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0-700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0A 700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='0-911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='3A 12 20 10 14 16 18 8 Time (s)16 McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' and b1 at shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2018) only observed an increase in Tc at shorter wavelengths for one of the six events that they analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This may be due to the fact that the other five flares did not provide the correct conditions for a solar bubble to form, or at the time of observation the bubbles had dissipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' CONCLUSIONS Using the F-CHROMA grid of RADYN models, we have shown that the LyC is greatly enhanced during solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The LyC spectral response is highly sensi- tive to the flux of the non-thermal electron beam, but is less dependent upon the spectral index or low-energy cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' LyC was found to radiate away between 1–3% of the total non-thermal electron energy injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' In- creases in solar irradiance associated with solar flares are known to drive dynamic and compositional changes in Earth’s ionosphere, which can have adverse implica- tions for modern technology on which society has be- come dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The 850–1027˚A range, in particular, is absorbed at an altitude of around 105–120km in the ionosphere (E-layer), where it drives the partial dissoci- ation of molecular oxygen (Robinson 1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This part of the spectrum is dominated by LyC, along with higher- order Lyman emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Both optically thin and thick layers of the LyC were found to form during solar flares, in agreement with the literature (Machado & Noyes 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Druett & Zharkova 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The optically thick layer is formed in NLTE in the QS and forms at the top of the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' During solar flares, this layer shifts deeper into the solar chromosphere due to the evapora- tion of the upper chromosphere, forming near the peak beam heating region, with < Te >≈ 10–40 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' It also forms closer to LTE conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The optically thin com- ponents of the LyC formed due to chromospheric evap- oration, with the number of optically thin components forming being dependent on the type of evaporation ob- served, gentle or explosive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' These optically thin com- ponents cause an enhancement in intensities away from the LyC head, resulting in increased b1 and Tc values determined at shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Fitting with the EB relation we find that Tc ≈ 10–16 kK, with b1 at times dropping to b1 ≈ few × 10, but which have a large scat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Our results suggest that the LyC spectral response is indicative of a chromospheric temperature and density enhancement largely probing the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Gra- dients in the derived Tc as a function of wavelength can also indicate the presence of regions of propagat- ing dense, cool material in the upper atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' The number of optically thin layers formed was found to be greater for stronger solar flares, and the LyC contribu- tion functions presented show that the optically thin lay- ers form and dissipate over a shorter period for stronger flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Model-data discrepancies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' the large scatter of b1 values that don’t always approach unity, contrary to Machado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' 2018), could result from an exagger- ated optically thin component to the LyC forming in the dense bubbles, if those bubbles are denser than in actual flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' For example, Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' (2020) mod- elled the ratio of the intensity of redshifted ‘satellite’ components of the Fe II line to the intensity of the sta- tionary component, noting that it was larger than the observed flare they were simulating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This could be due to the density in the modelled condensation (from which the satellite component originated) being larger than the condensation produced during the observed flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Our chromospheric bubbles may be similarly over-dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' EVE currently provides LyC observations with the greatest coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='However, these are Sun-as-a-star ob- servations with a cadence of 60 s (observations prior to 2014 had a 10 s cadence, but instrument degradation has forced a longer exposure time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Therefore, EVE LyC flare observations will observe a range of flaring loops at various heating or cooling stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Thus, some of the dynamic features may become smeared tempo- rally, especially for stronger flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' As the breakdown in the EB approximation and delayed enhancement in the LyC lightcurves also occur over second to sub-second timescales, it is unlikely that EVE can observe these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Additionally, as EVE provides Sun-as-a- star observations, EVE spectral irradiances can be con- verted to specific intensities by assuming the flaring area as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Varying this area shifts the LyC specific intensity up or down, while keeping the spectral slope fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This would vary the b1 value but not Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' SPICE on board the Solar Orbiter mission that was launched in 2020, provides EUV coverage in the wave- length ranges of λ = 704–790 ˚A and λ = 973–1049 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' This provides partial coverage of LyC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' It remains to be seen to what extent it is possible to extract continuum intensities, which will depend in part on how well we can resolve spectral lines with that passband, but SPICE observations may be used to determine b1 and Tc val- ues below the head the LyC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Unlike SDO/EVE, these observations would have spatial resolution and obtain higher cadences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' However, as SPICE only provides par- tial coverage of LyC, the derived b1 and Tc values may be elevated due to the presence of the optically thin LyC layers, enhancing the LyC spectrum at shorter wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' To determine if this is the case, b1 and Tc val- ues should be derived and compared from SPICE and EVE spectra over multiple wavelength ranges for any flares both instruments capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Our analysis paves the way for an interpretation of solar flare LyC observations taken by current and future missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' Formation of The Lyman Continuum During Solar Flares 17 We thank the anonymous referee for a careful review of our manuscript, and for useful comments and sug- gestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' SAM would like to thank the Science and Technology Facilities Council (UK) for the award of a PGR studentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' AJM acknowledges funding from the Science Technology Funding Council (STFC) Grant Code ST/T506369/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' ROM and GSK would like to acknowledge support from NASA Heliophysics Sup- porting Research grant NNH19ZDA001N and ROM thanks the Science and Technologies Facilities Council (UK) for the award of an Ernest Rutherford Fellowship (ST/N004981/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAzT4oBgHgl3EQfrv1r/content/2301.01648v1.pdf'} +page_content=' PJAS acknowledges support from the Fundo de Pesquisa Mackenzie (MackPesquisa), CNPq (contract 307612/2019-8), and FAPESP, the Sao Paulo Research 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Lyon, Université Lyon 1, CNRS, LIRIS UMR CNRS 5205, F-69621, Lyon, France +2Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada. +Abstract +We show that every 3-connected K2,ℓ-minor free graph with minimum degree at least +4 has maximum degree at most 7ℓ. As a consequence, we show that every 3-connected +K2,ℓ-minor free graph with minimum degree at least 5 and no twins of degree 5 has +bounded size. Our proofs use Steiner trees and nested cuts; in particular, they do +not rely on Ding’s characterization of K2,ℓ-minor free graphs. +1 +Introduction +Over the last decades minor-free graph classes received considerable attention. +Many com- +putationally difficult problems become tractable on minor-free graph classes, especially if the +excluded minor is planar. One of the simplest planar graphs is the complete bipartite graph +K2,ℓ. A first step to understanding the difficulty of a problem is to consider it on the class of +K2,ℓ-minor free graphs. The structure of K2,ℓ-minor free graphs has been widely studied, see +e.g. [1, 2, 3]. +An important graph class parameter is the edge density. One can easily notice that a K2,ℓ- +minor free graph G may contain Ω(ℓ·n) edges since the disjoint union of Kℓ+1 is K2,ℓ-minor free. +A similar example can be easily found for connected graphs. Chudnovsky, Reed and Seymour +proved in [1] that it is essentially tight since K2,ℓ-minor free graphs have at most 1 +2(ℓ+1)(n−1) +edges, and they showed that this bound can be reached (proving a conjecture of Myers [4]). +Ding [2] proposed a decomposition theorem for K2,ℓ-minor-free graphs (so far unpublished). +As a corollary, he deduced that all the 5-connected K2,ℓ-minor free graphs have bounded size, +as well as 3-connected K2,ℓ-minor free graphs of degree at least 6. One can remark that these +bounds are tight because of the strong product of an n-vertex cycle with an edge, which is +4-connected and 5-regular. +Ding also obtained as a corollary that 3-connected graphs with +minimum degree 6 have bounded maximum degree. +We reprove these results in a slightly stronger sense with simple self-contained proofs. +Namely, we show that: +Theorem 1.1. All 3-connected K2,ℓ-minor free graphs with minimum degree 5 containing no +twins1 of degree 5 have bounded size. +∗This work was supported by ANR project GrR (ANR-18-CE40-0032) +†The author is supported by the Vanier Canada Scholarship Program. +1We say two vertices u, v are twins in they have exactly the same closed neighbourhood, in particular u is +adjacent to v. +1 + +Observe that 5-connected graphs, and 3-connected graphs with minimum degree at least 6 +fall in the scope of the theorem. Many practical problems such as Maximum Independent Set +and Minimum Dominating Set can be reduced to twin-free graphs, which motivates their study. +The theorem is tight because of the family of twin-free 4-connected and 4-regular graphs depicted +in Figure 1. +Figure 1: An infinite family of twin-free 4-connected and 4-regular graphs without K2,5-minor. +Our proof consists of two parts. +The first, of independent interest, consists in proving +that 3-connected K2,ℓ-minor free graphs of minimum degree 4 have maximum degree O(ℓ) (it +strengthens a result of Ding [2] who only proved it for minimum degree 6). The second consists +in constructing a K2,ℓ-minor when there are too many cuts between two given vertices. +Chudnovsky, Reed and Seymour gave in [1] the sketch of a proof that 3-connected K2,ℓ-minor +free graphs contain at most 5 +2n+c(ℓ) edges for a large constant c (not explicit). Their technique, +based on an adaptation of a tree decomposition argument of [5] is completely different from ours. +Even if weaker, we can easily derive from our proof that 3-connected K2,ℓ-minor free graphs with +minimum degree 5 have at most 5 +2n + c(ℓ) edges. +2 +Maximum degree +In [2], Ding proved that K2,ℓ-minor free graphs that are 4-connected or 3-connected with mini- +mum degree 6 have bounded degree. We slightly improve these results (with a short proof): +Lemma 2.1. Every 3-connected K2,ℓ-minor free graph G with minimum degree 4 has maximum +degree at most 7ℓ. +Proof. Assume by contraction that there exists a 3-connected K2,ℓ-minor free graph G of min- +imum degree 4 with a vertex x of degree at least 7ℓ. Let X = N(x) and G′ = G \ x. Let T be +a Steiner tree of X in G′ (a subtree of G′ spanning all vertices in X with no leaf outside of X) +maximizing the number of leaves in X. Note that if T has at least ℓ leaves, then contracting all +internal nodes of T yields a K2,ℓ minor in G. +Therefore, we may assume that T has at most ℓ−1 leaves. Hence T has less than ℓ branching +nodes (that is nodes of degree at least 3). In particular, at least 6ℓ vertices of X have degree at +most 2 in T. They then belong to the set of paths of T minus its branching nodes. Note that +there are less than 2ℓ such paths since T has ℓ branching nodes and ℓ − 1 leaves. For each such +path containing the nodes u1, u2, . . . , us of X in that order, we say that ui is a strictly internal +vertex if i /∈ {1, s}, and that it is even (resp. odd) if furthermore i is even (resp. odd). Observe +that on each of the 2ℓ paths there are at most two non-strictly internal vertices and the number +of even vertices is at least the number of odd vertices. Therefore, denoting by e the number of +even vertices, we have 7ℓ ⩽ |X| ⩽ ℓ − 1 + 2e + 2 · 2ℓ hence e > ℓ. +2 + +Let u be an even vertex, and denote by v, w the previous and next vertices of X on the path +of T minus its internal nodes containing u. Since the minimum degree is at least 4, u has a +neighbor u′ which is not x nor its two neighbors in T. +Let C be the connected component of u′ in G \ T. Since neither {u, v} nor {u, w} are cuts +in G′, we get that C is adjacent to a vertex of T which is not in the vw-subpath of T, or C is +adjacent to at least one vertex v′ in the vu-subpath of T (distinct from u) and one vertex w′ in +the uw-subpath of T (distinct from u). +In the second case, let P be a path from v′ to w′ whose internal vertices are in C. We add +this path to T and remove an edge of T incident with u. This creates a Steiner tree of X in G′ +with one more leaf than T, a contradiction. +So we can assume that C is adjacent to a vertex which is not in the vw-subpath Q of T. So +there exists a path Pu from u to a vertex of T − Q with internal vertices in C. Thus adding Pu +to T creates a cycle containing either v or w, say v by symmetry. Now we remove an edge of this +cycle incident to v, and the result is a Steiner tree of X in G′ where v is a leaf. By maximality +of T, this means that the endpoint of Pu in T must be a leaf. +Since there are at least ℓ even vertices and at most ℓ − 1 leaves in T, at least two paths Pu +and Pu′ must end on the same leaf of T. Define v (resp. v′) as the vertex of X preceding or +following u (resp. u′) in the path containing u (resp. u′) in T minus its internal nodes, such +that v (resp. v′) lies in the unique cycle of T ∪ Pu (resp. T ∪ Pu′). +Let ev (resp. ev′) be the edge incident with v (resp. v′) in the unique cycle in T ∪ Pu (resp. +T ∪ Pu′). Then (T ∪ Pu ∪ Pu′) − {ev, ev′} is a Steiner tree of X in G′, but with one more leaf +than T, a contradiction. +Note that both hypothesis are required, as shown by the following examples with arbitrary +large maximum degree: +• Wheels have minimum degree 3 and the central vertex has arbitrary large degree. Wheels +are 3-connected and K2,4-minor-free. +• An even wheel u, u1, . . . , u2n with an attached copy of K5 on each edge u2i−1u2i is 2- +connected and u has arbitrary high degree. +The graph has minimum degree 4 and is +K2,7-minor-free. (Note that this can be extended so that the graph has minimum degree +δ and is K2,2δ−1-minor-free.) +3 +Proof of Theorem 1.1 +Let s, t be two vertices of the graph G. An st-cut is a subset of vertices C not containing s nor t +and whose deletion separates s from t. We say that C is a k-cut if |C| = k, we denote by η(s, t) +the st-connectivity, that is the minimum size of an st-cut. For every st-cut C, we denote by SC +the connected component of s in C. A sequence of cuts is nested if for every two cuts C, C′ in +the sequence, SC ⊆ SC′ or SC′ ⊆ SC. Finally, we say a sequence of cuts is 2-nested if the cuts +are nested and the distance between any two cuts is at least 2, i.e. they are disjoint and there is +no edge between distinct cuts. All these notions can be extended when s and t are replaced by +sets of vertices. Note that in particular we consider that if S, T are disjoint sets of vertices, an +ST-cut is disjoint from S ∪ T. The proof of Theorem 1.1 relies on the following lemma, which +shows that if we have a large enough 2-nested sequence of st-cuts, then G admits a K2,ℓ-minor. +Lemma 3.1. Suppose that G is 3-connected, has minimum degree 5 and no twins of degree 5. +Let S, T be two subsets of vertices inducing connected subgraphs of G. If G contains a 2-nested +sequence of more than ℓ · η(S, T)2 ST-cuts of size η(S, T) then G admits a K2,ℓ-minor. +3 + +Proof. Let η = η(S, T), d = ℓ · η2 and denote by C1, . . . , Cd the 2-nested sequence of η-cuts. +Consider η internally disjoint ST-paths P1, . . . , Pη and write P = P1 ∪ · · · ∪ Pη. For each j ∈ +[1, d − 1], denote by Yj the set SCj+1 − (SCj ∪ Cj) of vertices between the cuts Cj and Cj+1. +Observe that exactly one of the following may happen: +(1) for some a ̸= b, there exists a path of length at least two between Pa ∩ (Yj ∪ Cj ∪ Cj+1) +and Pb ∩ (Yj ∪ Cj ∪ Cj+1) with internal nodes in Yj \ (Pa ∪ Pb). +(2) Yj \ P is not empty and (1) does not hold. In particular, each connected component of +Yj \ P is adjacent to exactly one path. +(3) Yj \ P is empty and (1) does not hold. +We choose P1, . . . , Pη such that the number of times (1) holds is maximized, and with respect +to that, the number of times (2) holds is maximized. Fix j ∈ [1, d − 1]. +If (2) holds, then there exists a component C in Yj \ P which has only neighbours on some +path, say P1. Let v1, v2, . . . , vr be its neighbours on P1 in order (note that r ⩾ 3 otherwise +{v1, v2} is a 2-cut in G). Since C is connected, there exists a path PC in C from a neighbour of +v1 to a neighbour of vr. We replace P1 by the path P ′ +1 following P1 up to v1, then taking PC, +and following P1 starting from vr. Since G is 3-connected, deleting v1, vr does not disconnect +the graph. So there is a path from C to some other path in Yj \ {v1, vr}, say P2, and this path +does not intersect P3, . . . , Pη. Since C does not have neighbors on any Pi for i > 1, this path +must go through some vi with i /∈ {1, r}. Now the component of vi in Yj \ (P ′ +1 ∪ P2 ∪ · · · ∪ Pη) +has neighbors both in P ′ +1 and P2, hence satisfies (1), a contradiction with our choice of P. +Assume now that (3) holds. Since the sequence is 2-nested, Yj ∩ P1 is not empty. Let v +be the first vertex of P1 in Yj. Note that Yj ∩ P1 is induced, otherwise we can replace it by a +shorter subpath and end up in case (1) or (2), a contradiction with our choice of P. Therefore, +since v has degree at least 5, v must have at least three neighbors outside of P1. They should +lie on exactly one other path since Yj \ P is empty and (1) does not hold, say P2 by symmetry. +Let w be the first neighbor of v in P2 ∩ Yj. +Note that w does not belong to Cj nor Cj+1. +Assume v, w have only neighbours on P1 ∪ P2, and denote by w1, . . . , wt the neighbors of v in +P2 (following their order in P2) and by v1, . . . , vs the neighbors of w in P1. Note that s, t ⩾ 3 +and that v ∈ {v1, v2} and w ∈ {w1, w2} depending on whether v1 or w1 lie in Cj or not. +Suppose that for some i > 2, vi /∈ N[v]. Let P −v +1 +be the subpath of P1 where we delete +all vertices after v and let P +vi +1 +be the subpath of P1 where we delete all vertices before vi. +Similarly, let P −w +2 +be the subpath of P2 where we delete all vertices after w and let P +wt +2 +be the +subpath of P2 where we delete all vertices before wt. Now replacing P1 by P −v +1 +P +wt +2 +and P2 by +P −w +2 +P +vi +1 +gives a new collection of η internally disjoint paths from S to T where the total length +of the paths in the section j has been reduced. So (3) does not hold anymore and the number +of sections satisfying (1) or (2) has increased, a contradiction with the choice of P. +Suppose now that vi ∈ N[v] and by symmetry wi ∈ N[w] for all i > 2. Recall that P1 is +induced, hence vvi cannot be an edge for i ⩾ 4. Therefore, s = 3 and v = v2. By symmetry, we +also get t = 3 and w = w2. Now, if vv1 is not an edge, then we can again replace P1 by P −v1 +1 +P +w +2 +and P2 by P −w1 +2 +P +v +1 +and get a contradiction. Hence vv1 is an edge and by symmetry ww1 is an +edge. Therefore, v and w have both degree 5 and are twins, which violates the hypothesis on G. +Therefore, we may assume that (1) holds for every j ∈ [1, d−1]. By the pigeonhole principle, +there is a, b ∈ [1, η] with a ̸= b and at least 2ℓ paths of length at least 2 between Pa and Pb +with internal nodes outside of Pa ∪ Pb. Since the collection of cuts is 2-nested, these paths can +only intersect on the vertices of the cuts and then at least ℓ of them are disjoint. Therefore +contracting S ∪ T ∪ Pa and Pb into two vertices yields a K2,ℓ minor in G. +4 + +We now conclude this section with the proof of Theorem 1.1, obtained by combining Lemma 2.1 +and Lemma 3.1. +Proof of Theorem 1.1. Let d = ℓ3, nℓ ⩾ (7ℓ)2·(d+1)ℓ and G be a graph satisfying the hypothesis +of the theorem. We may assume that G has degree at most 7ℓ otherwise we get a K2,ℓ-minor by +Lemma 2.1. Therefore, there must be two vertices s, t at distance at least 2 · (d + 1)ℓ in G. Let +us denote by η(s, t) the st-connectivity. +Let Vi be the vertices at distance i to s and let Ci ⊆ Vi be an inclusion-wise minimal st-cut. +In V \ Ci, denote by Si (resp. Ti) the set of vertices in the connected component of s (resp. t). +Let k be the maximum integer such that k ⩽ ℓ and there exist Ci, Cj at distance j − i ⩾ +2 · (d + 1)ℓ−k such that η(Ci, Cj) ⩾ k. If k = ℓ, we obtain a K2,ℓ-minor by contracting Si and Tj +into two vertices and contracting the internal vertices of each of the ℓ pairwise disjoint paths of +length at least 2 between Si and Tj (and possibly deleting some edges). +Hence we can assume that k ⩽ ℓ − 1. Let Ci and Cj be at distance j − i ⩾ 2 · (d + 1)ℓ−k, +and let d′ = 2 · (d + 1)ℓ−k−1. We consider the d + 1 cuts Ci, Ci+d′, Ci+2d′, . . . , Ci+d·d′ (which +are pairwise at distance at least d′). By maximality of k, we cannot have η(Ci, Cj) > k, nor +η(Ci+i′d′, Ci+(i′+1)d′) > k for any 0 ⩽ i′ ⩽ d − 1. +Hence η(Vi+i′d′, Vi+(i′+1)d′) = k = η(Ci, Cj) +for each i′ ⩽ d − 1. But then there exists a cut of size k between Ci+i′d′, Ci+(i′+1)d′ for each +0 ⩽ i′ ⩽ d − 1. These d k-cuts between Ci and Cj are 2-nested, hence by Lemma 3.1, G then +contains a K2,ℓ-minor, which concludes the proof. +One can easily adapt the proofs of Lemma 3.1 and Theorem 1.1 to show that every large +enough 3-connected graph G with minimum degree 5 either contains a K2,ℓ-minor or contains +the 2 × 4 king’s graph2 as an induced subgraph where the middle vertices have degree 5 in G +(the current proof of Lemma 3.1 only ensures the existence of a 2×3 king’s graph). Contracting +the K4 in the middle into a single edge to obtain a 2 × 3 king’s graph as in Figure 2 keeps +the minimum degree and the connectivity of the graph. Hence, we can obtain the following +weakening of the result by Chudnovsky et al. [1]: every 3-connected K2,ℓ-minor free graph of +minimum degree 5 contains at most 5 +2n + c(ℓ) edges. +Figure 2: Contracting the red thick edges reduces a 2 × 4 king’s graph to a 2 × 3 king’s graph. +References +[1] M. Chudnovsky, B. A. Reed, and P. D. Seymour. The edge-density for K2,t-minors. J. Comb. +Theory, Ser. B, 101(1):18–46, 2011. +[2] G. Ding. Graphs without large K2,n-minors. arXiv, abs/1702.01355, 2017. +[3] M. N. Ellingham, E. A. Marshall, K. Ozeki, and S. Tsuchiya. A characterization of K2,4- +minor-free graphs. SIAM J. Discret. Math., 30(2):955–975, 2016. +[4] J. S. Myers. The extremal function for unbalanced bipartite minors. Discret. Math., 271(1- +3):209–222, 2003. +[5] B. Oporowski, J. Oxley, and R. Thomas. Typical subgraphs of 3-and 4-connected graphs. +Journal of Combinatorial Theory, Series B, 57(2):239–257, 1993. +2A king’s graph is a grid with diagonals. +5 + diff --git a/w9A0T4oBgHgl3EQfMf9w/content/tmp_files/load_file.txt b/w9A0T4oBgHgl3EQfMf9w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f36769162198cf209ee480a9f6d8556f7819dfb4 --- /dev/null +++ b/w9A0T4oBgHgl3EQfMf9w/content/tmp_files/load_file.txt @@ -0,0 +1,256 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf,len=255 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='02133v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='CO] 5 Jan 2023 A note on highly connected K2,ℓ-minor free graphs∗ Nicolas Bousquet1, Théo Pierron1, and Alexandra Wesolek†2 1Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Lyon, Université Lyon 1, CNRS, LIRIS UMR CNRS 5205, F-69621, Lyon, France 2Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Abstract We show that every 3-connected K2,ℓ-minor free graph with minimum degree at least 4 has maximum degree at most 7ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' As a consequence, we show that every 3-connected K2,ℓ-minor free graph with minimum degree at least 5 and no twins of degree 5 has bounded size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Our proofs use Steiner trees and nested cuts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' in particular, they do not rely on Ding’s characterization of K2,ℓ-minor free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 1 Introduction Over the last decades minor-free graph classes received considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Many com- putationally difficult problems become tractable on minor-free graph classes, especially if the excluded minor is planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' One of the simplest planar graphs is the complete bipartite graph K2,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' A first step to understanding the difficulty of a problem is to consider it on the class of K2,ℓ-minor free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' The structure of K2,ℓ-minor free graphs has been widely studied, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' An important graph class parameter is the edge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' One can easily notice that a K2,ℓ- minor free graph G may contain Ω(ℓ·n) edges since the disjoint union of Kℓ+1 is K2,ℓ-minor free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' A similar example can be easily found for connected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Chudnovsky, Reed and Seymour proved in [1] that it is essentially tight since K2,ℓ-minor free graphs have at most 1 2(ℓ+1)(n−1) edges, and they showed that this bound can be reached (proving a conjecture of Myers [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Ding [2] proposed a decomposition theorem for K2,ℓ-minor-free graphs (so far unpublished).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' As a corollary, he deduced that all the 5-connected K2,ℓ-minor free graphs have bounded size, as well as 3-connected K2,ℓ-minor free graphs of degree at least 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' One can remark that these bounds are tight because of the strong product of an n-vertex cycle with an edge, which is 4-connected and 5-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Ding also obtained as a corollary that 3-connected graphs with minimum degree 6 have bounded maximum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' We reprove these results in a slightly stronger sense with simple self-contained proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Namely, we show that: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' All 3-connected K2,ℓ-minor free graphs with minimum degree 5 containing no twins1 of degree 5 have bounded size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' ∗This work was supported by ANR project GrR (ANR-18-CE40-0032) †The author is supported by the Vanier Canada Scholarship Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 1We say two vertices u, v are twins in they have exactly the same closed neighbourhood, in particular u is adjacent to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 1 Observe that 5-connected graphs, and 3-connected graphs with minimum degree at least 6 fall in the scope of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Many practical problems such as Maximum Independent Set and Minimum Dominating Set can be reduced to twin-free graphs, which motivates their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' The theorem is tight because of the family of twin-free 4-connected and 4-regular graphs depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Figure 1: An infinite family of twin-free 4-connected and 4-regular graphs without K2,5-minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Our proof consists of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' The first, of independent interest, consists in proving that 3-connected K2,ℓ-minor free graphs of minimum degree 4 have maximum degree O(ℓ) (it strengthens a result of Ding [2] who only proved it for minimum degree 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' The second consists in constructing a K2,ℓ-minor when there are too many cuts between two given vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Chudnovsky, Reed and Seymour gave in [1] the sketch of a proof that 3-connected K2,ℓ-minor free graphs contain at most 5 2n+c(ℓ) edges for a large constant c (not explicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Their technique, based on an adaptation of a tree decomposition argument of [5] is completely different from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Even if weaker, we can easily derive from our proof that 3-connected K2,ℓ-minor free graphs with minimum degree 5 have at most 5 2n + c(ℓ) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 2 Maximum degree In [2], Ding proved that K2,ℓ-minor free graphs that are 4-connected or 3-connected with mini- mum degree 6 have bounded degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' We slightly improve these results (with a short proof): Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Every 3-connected K2,ℓ-minor free graph G with minimum degree 4 has maximum degree at most 7ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Assume by contraction that there exists a 3-connected K2,ℓ-minor free graph G of min- imum degree 4 with a vertex x of degree at least 7ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let X = N(x) and G′ = G \\ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let T be a Steiner tree of X in G′ (a subtree of G′ spanning all vertices in X with no leaf outside of X) maximizing the number of leaves in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Note that if T has at least ℓ leaves, then contracting all internal nodes of T yields a K2,ℓ minor in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Therefore, we may assume that T has at most ℓ−1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Hence T has less than ℓ branching nodes (that is nodes of degree at least 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' In particular, at least 6ℓ vertices of X have degree at most 2 in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' They then belong to the set of paths of T minus its branching nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Note that there are less than 2ℓ such paths since T has ℓ branching nodes and ℓ − 1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' For each such path containing the nodes u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , us of X in that order, we say that ui is a strictly internal vertex if i /∈ {1, s}, and that it is even (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' odd) if furthermore i is even (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Observe that on each of the 2ℓ paths there are at most two non-strictly internal vertices and the number of even vertices is at least the number of odd vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Therefore, denoting by e the number of even vertices, we have 7ℓ ⩽ |X| ⩽ ℓ − 1 + 2e + 2 · 2ℓ hence e > ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 2 Let u be an even vertex, and denote by v, w the previous and next vertices of X on the path of T minus its internal nodes containing u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Since the minimum degree is at least 4, u has a neighbor u′ which is not x nor its two neighbors in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let C be the connected component of u′ in G \\ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Since neither {u, v} nor {u, w} are cuts in G′, we get that C is adjacent to a vertex of T which is not in the vw-subpath of T, or C is adjacent to at least one vertex v′ in the vu-subpath of T (distinct from u) and one vertex w′ in the uw-subpath of T (distinct from u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' In the second case, let P be a path from v′ to w′ whose internal vertices are in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' We add this path to T and remove an edge of T incident with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' This creates a Steiner tree of X in G′ with one more leaf than T, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' So we can assume that C is adjacent to a vertex which is not in the vw-subpath Q of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' So there exists a path Pu from u to a vertex of T − Q with internal vertices in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Thus adding Pu to T creates a cycle containing either v or w, say v by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Now we remove an edge of this cycle incident to v, and the result is a Steiner tree of X in G′ where v is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' By maximality of T, this means that the endpoint of Pu in T must be a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Since there are at least ℓ even vertices and at most ℓ − 1 leaves in T, at least two paths Pu and Pu′ must end on the same leaf of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Define v (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' v′) as the vertex of X preceding or following u (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' u′) in the path containing u (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' u′) in T minus its internal nodes, such that v (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' v′) lies in the unique cycle of T ∪ Pu (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' T ∪ Pu′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let ev (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' ev′) be the edge incident with v (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' v′) in the unique cycle in T ∪ Pu (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' T ∪ Pu′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Then (T ∪ Pu ∪ Pu′) − {ev, ev′} is a Steiner tree of X in G′, but with one more leaf than T, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Note that both hypothesis are required, as shown by the following examples with arbitrary large maximum degree: Wheels have minimum degree 3 and the central vertex has arbitrary large degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Wheels are 3-connected and K2,4-minor-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' An even wheel u, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , u2n with an attached copy of K5 on each edge u2i−1u2i is 2- connected and u has arbitrary high degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' The graph has minimum degree 4 and is K2,7-minor-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' (Note that this can be extended so that the graph has minimum degree δ and is K2,2δ−1-minor-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=') 3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1 Let s, t be two vertices of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' An st-cut is a subset of vertices C not containing s nor t and whose deletion separates s from t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' We say that C is a k-cut if |C| = k, we denote by η(s, t) the st-connectivity, that is the minimum size of an st-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' For every st-cut C, we denote by SC the connected component of s in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' A sequence of cuts is nested if for every two cuts C, C′ in the sequence, SC ⊆ SC′ or SC′ ⊆ SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Finally, we say a sequence of cuts is 2-nested if the cuts are nested and the distance between any two cuts is at least 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' they are disjoint and there is no edge between distinct cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' All these notions can be extended when s and t are replaced by sets of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Note that in particular we consider that if S, T are disjoint sets of vertices, an ST-cut is disjoint from S ∪ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1 relies on the following lemma, which shows that if we have a large enough 2-nested sequence of st-cuts, then G admits a K2,ℓ-minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Suppose that G is 3-connected, has minimum degree 5 and no twins of degree 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let S, T be two subsets of vertices inducing connected subgraphs of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' If G contains a 2-nested sequence of more than ℓ · η(S, T)2 ST-cuts of size η(S, T) then G admits a K2,ℓ-minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let η = η(S, T), d = ℓ · η2 and denote by C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , Cd the 2-nested sequence of η-cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Consider η internally disjoint ST-paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , Pη and write P = P1 ∪ · · · ∪ Pη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' For each j ∈ [1, d − 1], denote by Yj the set SCj+1 − (SCj ∪ Cj) of vertices between the cuts Cj and Cj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Observe that exactly one of the following may happen: (1) for some a ̸= b, there exists a path of length at least two between Pa ∩ (Yj ∪ Cj ∪ Cj+1) and Pb ∩ (Yj ∪ Cj ∪ Cj+1) with internal nodes in Yj \\ (Pa ∪ Pb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' (2) Yj \\ P is not empty and (1) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' In particular, each connected component of Yj \\ P is adjacent to exactly one path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' (3) Yj \\ P is empty and (1) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' We choose P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , Pη such that the number of times (1) holds is maximized, and with respect to that, the number of times (2) holds is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Fix j ∈ [1, d − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' If (2) holds, then there exists a component C in Yj \\ P which has only neighbours on some path, say P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , vr be its neighbours on P1 in order (note that r ⩾ 3 otherwise {v1, v2} is a 2-cut in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Since C is connected, there exists a path PC in C from a neighbour of v1 to a neighbour of vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' We replace P1 by the path P ′ 1 following P1 up to v1, then taking PC, and following P1 starting from vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Since G is 3-connected, deleting v1, vr does not disconnect the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' So there is a path from C to some other path in Yj \\ {v1, vr}, say P2, and this path does not intersect P3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , Pη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Since C does not have neighbors on any Pi for i > 1, this path must go through some vi with i /∈ {1, r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Now the component of vi in Yj \\ (P ′ 1 ∪ P2 ∪ · · · ∪ Pη) has neighbors both in P ′ 1 and P2, hence satisfies (1), a contradiction with our choice of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Assume now that (3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Since the sequence is 2-nested, Yj ∩ P1 is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let v be the first vertex of P1 in Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Note that Yj ∩ P1 is induced, otherwise we can replace it by a shorter subpath and end up in case (1) or (2), a contradiction with our choice of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Therefore, since v has degree at least 5, v must have at least three neighbors outside of P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' They should lie on exactly one other path since Yj \\ P is empty and (1) does not hold, say P2 by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let w be the first neighbor of v in P2 ∩ Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Note that w does not belong to Cj nor Cj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Assume v, w have only neighbours on P1 ∪ P2, and denote by w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , wt the neighbors of v in P2 (following their order in P2) and by v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , vs the neighbors of w in P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Note that s, t ⩾ 3 and that v ∈ {v1, v2} and w ∈ {w1, w2} depending on whether v1 or w1 lie in Cj or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Suppose that for some i > 2, vi /∈ N[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let P −v 1 be the subpath of P1 where we delete all vertices after v and let P +vi 1 be the subpath of P1 where we delete all vertices before vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Similarly, let P −w 2 be the subpath of P2 where we delete all vertices after w and let P +wt 2 be the subpath of P2 where we delete all vertices before wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Now replacing P1 by P −v 1 P +wt 2 and P2 by P −w 2 P +vi 1 gives a new collection of η internally disjoint paths from S to T where the total length of the paths in the section j has been reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' So (3) does not hold anymore and the number of sections satisfying (1) or (2) has increased, a contradiction with the choice of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Suppose now that vi ∈ N[v] and by symmetry wi ∈ N[w] for all i > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Recall that P1 is induced, hence vvi cannot be an edge for i ⩾ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Therefore, s = 3 and v = v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' By symmetry, we also get t = 3 and w = w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Now, if vv1 is not an edge, then we can again replace P1 by P −v1 1 P +w 2 and P2 by P −w1 2 P +v 1 and get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Hence vv1 is an edge and by symmetry ww1 is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Therefore, v and w have both degree 5 and are twins, which violates the hypothesis on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Therefore, we may assume that (1) holds for every j ∈ [1, d−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' By the pigeonhole principle, there is a, b ∈ [1, η] with a ̸= b and at least 2ℓ paths of length at least 2 between Pa and Pb with internal nodes outside of Pa ∪ Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Since the collection of cuts is 2-nested, these paths can only intersect on the vertices of the cuts and then at least ℓ of them are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Therefore contracting S ∪ T ∪ Pa and Pb into two vertices yields a K2,ℓ minor in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 4 We now conclude this section with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1, obtained by combining Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let d = ℓ3, nℓ ⩾ (7ℓ)2·(d+1)ℓ and G be a graph satisfying the hypothesis of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' We may assume that G has degree at most 7ℓ otherwise we get a K2,ℓ-minor by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Therefore, there must be two vertices s, t at distance at least 2 · (d + 1)ℓ in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let us denote by η(s, t) the st-connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let Vi be the vertices at distance i to s and let Ci ⊆ Vi be an inclusion-wise minimal st-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' In V \\ Ci, denote by Si (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Ti) the set of vertices in the connected component of s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let k be the maximum integer such that k ⩽ ℓ and there exist Ci, Cj at distance j − i ⩾ 2 · (d + 1)ℓ−k such that η(Ci, Cj) ⩾ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' If k = ℓ, we obtain a K2,ℓ-minor by contracting Si and Tj into two vertices and contracting the internal vertices of each of the ℓ pairwise disjoint paths of length at least 2 between Si and Tj (and possibly deleting some edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Hence we can assume that k ⩽ ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Let Ci and Cj be at distance j − i ⩾ 2 · (d + 1)ℓ−k, and let d′ = 2 · (d + 1)ℓ−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' We consider the d + 1 cuts Ci, Ci+d′, Ci+2d′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' , Ci+d·d′ (which are pairwise at distance at least d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' By maximality of k, we cannot have η(Ci, Cj) > k, nor η(Ci+i′d′, Ci+(i′+1)d′) > k for any 0 ⩽ i′ ⩽ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Hence η(Vi+i′d′, Vi+(i′+1)d′) = k = η(Ci, Cj) for each i′ ⩽ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' But then there exists a cut of size k between Ci+i′d′, Ci+(i′+1)d′ for each 0 ⩽ i′ ⩽ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' These d k-cuts between Ci and Cj are 2-nested, hence by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1, G then contains a K2,ℓ-minor, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' One can easily adapt the proofs of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1 to show that every large enough 3-connected graph G with minimum degree 5 either contains a K2,ℓ-minor or contains the 2 × 4 king’s graph2 as an induced subgraph where the middle vertices have degree 5 in G (the current proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content='1 only ensures the existence of a 2×3 king’s graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Contracting the K4 in the middle into a single edge to obtain a 2 × 3 king’s graph as in Figure 2 keeps the minimum degree and the connectivity of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Hence, we can obtain the following weakening of the result by Chudnovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' [1]: every 3-connected K2,ℓ-minor free graph of minimum degree 5 contains at most 5 2n + c(ℓ) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Figure 2: Contracting the red thick edges reduces a 2 × 4 king’s graph to a 2 × 3 king’s graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' Chudnovsky, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 2A king’s graph is a grid with diagonals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9A0T4oBgHgl3EQfMf9w/content/2301.02133v1.pdf'} diff --git a/w9FIT4oBgHgl3EQfzit1/vector_store/index.pkl b/w9FIT4oBgHgl3EQfzit1/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7c0372244622c4ab198a5132f28a310054597e9e --- /dev/null +++ b/w9FIT4oBgHgl3EQfzit1/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cdb515458c106ee4c229ee1ee20720ca752d33e43752abaa4b51e9964d558a9f +size 199969 diff --git a/wNFPT4oBgHgl3EQfOzT5/content/tmp_files/2301.13036v1.pdf.txt b/wNFPT4oBgHgl3EQfOzT5/content/tmp_files/2301.13036v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e76b2bc736af5ae427b32d3c454c07219d00ac1e --- /dev/null +++ b/wNFPT4oBgHgl3EQfOzT5/content/tmp_files/2301.13036v1.pdf.txt @@ -0,0 +1,521 @@ +Federated Learning for Water Consumption +Forecasting in Smart Cities +Mohammed El Hanjri‡, Hibatallah Kabbaj‡, Abdellatif Kobbane‡, Amine Abouaomar∗ +‡ENSIAS, Mohammed V University in Rabat, Morocco. +∗School of Science and Engineering, Al Akhawayn University in Ifrane, Morocco. +Abstract—Water consumption remains a major concern among +the world’s future challenges. For applications like load moni- +toring and demand response, deep learning models are trained +using enormous volumes of consumption data in smart cities. +On the one hand, the information used is private. For instance, +the precise information gathered by a smart meter that is a +part of the system’s IoT architecture at a consumer’s residence +may give details about the appliances and, consequently, the +consumer’s behavior at home. On the other hand, enormous data +volumes with sufficient variation are needed for the deep learning +models to be trained properly. This paper introduces a novel +model for water consumption prediction in smart cities while +preserving privacy regarding monthly consumption. The pro- +posed approach leverages federated learning (FL) as a machine +learning paradigm designed to train a machine learning model in +a distributed manner while avoiding sharing the users data with a +central training facility. In addition, this approach is promising to +reduce the overhead utilization through decreasing the frequency +of data transmission between the users and the central entity. +Extensive simulation illustrate that the proposed approach shows +an enhancement in predicting water consumption for different +households. +Index Terms—Federated learning, edge learning, water con- +sumption prediction +I. INTRODUCTION +In order to develop smart cities, load forecasting is a +crucial step. Long-term consumption forecasts of various +vital resources are essential for infrastructure management +and planning [1], [2]. In addition, short and medium term +consumption forecasts are critical for system performance. The +daily operational efficiency of water distribution, in particular, +requires accurate short-term forecasting, which relies on the +collection and analysis of large volumes of data from the +various domains [3]. However, the forecasting of individual +short-term loads is a challenging task to accomplish due to the +high volatility of user profiles. Indeed, the water demand of a +household or building is strongly influenced by the behavior +of its residents, which is too stochastic and therefore difficult +to anticipate. +To predict load and consumption, different approaches have +been proposed [4], including linear and dynamic program- +ming, heuristic methods, game theory, and fuzzy methods +[5]. The problem is that existing centralized and decentralized +approaches do not consider online solutions for large-scale +real-world databases, which is usually the case for residential +consumption [6], [7]. Due to large amounts of data from water +meters, these approaches generate notable slowdowns, which +are not optimal for real-time applications. In the era of big +data, many machine learning methods are emerging as suitable +for overcoming this limitation by automatically exploiting, +controlling, and optimizing load prediction models [8]. This +can be achieved by applying successive transformations to +historical data to train robust machine learning models to +accommodate the high uncertainty of water patterns, and +control demand based on load patterns and market cost signals. +By utilizing the caching capabilities coupled with the Fog +characteristics, the authors provide a green method to enhance +performance of the energy efficiency in a fog computing envi- +ronment and Internet of Everything (IoE) devices. To handle +distributed networking, processing, and storage resources, they +make use of fog computing. As content requesters with limited +resources, IoT devices will make use of cutting-edge caching +techniques to deliver high-quality services that demand a lot +of computational power and a high throughput [9]–[11]. +Recently, the machine learning community proposed Fed- +erated Learning (FL) as a novel on-device method to address +privacy issues while also growing the number and variety of +data sets [12]. A decentralized machine learning method called +federated learning allows any device to train a central model +without delivering any data [1], [7]. The model is initially +initialized by the server either randomly or using data that is +readily accessible to the public [13]. The model is then given +to a group of clients that have been chosen at random, where it +will be trained locally using their data. Each client updates the +model’s weights, which are then communicated to the server, +averaged, and used to update the overall model. This process +will be continued until the global model stabilizes [14]. +This paper’s primary goal is to assess how well the Fed- +erated Learning approach worked for the STLF challenge on +home water use. For time series forecasting, we use a deep +neural network called Long Short Term Memory (LSTM) +[15], which makes predictions about future measurements of +the water consumption of the home based on past ones. We +investigate a set of structures with comparable characteristics +(geographical location, type of building). Federated learning is +carried out using home grid Edge technology [16], [17]. Edge +equipment, such as smart meters or more sophisticated devices, +is frequently installed at the end of the water distribution +system to serve as a smart interface between the customer +and the water supply. We made the following contributions to +this effort, in brief: Using Edge technology in smart cities, We +first suggest an enabling architecture for FL, then we simulate +arXiv:2301.13036v1 [cs.LG] 30 Jan 2023 + +the potential accuracy gain of FL and last we use numerical +findings to simulate the potential network load gain. We also +include in these contributions the improvement in privacy +made available by decentralization and Edge computing. +The rest of this paper is organized as follows, with +section II discussing related works with a focus on load +forecasting and privacy. The system model is described in +Section III, while Section IV presents the simulation and +numerical findings. Section V serves as the paper’s conclusion. +II. RELATED WORK +Deep neural networks, especially in particular Long-short +term memory (LSTM), have been employed extensively in +recent research to address the difficulty of short-term load +forecasting. However, the results fall short of the required +level of precision in terms of Mean Average Percentage +Error (MAPE) and Root Mean Square Error (RMSE), despite +benchmarks demonstrating LSTM’s capability in comparison +to other methods [18]. The authors of [19] suggest using a +sequence-to-sequence LSTM version of LSTM to increase +forecasting accuracy. This variant performs better for data with +a one-minute resolution but does not significantly outperform +regular LSTM for data with an hour resolution. Additionally, +some writers [20] employ the evolutionary algorithm to solve +the challenge of selecting the optimal LSTM network because +they view it as a hyperparameter tuning problem. +According to some other studies, the issue is actually with +data-driven forecasting models capacity to generalize, not just +with neural network architecture. In fact, when applied to fresh +datasets, several of the presented models accuracy decreases +[18]. +Grouping data from several clients is another method for +enhancing the training data. By grouping people with com- +parable profiles, the authors of [21] are able to lower the +variation of uncertainty within groups. To solve the overfitting +issue, authors in [22] suggest a pooling strategy that broadens +the diversity of the data. However, these approaches are very +centralized and prone to privacy problems. +In the proposed study, we recommend using the federated +learning strategy to carry out client selection and neural +network training at the edge. This will allow the use of data +to build a reference model without sacrificing resident privacy. +In order to do this, the Home Area Network (HAN) Edge +Equipment is utilized. +III. SYSTEM MODEL +We suggest the network architecture shown in Figure 1, +which has a Multi-access Edge Computing (MEC) server +and users as its two primary parts. The users are buildings +having Edge equipment made up primarily of smart meters +and other HAN devices. For STLF, FL is utilized to create a +global model based on LSTM. The MEC server orchestrates +the training rounds, which are carried out by the users using +information about their local water consumption. We go into +greater detail regarding LSTM and its use in forecasting, as +well as FL and its use in our system model, in this part. +A. Time series forecasting using LSTM +In this work, the time series forecasting technique with +LSTM is used to predict the future water load. A time series is +an organized collection of equally spaced data sets that shows +how a certain variable has changed over time. By modeling +the connections between the points of the current data and the +points of the past, time series forecasting is made possible, +but the precision of the forecasts is greatly influenced by the +model that is used and the caliber of the historical data points. +Recurrent neural networks (RNNs) like LSTM are more ef- +fective than typical RNNs and fundamentally different from +conventional feedforward RNNs. The strength of LSTM is +sequence learning. It can resolve vanishing and exploding +gradient difficulties, which are frequent in RNNs, and establish +temporal correlations between earlier data points and the +current situation. Gradient exploding relates to the opposite +process, whereas gradient vanishing describes how the norm +of the gradient for long-term components gets less, leading +weights to never change at lower levels. This is made possible +by two of its essential elements: the memory cell, which +is utilized to recall significant prior events, and the gates, +which control the information flow. The input gate, output +gate, and forget gate are the three gates that make up an +LSTM. During the learning process, they acquire the skill +of clearing the memory of irrelevant data. Utilizing LSTM +and its derivatives, nearly all state-of-the-art outcomes in +sequence learning are accomplished. For residential STLF, it +is anticipated that the LSTM network will be able to track the +states of the inhabitants, abstract some states from the provided +consumption profile, and forecast future consumption using the +learned data. +B. Federated Learning +A type of machine learning called federated learning in- +volves distributing the majority of the training process across +various clients, or computer devices. On mobile device key- +boards for next word prediction, Google first put out and +executed the idea [23]. For many situations, this strategy is +ideal: 1) In highly distributed systems, when there are orders of +magnitude more devices than data center nodes, 2) When data +privacy is important, and 3) When data is huge in comparison +to model updates. In cases when datasets are uneven or not +dispersed evenly, federated learning has been shown to be quite +effective. +Following is an example of federated learning iteration: First, +the current model is given to each client in a selected subset. +In this instance, clients are housed using Edge equipment (e.g. +smart meters). On locally stored data, a few clients choose to +compute Stochastic Gradient Descent (SGD) updates. A server +then integrates the client updates to produce a new global +model. A different group of clients is sent the updated model. +Until the necessary forecast accuracy is attained, this process +is repeated. Algorithm 1 describes the procedures in detail. + +Fig. 1. Network architecture and components +The server employs the Federated Averaging algorithm to +integrate the client updates [12]. The starting global model is +first randomly initialized or trained beforehand using publicly +accessible data. A subset S of clients with sufficient data +records and diverse enough consumption loads to augment the +training data are sent a global model ωt by the server in each +training round t. This requirement was introduced to make sure +we have enough variety among the data points to accurately +depict the occupants’ typical consumption. Each client s in +the subset then employs ms samples from its own local data. +In our case, the volume is based on the amount of data that +is locally stored and the length of time that the smart meter +has been providing data. The dataset used consists of sliding +windows with a predetermined number of look-back steps. The +average gradient gs is then calculated by each client s using +SGD, with a learning rate η. The server receives the updated +models ωs and aggregates them. +ωt+1 = +S +� +s=0 +ms +M ωs +t+1 +(1) +The centralized strategy might not, however, account for +every user’s water consumption. Personalization is one sug- +gestion for resolving this issue. Many applications, which call +for analyzing and responding to user behavior, are focused on +personalization. To create a customized model for each user, +the centralized model is retrained using user-specific data. This +can be accomplished by retraining the model locally over a +Algorithm 1: Federated Averaging Algorithm. tmax is +the maximum number of rounds and M = � +s ms. +Initialize the model in training round t = 0; +while t ≤ tmax do +Select subset S of clients; +for client s in S do +if ∆(monthlyload) > threshold then +s obtains model ωt; +s calculates average gradient gs with SGD; +s updates local model ωs +t+1 ← ωs +t − ηgs; +s communicates updated model to server; +end +end +server updates global model using the formula in +eq (1); begin the following round t ← t + 1. +end +limited number of epochs using only the customer’s data. +Since user identities are still at risk and can be ascertained +through reverse engineering, federated learning poses less of a +privacy threat than centralized server storage. Weight changes +are processed in memory and are deleted after aggregation, +whereas model updates received by each client are transient +and never retained on the server. Individual weight uploads +must not be reviewed or assessed according to the federated +learning approach. Because the network and server cannot be + +Cloud layer +Cloud infrastructure +Cloud aggregation +AC=WUW2U...UWN +Uplink comm. +Edge layer +Downlink comm. +Edge aggregation +gNodeB +Awe = wi Uw2 U.. UwN +Water counter +Local computing +devices +W2 +WN +w) +layer +End users' +Model 1 +Model 2 +Model N +House 1 +House 2 +House N2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +Time (Month) +0 +500 +1000 +1500 +2000 +Consumption (HCF) +Water Consumption of building 04 (BLD 04) +Fig. 2. The distribution of water consumption data in building 04 (BLD 04). (HCF; Hundred-Cubic Foot) +trusted with precise user data, this is nevertheless more secure +than server training. For billing purposes, some data must still +be supplied in an aggregated form, but these data are not +particularly revealing. +C. Reward in the network load +First, we establish the network load QC for a server in +centralized training in Eq. 1 and the network load in FL QF +in Eq. 2, in order to compare the reward in network load in +FL to centralized training. +ℓG is the length of the global model, and ℓs is the length of the +data that the client s supplied. The number of hops between +client s and the server in the centralized training is denoted +by hs. +QC = +M +� +s=1 +ℓs × hs +(2) +and +QF = ℓG × +tmax +� +t=1 +S +� +s=1 +hs,t +(3) +where S is the total number of users in each batch, and hs,t +is the number of hops between the client s chosen in round t +and the server. +We define the reward in networking load using Eq.1 and Eq.2, +as follows: +R = 1 − QF +QC +(4) +IV. SIMULATION AND RESULTS +A. Dataset Pre-Processing +This study made use of data from the Kaggle website. +Data about New York’s water consumption and costs include +Consumption and expense data broken down by borough and +development monthly. Utility vendor and meter information is +included in the data set. For over 300 buildings in the New +York city, New York Water data provides water consumption +and cost statistics at ”monthly” intervals between 2013 and +2020. +B. Numerical Results +For deep learning models to perform the best in forecasting, +hyper-parameter adjustment is crucial. However, we only +evaluate the federated learning approach in this paper. +This +section +examines +how +personalization +affects +the +efficiency of the models. First, we examine if localizing +the model’s training for the participating clients improves +outcomes. +We chose a client (building; BLD 04) randomly from +the sample of participants to show how personalization can +improve predictions. The distribution of water consumption +data in building 04 (BLD 04) between 2013 and 2020 is given +as shown in Figure 2. +We have used the matching personalized models as well as the +global model. Figure 3 illustrates the actual and anticipated +consumption profiles between 2018 and 2020. +Using just a portion of the users that make up the popu- +lation, we come to the conclusion that it is possible to train +robust models for the consumption profiles of that popula- +tion. Retraining the model can produce a customized model +that better follows the profile’s characteristics and produces +predictions with higher accuracy for applications with high +accuracy needs. Even said, for brand-new customers lacking +sufficient data for customisation, the predictions made using +the global model can be a useful starting point. + +2018-04 +2018-07 +2018-10 +2019-01 +2019-04 +2019-07 +2019-10 +2020-01 +2020-04 +Time (Month) +150 +200 +250 +300 +350 +400 +450 +Consumption (HCF) +real data of consumption +forcasts for next month consumption using FL +forcasts for next month consumption with global model +Fig. 3. predictions for the client’s (BLD 04) consumption for the following month for those who took part in training the global model. +(HCF; Hundred-Cubic Foot) +The edge devices ability to provide local training is a necessity +for the suggested approach’s feasibility. The performance of +new IoT devices will very certainly be compromised by +training a neural network, even though they have the CPU +power to execute complex machine learning models. +V. CONCLUSION +Given +the +stochastic +nature +of +consumption +profiles, +projecting individual short-term loads is a difficult challenge. +For the purpose of short-term load forecasting in smart +cities, we proposed a system model in this research that +uses federated learning to address issues with privacy and +data diversity. Contrary to centralized techniques, federated +learning in the proposed system trains models on edge +devices, limiting security concerns to those specific to the +device. We ran tests to assess how well centralized and +customized models performed in federated contexts. The +simulation results show that it is a promising approach +for developing high-performing models with a significantly +reduced networking load than a centralised model, while +preserving the privacy of consumption data. +REFERENCES +[1] A. Tak and S. Cherkaoui, “Federated edge learning: Design issues and +challenges,” IEEE Network, vol. 35, no. 2, pp. 252–258, 2021. +[2] A. Ta¨ık and S. 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Manic, “Building energy load +forecasting using deep neural networks,” in IECON 2016-42nd Annual +Conference of the IEEE Industrial Electronics Society. +IEEE, 2016, +pp. 7046–7051. +[20] A. Almalaq and J. J. Zhang, “Evolutionary deep learning-based energy +consumption prediction for buildings,” ieee access, vol. 7, pp. 1520– +1531, 2018. +[21] B. Stephen, X. Tang, P. R. Harvey, S. Galloway, and K. I. Jennett, +“Incorporating practice theory in sub-profile models for short term +aggregated residential load forecasting,” IEEE Transactions on Smart +Grid, vol. 8, no. 4, pp. 1591–1598, 2015. +[22] H. Shi, M. Xu, and R. Li, “Deep learning for household load fore- +casting—a novel pooling deep rnn,” IEEE Transactions on Smart Grid, +vol. 9, no. 5, pp. 5271–5280, 2017. +[23] A. Hard, K. Rao, R. Mathews, S. Ramaswamy, F. Beaufays, S. Augen- +stein, H. Eichner, C. Kiddon, and D. Ramage, “Federated learning for +mobile keyboard prediction,” arXiv preprint arXiv:1811.03604, 2018. + diff --git a/wNFPT4oBgHgl3EQfOzT5/content/tmp_files/load_file.txt b/wNFPT4oBgHgl3EQfOzT5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8fbd6e2da3a394fa4b727a27958df1619541a23d --- /dev/null +++ b/wNFPT4oBgHgl3EQfOzT5/content/tmp_files/load_file.txt @@ -0,0 +1,354 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf,len=353 +page_content='Federated Learning for Water Consumption Forecasting in Smart Cities Mohammed El Hanjri‡, Hibatallah Kabbaj‡, Abdellatif Kobbane‡, Amine Abouaomar∗ ‡ENSIAS, Mohammed V University in Rabat, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' ∗School of Science and Engineering, Al Akhawayn University in Ifrane, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Abstract—Water consumption remains a major concern among the world’s future challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For applications like load moni- toring and demand response, deep learning models are trained using enormous volumes of consumption data in smart cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' On the one hand, the information used is private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For instance, the precise information gathered by a smart meter that is a part of the system’s IoT architecture at a consumer’s residence may give details about the appliances and, consequently, the consumer’s behavior at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' On the other hand, enormous data volumes with sufficient variation are needed for the deep learning models to be trained properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This paper introduces a novel model for water consumption prediction in smart cities while preserving privacy regarding monthly consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The pro- posed approach leverages federated learning (FL) as a machine learning paradigm designed to train a machine learning model in a distributed manner while avoiding sharing the users data with a central training facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In addition, this approach is promising to reduce the overhead utilization through decreasing the frequency of data transmission between the users and the central entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Extensive simulation illustrate that the proposed approach shows an enhancement in predicting water consumption for different households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Index Terms—Federated learning, edge learning, water con- sumption prediction I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' INTRODUCTION In order to develop smart cities, load forecasting is a crucial step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Long-term consumption forecasts of various vital resources are essential for infrastructure management and planning [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In addition, short and medium term consumption forecasts are critical for system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The daily operational efficiency of water distribution, in particular, requires accurate short-term forecasting, which relies on the collection and analysis of large volumes of data from the various domains [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' However, the forecasting of individual short-term loads is a challenging task to accomplish due to the high volatility of user profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Indeed, the water demand of a household or building is strongly influenced by the behavior of its residents, which is too stochastic and therefore difficult to anticipate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' To predict load and consumption, different approaches have been proposed [4], including linear and dynamic program- ming, heuristic methods, game theory, and fuzzy methods [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The problem is that existing centralized and decentralized approaches do not consider online solutions for large-scale real-world databases, which is usually the case for residential consumption [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Due to large amounts of data from water meters, these approaches generate notable slowdowns, which are not optimal for real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In the era of big data, many machine learning methods are emerging as suitable for overcoming this limitation by automatically exploiting, controlling, and optimizing load prediction models [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This can be achieved by applying successive transformations to historical data to train robust machine learning models to accommodate the high uncertainty of water patterns, and control demand based on load patterns and market cost signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' By utilizing the caching capabilities coupled with the Fog characteristics, the authors provide a green method to enhance performance of the energy efficiency in a fog computing envi- ronment and Internet of Everything (IoE) devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' To handle distributed networking, processing, and storage resources, they make use of fog computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' As content requesters with limited resources, IoT devices will make use of cutting-edge caching techniques to deliver high-quality services that demand a lot of computational power and a high throughput [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Recently, the machine learning community proposed Fed- erated Learning (FL) as a novel on-device method to address privacy issues while also growing the number and variety of data sets [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' A decentralized machine learning method called federated learning allows any device to train a central model without delivering any data [1], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The model is initially initialized by the server either randomly or using data that is readily accessible to the public [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The model is then given to a group of clients that have been chosen at random, where it will be trained locally using their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Each client updates the model’s weights, which are then communicated to the server, averaged, and used to update the overall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This process will be continued until the global model stabilizes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This paper’s primary goal is to assess how well the Fed- erated Learning approach worked for the STLF challenge on home water use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For time series forecasting, we use a deep neural network called Long Short Term Memory (LSTM) [15], which makes predictions about future measurements of the water consumption of the home based on past ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' We investigate a set of structures with comparable characteristics (geographical location, type of building).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Federated learning is carried out using home grid Edge technology [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Edge equipment, such as smart meters or more sophisticated devices, is frequently installed at the end of the water distribution system to serve as a smart interface between the customer and the water supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' We made the following contributions to this effort, in brief: Using Edge technology in smart cities, We first suggest an enabling architecture for FL, then we simulate arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content='13036v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content='LG] 30 Jan 2023 the potential accuracy gain of FL and last we use numerical findings to simulate the potential network load gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' We also include in these contributions the improvement in privacy made available by decentralization and Edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The rest of this paper is organized as follows, with section II discussing related works with a focus on load forecasting and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The system model is described in Section III, while Section IV presents the simulation and numerical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Section V serves as the paper’s conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' RELATED WORK Deep neural networks, especially in particular Long-short term memory (LSTM), have been employed extensively in recent research to address the difficulty of short-term load forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' However, the results fall short of the required level of precision in terms of Mean Average Percentage Error (MAPE) and Root Mean Square Error (RMSE), despite benchmarks demonstrating LSTM’s capability in comparison to other methods [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The authors of [19] suggest using a sequence-to-sequence LSTM version of LSTM to increase forecasting accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This variant performs better for data with a one-minute resolution but does not significantly outperform regular LSTM for data with an hour resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Additionally, some writers [20] employ the evolutionary algorithm to solve the challenge of selecting the optimal LSTM network because they view it as a hyperparameter tuning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' According to some other studies, the issue is actually with data-driven forecasting models capacity to generalize, not just with neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In fact, when applied to fresh datasets, several of the presented models accuracy decreases [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Grouping data from several clients is another method for enhancing the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' By grouping people with com- parable profiles, the authors of [21] are able to lower the variation of uncertainty within groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' To solve the overfitting issue, authors in [22] suggest a pooling strategy that broadens the diversity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' However, these approaches are very centralized and prone to privacy problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In the proposed study, we recommend using the federated learning strategy to carry out client selection and neural network training at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This will allow the use of data to build a reference model without sacrificing resident privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In order to do this, the Home Area Network (HAN) Edge Equipment is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' SYSTEM MODEL We suggest the network architecture shown in Figure 1, which has a Multi-access Edge Computing (MEC) server and users as its two primary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The users are buildings having Edge equipment made up primarily of smart meters and other HAN devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For STLF, FL is utilized to create a global model based on LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The MEC server orchestrates the training rounds, which are carried out by the users using information about their local water consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' We go into greater detail regarding LSTM and its use in forecasting, as well as FL and its use in our system model, in this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Time series forecasting using LSTM In this work, the time series forecasting technique with LSTM is used to predict the future water load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' A time series is an organized collection of equally spaced data sets that shows how a certain variable has changed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' By modeling the connections between the points of the current data and the points of the past, time series forecasting is made possible, but the precision of the forecasts is greatly influenced by the model that is used and the caliber of the historical data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Recurrent neural networks (RNNs) like LSTM are more ef- fective than typical RNNs and fundamentally different from conventional feedforward RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The strength of LSTM is sequence learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' It can resolve vanishing and exploding gradient difficulties, which are frequent in RNNs, and establish temporal correlations between earlier data points and the current situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Gradient exploding relates to the opposite process, whereas gradient vanishing describes how the norm of the gradient for long-term components gets less, leading weights to never change at lower levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This is made possible by two of its essential elements: the memory cell, which is utilized to recall significant prior events, and the gates, which control the information flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The input gate, output gate, and forget gate are the three gates that make up an LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' During the learning process, they acquire the skill of clearing the memory of irrelevant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Utilizing LSTM and its derivatives, nearly all state-of-the-art outcomes in sequence learning are accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For residential STLF, it is anticipated that the LSTM network will be able to track the states of the inhabitants, abstract some states from the provided consumption profile, and forecast future consumption using the learned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Federated Learning A type of machine learning called federated learning in- volves distributing the majority of the training process across various clients, or computer devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' On mobile device key- boards for next word prediction, Google first put out and executed the idea [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For many situations, this strategy is ideal: 1) In highly distributed systems, when there are orders of magnitude more devices than data center nodes, 2) When data privacy is important, and 3) When data is huge in comparison to model updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In cases when datasets are uneven or not dispersed evenly, federated learning has been shown to be quite effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Following is an example of federated learning iteration: First, the current model is given to each client in a selected subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In this instance, clients are housed using Edge equipment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' smart meters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' On locally stored data, a few clients choose to compute Stochastic Gradient Descent (SGD) updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' A server then integrates the client updates to produce a new global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' A different group of clients is sent the updated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Until the necessary forecast accuracy is attained, this process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Algorithm 1 describes the procedures in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Network architecture and components The server employs the Federated Averaging algorithm to integrate the client updates [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The starting global model is first randomly initialized or trained beforehand using publicly accessible data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' A subset S of clients with sufficient data records and diverse enough consumption loads to augment the training data are sent a global model ωt by the server in each training round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This requirement was introduced to make sure we have enough variety among the data points to accurately depict the occupants’ typical consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Each client s in the subset then employs ms samples from its own local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' In our case, the volume is based on the amount of data that is locally stored and the length of time that the smart meter has been providing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The dataset used consists of sliding windows with a predetermined number of look-back steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The average gradient gs is then calculated by each client s using SGD, with a learning rate η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The server receives the updated models ωs and aggregates them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' ωt+1 = S � s=0 ms M ωs t+1 (1) The centralized strategy might not, however, account for every user’s water consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Personalization is one sug- gestion for resolving this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Many applications, which call for analyzing and responding to user behavior, are focused on personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' To create a customized model for each user, the centralized model is retrained using user-specific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This can be accomplished by retraining the model locally over a Algorithm 1: Federated Averaging Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' tmax is the maximum number of rounds and M = � s ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Initialize the model in training round t = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' while t ≤ tmax do Select subset S of clients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' for client s in S do if ∆(monthlyload) > threshold then s obtains model ωt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' s calculates average gradient gs with SGD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' s updates local model ωs t+1 ← ωs t − ηgs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' s communicates updated model to server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' end end server updates global model using the formula in eq (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' begin the following round t ← t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' end limited number of epochs using only the customer’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Since user identities are still at risk and can be ascertained through reverse engineering, federated learning poses less of a privacy threat than centralized server storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Weight changes are processed in memory and are deleted after aggregation, whereas model updates received by each client are transient and never retained on the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Individual weight uploads must not be reviewed or assessed according to the federated learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Because the network and server cannot be Cloud layer Cloud infrastructure Cloud aggregation AC=WUW2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content='UWN Uplink comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Edge layer Downlink comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Edge aggregation gNodeB Awe = wi Uw2 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=". UwN Water counter Local computing devices W2 WN w) layer End users' Model 1 Model 2 Model N House 1 House 2 House N2013 2014 2015 2016 2017 2018 2019 2020 Time (Month) 0 500 1000 1500 2000 Consumption (HCF) Water Consumption of building 04 (BLD 04) Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The distribution of water consumption data in building 04 (BLD 04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' (HCF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Hundred-Cubic Foot) trusted with precise user data, this is nevertheless more secure than server training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For billing purposes, some data must still be supplied in an aggregated form, but these data are not particularly revealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Reward in the network load First, we establish the network load QC for a server in centralized training in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' 1 and the network load in FL QF in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' 2, in order to compare the reward in network load in FL to centralized training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' ℓG is the length of the global model, and ℓs is the length of the data that the client s supplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The number of hops between client s and the server in the centralized training is denoted by hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' QC = M � s=1 ℓs × hs (2) and QF = ℓG × tmax � t=1 S � s=1 hs,t (3) where S is the total number of users in each batch, and hs,t is the number of hops between the client s chosen in round t and the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' We define the reward in networking load using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content='1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content='2, as follows: R = 1 − QF QC (4) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' SIMULATION AND RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Dataset Pre-Processing This study made use of data from the Kaggle website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Data about New York’s water consumption and costs include Consumption and expense data broken down by borough and development monthly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Utility vendor and meter information is included in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For over 300 buildings in the New York city, New York Water data provides water consumption and cost statistics at ”monthly” intervals between 2013 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Numerical Results For deep learning models to perform the best in forecasting, hyper-parameter adjustment is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' However, we only evaluate the federated learning approach in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' This section examines how personalization affects the efficiency of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' First, we examine if localizing the model’s training for the participating clients improves outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' We chose a client (building;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' BLD 04) randomly from the sample of participants to show how personalization can improve predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The distribution of water consumption data in building 04 (BLD 04) between 2013 and 2020 is given as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' We have used the matching personalized models as well as the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Figure 3 illustrates the actual and anticipated consumption profiles between 2018 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Using just a portion of the users that make up the popu- lation, we come to the conclusion that it is possible to train robust models for the consumption profiles of that popula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Retraining the model can produce a customized model that better follows the profile’s characteristics and produces predictions with higher accuracy for applications with high accuracy needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Even said, for brand-new customers lacking sufficient data for customisation, the predictions made using the global model can be a useful starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' 2018-04 2018-07 2018-10 2019-01 2019-04 2019-07 2019-10 2020-01 2020-04 Time (Month) 150 200 250 300 350 400 450 Consumption (HCF) real data of consumption forcasts for next month consumption using FL forcasts for next month consumption with global model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' predictions for the client’s (BLD 04) consumption for the following month for those who took part in training the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' (HCF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Hundred-Cubic Foot) The edge devices ability to provide local training is a necessity for the suggested approach’s feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The performance of new IoT devices will very certainly be compromised by training a neural network, even though they have the CPU power to execute complex machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' CONCLUSION Given the stochastic nature of consumption profiles, projecting individual short-term loads is a difficult challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' For the purpose of short-term load forecasting in smart cities, we proposed a system model in this research that uses federated learning to address issues with privacy and data diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Contrary to centralized techniques, federated learning in the proposed system trains models on edge devices, limiting security concerns to those specific to the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' We ran tests to assess how well centralized and customized models performed in federated contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' The simulation results show that it is a promising approach for developing high-performing models with a significantly reduced networking load than a centralised model, while preserving the privacy of consumption data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': 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and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content=' Ramage, “Federated learning for mobile keyboard prediction,” arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} +page_content='03604, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNFPT4oBgHgl3EQfOzT5/content/2301.13036v1.pdf'} diff --git a/y9FJT4oBgHgl3EQfiSxv/content/tmp_files/2301.11569v1.pdf.txt b/y9FJT4oBgHgl3EQfiSxv/content/tmp_files/2301.11569v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2590a06d95ae377539f50108285b4a3c8f38bab --- /dev/null +++ b/y9FJT4oBgHgl3EQfiSxv/content/tmp_files/2301.11569v1.pdf.txt @@ -0,0 +1,468 @@ + +Vulnerability Analysis of Azure Blockchain +Workbench Key Management System + +Dmitry Tanana +Laboratory of Combinatorial Algebra +Ural Federal University +Yekaterinburg, Russia +ddtanana@urfu.ru + +Abstract—With rise of blockchain popularity, more and +more people seek to implement blockchain technology into their +projects. Most common way is to take existing blockchain stack, +such as Azure Blockchain Workbench or Oracle Blockchain +Platform. While the blockchain technology is well-protected by +its algorithms it is still vulnerable because its privacy relies on +regular cryptography. And mistakes or vulnerabilities in key +management protocols can affect even the most secure +blockchain projects. This article considers question of +vulnerabilities within Azure Blockchain Workbench key +management system. We describe potential threats for each +stage of key management lifecycle based on public reports and +then assess how likely are those threats to realize within Azure +Blockchain Workbench environment based on the technical +documentation for Azure Blockchain Workbench and Azure +Key Vault. Finally, we compile results of our assessment into the +key management threat table with three distinct degrees of +protection – fully protected, partially protected and not +protected. +Keywords—Cryptography, key management, threat analysis, +vulnerability analysis, Azure Blockchain Workbench +I. INTRODUCTION +Nowadays one of the most discussed topics in the field of +information technology are distributed ledger systems, also +known as blockchain. Such systems are being introduced +around the world in many areas including medicine, +education, finance and many others. +End user security in those systems relies primarily on +cryptographic encryption, therefore the blockchain is only as +secure as the encryption keys used. Indeed, many well-known +incidents in which data from blockchain was somehow +disclosed, happened only because of key mismanagement [1]. +Therefore, the safety of information in distributed ledger +systems directly depends on the key strength and the +effectiveness of mechanisms and protocols associated with +their management. Our paper will focus on the vulnerability +analysis of blockchain key management systems on the +example of Azure Blockchain Workbench. To the best of our +knowledge, this would be the first analysis of this type for +Azure Blockchain Workbench. +II. CRYPTOGRAPHIC KEY LIFECYCLE STAGES AND THREATS +A. Lifecycle +In order to evaluate specific security measures, we must +consider the whole lifecycle of cryptographic key within the +distributed ledger system. +The cryptographic key lifecycle is a set of states through +which the key passes during its existence in automated system. +Cryptographic key lifecycle can be divided into following +stages [2], as seen on Fig. 1: +• Generation – key is generated with required +cryptographic properties. +• Registration – key is registered in the database and +issuing key certificate. +• Distribution – keys are distributed to the owners via +secure channel. +• User initialization – user is authenticated for the purpose +of key reception. +• Operation – keys are used to protect the information +during regular operations. +• Update – key is replaced with a new one after a certain +period of time. +• Recovery – key is restored after being lost, if it was lost +without compromise possibility. +• Cancellation – key is revoked as a result of expiration or +compromise. +• Storage and archiving – on this stage keys are stored in +proper conditions, ensuring their safety until replacement, +discontinued keys are archived and stored until their complete +removal. +• Erasure – key is completely deleted from the system after +expiration or compromise. + +Fig. 1. Block diagram of a blockchain cryptographic key lifecycle stages. + + +This work was supported by the Ministry of Science and Higher +Education of the Russian Federation, project no. FEUZ-2020-0016 . + +Generation +Registration +User initialization +Distribution +Operation +Update +Recovery +Cancellation +Storage & +archiving +ErasureB. Threats +We’ll consider potential threats for each of the stages +based on public reports about implemented attacks or +attempted attacks. +We propose the following structure for describing threats: +• Name of the threat. +• Description of the threat. +• Mitigation measures to reduce the likelihood of the +threat. +The main threats at the generation stage: +– Threat of generating weak cryptographic key [3]. This +threat is a result of unsuccessful random number generator +choice. It can produce a number which will result in +generation of weak cryptographic key. +To mitigate this threat, strength of cryptographic keys +must match the value of data it protects, and the period of +time during which the data is protected by that key. +Cryptographic key must be long enough for its purpose and +generated using high quality random number generator, +ideally the one that collects entropy from a suitable hardware +source. +The main threats at the distribution and operation stages: +– Threat of insecure distribution of cryptographic keys +[4]. Key distribution is one of the most vulnerable stages in +the whole lifecycle. If this process is not secure enough, keys +can easily become compromised. +To mitigate this threat only secure protocols should be +used for the distribution of cryptographic keys and certain +rules of conduct should be followed by end users (do not +transfer keys by open channels, etc.). +– Threat of key misuse [4]. If cryptographic keys are used +for tasks that do not correspond with its generation purpose, +that may lead to problems with maintenance of required +protection level protection. +To mitigate this threat each cryptographic key should be +generated for one specific purpose only. +– Threat of key reuse [4]. Key being reused after its +expiration can lead to compromise. +To mitigate this threat reuse of cryptographic keys should +not be allowed. For a new operation new cryptographic key +should be generated instead. +– Threat of using a single cryptographic key to encrypt +large amounts of data [4]. Using one single cryptographic key +to encrypt large amounts of data (at the same time or at +multiple instances) can make this key more vulnerable for +attacks. +To mitigate this threat a single cryptographic key should +not be used for high amount of data encryption. If necessary, +a new cryptographic key should be generated instead. +– Internal threats (attacker can have access to +cryptographic keys) [5]. If attacker has constant access to the +cryptographic keys, that access can be used for malicious +purposes or keys can be transferred and/or sold to a third +party. +To mitigate this threat, it is necessary to employ user +authentication, dual control and separation of roles. +The main threats at the storage stage: +– Threat of improper storage [6]. Any bypass attack with +filtration of protected data can compromise cryptographic +keys. +To mitigate this threat cryptographic keys must never be +stored along with data they are protecting (for example, on a +same server, in a same database, etc.). +– Threat of weak protection [6]. The cryptographic keys, +even if they are stored only in server memory, may be +vulnerable to compromise. +To mitigate this threat: regardless of the storage, keys +must be stored only in encrypted form (exception: if they are +stored in an environment which is protected from +unauthorized access). +– Threat of inaccessibility [6]. If cryptographic keys are +not available when they are required or are lost due to any +malfunction, and the backup is not available, data encrypted +with these keys may also be inaccessible or lost. +To mitigate this threat storage location of cryptographic +keys must be reliably protected from failure and backups +should be kept for the keys themselves if they need to be +recovered. +– Threat of logging absence [6]. If cryptographic key +lifecycle is not fully recorded or logged it will be more +difficult to determine when exactly the security incident +happened, which will affect the provision of security +information for other cryptographic keys in the future. +To mitigate this threat, it is necessary to keep full action +logs for cryptographic keys. The journal should be protected +from third-party access and ideally stored on a distributed +ledger so it would be impossible to remove entries from it. +The main threats at the key erasure stage: +– Threat of leaving cryptographic keys intact after their +expiration [7]. Keys which are not properly destroyed can be +compromised and used by attackers. +To mitigate this threat cryptographic keys must be +destroyed (safely removed without leaving any traces) after +their expiration, if they are not clearly required for later use +(for example, to decrypt data). +– Threat of manual management [7]. Manual management +of cryptographic keys using paper or unsuitable electronic +tools, such as Excel spreadsheets, and accompanying manual +input of cryptographic keys will lead to mistakes that can +make keys extremely vulnerable. +To mitigate this threat management of cryptographic keys +should be entrusted to an automated system, which is +designated for key management and corresponds to modern +information security requirements. +Overall block diagram of cryptographic key management +system threats with corresponding stages can be seen on Fig. +2. + + + + +Fig. 2. Block diagram of a blockchain cryptographic key lifecycle threats +with corresponding stages. +After analyzing potential threats to cryptographic keys +management process, we will consider them specifically for +the Azure Blockchain Workbench [8]. +III. AZURE BLOCKCHAIN WORKBENCH SECURITY +SPECIFICATIONS +The Azure Blockchain Workbench key management +system is based on the Azure Key Vault [9]. Blockchain +participants are using Azure Key Vault or third-party +certificate authority in order to create identifiers and +associated certificates which are required for all nodes to carry +out transactions in blockchain network. +Here are the rules and recommendations described in +Azure Blockchain Workbench documentation [8,9] for +cryptographic key management. +Generation Stage: +• Generation of public and private keys occurs according +to the EC-HSM (Elliptic Curve - Hardware Security Module) +algorithm with P-256 elliptic curve key in base version. +• If the private key is lost and cannot be restored or +updated, then it will be necessary to generate a new private +key and register a new identity at the certificate authority. +Storage stage: +• Keys are stored in the browser cache when they are +initially added to the participant's blockchain wallet so that the +console could use them to control blockchain components. +• Customers are encouraged to export keys and import +them into their own cryptographic key management system in +case they clear browser cache or switch browsers. +• Customers are responsible for storage, backup and +recovery of all exported keys. +For generating and storing a cryptographic key Azure +Blockchain Workbench also offers the use of HSM (Hardware +Security Module). It complies with the FIPS 140-2 Level 3 +standard. The blockchain network node is responsible for the +correct configuration and operation of the HSM. +Also in the HSM case the following cryptographic key +management rules are used [6]: +• The private key for the node should not be stored in +browser cache. +• The private key must be accessible from the HSM via +proxy. +• When registering administrator IDs of another node or +client application in certification authority using the console, +their private keys should not be stored in the HSM because +they need the private key to operate in the network. +IV. RESULTS AND CONCLUSION +Based on the analysis of Azure Blockchain Workbench, +technical documentation for Azure Blockchain Workbench +and Azure Key Vault and the compiled description we will +analyze threats for Azure Blockchain Workbench key +management system and assess the degree of protection from +them. +In our study we’ll use the following degrees of protection: +• Fully protected – security measures of Azure Blockchain +Workbench completely eliminate the possibility of threat. +• Partially protected – security measures of Azure +Blockchain Workbench partially (or completely, but not in all +cases) eliminate the possibility of a threat. The likelihood of a +threat is still present. +• Not protected – security measures of Azure Blockchain +Workbench for this threat do not exist or do not fulfill their +functions. High likelihood of a threat present. +The results of our threat analysis can be seen in table 1. +TABLE I. +KEY MANAGEMENT THREAT TABLE FOR AZURE +BLOCKCHAIN WORKBENCH +№ Threat +Security element of Azure +Blockchain key management +system for this threat +Degree of +protection +1 +Threat of generating +weak cryptographic +key +Generation of public and private +keys occurs according to EC-HSM +algorithm with P-256 elliptic curve +key in base version +Fully +protected +2 +Threat of key misuse Not found +Not +protected +3 +Threat of key reuse +Key management rules +Fully +protected +4 +Threat of using a +single cryptographic +key to encrypt large +amounts of data +Not found +Not +protected +5 +Threat of improper +storage +Key management system or +certified Hardware Security +Module +Fully +protected +6 +Threat of weak +protection +Key management system or +certified Hardware Security +Module +Fully +protected +7 +Threat of insecure +distribution of +cryptographic keys +Key management system or +certified Hardware Security +Module +Fully +protected + +Cryptographic key +lifecycle threats +Generation +Distribution & +Storage stage +Erasure stage +stage +operation stage +Weak key +Insecure key +Improper +Leaving keys +generation +distribution +storage +intact after +expiration +Key misuse +Weak protection +Manual +management +Key reuse +Threat of +inaccessibility +Using single key +to encrypt large +Absence of logs +amounts of data +Internal threats8 +Threat of leaving +cryptographic keys +intact after their +expiration +Not found +Not +protected +9 +Internal threats +(attacker can have +access to +cryptographic keys) +Certified Hardware Security +Module system. No protection if +Hardware Security Module is not +used +Partially +protected +10 Threat of +inaccessibility +Clients are responsible for storage, +backup and emergency recovery of +all exported keys +Partially +protected +11 Threat of logging +absence +Certified Hardware Security +Module system. No protection if +Hardware Security Module is not +used +Partially +protected +12 Threat of manual +management +Key management system or +certified Hardware Security +Module +Fully +protected + + As a result of the threat analysis for the Azure Blockchain +Workbench, we were able to assess the degree of protection +from various attack vectors within its key management system +and identify potential threats for the information security of +Azure Blockchain Workbench key management system. +We’d like to underline that for many attack vectors +protection relies solely on certified Hardware Security +Module, which might be unavailable for broader user base. +Additionally, for a few potential vulnerabilities – using single +cryptographic key to encrypt large amounts of data and +leaving cryptographic key intact after its expiration there were +no security measures at all, which means that end users must +be extra careful to prevent those. + Identified threats can be used by Azure Blockchain +developers to further mitigate them or by end users in their +risk minimization strategy. +REFERENCES + +[1] P. Gaudry, A. Golovnev, “Breaking the encryption scheme of the +Moscow +Internet +voting +system”, +Available +at: +https://arxiv.org/abs/1908.05127. +[2] S. Rafaeli, D. Hutchison, 2003, "A Survey of Key Management for +Secure Group Communication", ACM Computing Surveys, vol. 35, no. +3, pp. 309-329. +[3] F. Casino, T.K. Dasaklis, C. Patsakis,. 2019, "A systematic literature +review of blockchain-based applications: Current status, classification +and open issues", Telematics and Informatics, vol. 36, pp. 55-81. +[4] M. Ma, G. Shi, F. Li, 2019, "Privacy-Oriented Blockchain-Based +Distributed Key Management Architecture for Hierarchical Access +Control in the IoT Scenario", IEEE Access, vol. 7, pp. 34045-34059. +[5] L. Veltri, S. Cirani, S. Busanelli, G. Ferrari, 2013, "A novel batch- +based group key management protocol applied to the Internet of +Things", Ad Hoc Networks, vol. 11, no. 8, pp. 2724-2737. +[6] N.Z. Aitzhan, D. Svetinovic, 2018, "Security and Privacy in +Decentralized Energy Trading Through Multi-Signatures, Blockchain +and Anonymous Messaging Streams", IEEE Transactions on +Dependable and Secure Computing, vol. 15, no. 5, pp. 840-852. +[7] Y. Sun, W. Trappe, K.J.R. Liu, 2004, "A scalable multicast key +management +scheme +for +heterogeneous +wireless +networks", +IEEE/ACM Transactions on Networking, vol. 12, no. 4, pp. 653-666. +[8] “Azure Blockchain Workbench Architecture”, Available at: +https://docs.microsoft.com/en- +us/azure/blockchain/workbench/architecture +[9] “About +Azure +Key +Vault”, +Available +at: +https://docs.microsoft.com/en-us/azure/key-vault/general/overview + + diff --git a/y9FJT4oBgHgl3EQfiSxv/content/tmp_files/load_file.txt b/y9FJT4oBgHgl3EQfiSxv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..79533748060d017f24c0c3d7d70eea2bab031498 --- /dev/null +++ b/y9FJT4oBgHgl3EQfiSxv/content/tmp_files/load_file.txt @@ -0,0 +1,279 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf,len=278 +page_content='Vulnerability Analysis of Azure Blockchain Workbench Key Management System Dmitry Tanana Laboratory of Combinatorial Algebra Ural Federal University Yekaterinburg, Russia ddtanana@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='ru Abstract—With rise of blockchain popularity, more and more people seek to implement blockchain technology into their projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Most common way is to take existing blockchain stack, such as Azure Blockchain Workbench or Oracle Blockchain Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' While the blockchain technology is well-protected by its algorithms it is still vulnerable because its privacy relies on regular cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' And mistakes or vulnerabilities in key management protocols can affect even the most secure blockchain projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' This article considers question of vulnerabilities within Azure Blockchain Workbench key management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' We describe potential threats for each stage of key management lifecycle based on public reports and then assess how likely are those threats to realize within Azure Blockchain Workbench environment based on the technical documentation for Azure Blockchain Workbench and Azure Key Vault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Finally, we compile results of our assessment into the key management threat table with three distinct degrees of protection – fully protected, partially protected and not protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Keywords—Cryptography, key management, threat analysis, vulnerability analysis, Azure Blockchain Workbench I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' INTRODUCTION Nowadays one of the most discussed topics in the field of information technology are distributed ledger systems, also known as blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Such systems are being introduced around the world in many areas including medicine, education, finance and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' End user security in those systems relies primarily on cryptographic encryption, therefore the blockchain is only as secure as the encryption keys used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Indeed, many well-known incidents in which data from blockchain was somehow disclosed, happened only because of key mismanagement [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Therefore, the safety of information in distributed ledger systems directly depends on the key strength and the effectiveness of mechanisms and protocols associated with their management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Our paper will focus on the vulnerability analysis of blockchain key management systems on the example of Azure Blockchain Workbench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To the best of our knowledge, this would be the first analysis of this type for Azure Blockchain Workbench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' CRYPTOGRAPHIC KEY LIFECYCLE STAGES AND THREATS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Lifecycle In order to evaluate specific security measures, we must consider the whole lifecycle of cryptographic key within the distributed ledger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The cryptographic key lifecycle is a set of states through which the key passes during its existence in automated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Cryptographic key lifecycle can be divided into following stages [2], as seen on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' 1: Generation – key is generated with required cryptographic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Registration – key is registered in the database and issuing key certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Distribution – keys are distributed to the owners via secure channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' User initialization – user is authenticated for the purpose of key reception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Operation – keys are used to protect the information during regular operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Update – key is replaced with a new one after a certain period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Recovery – key is restored after being lost, if it was lost without compromise possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Cancellation – key is revoked as a result of expiration or compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Storage and archiving – on this stage keys are stored in proper conditions, ensuring their safety until replacement, discontinued keys are archived and stored until their complete removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Erasure – key is completely deleted from the system after expiration or compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Block diagram of a blockchain cryptographic key lifecycle stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' This work was supported by the Ministry of Science and Higher Education of the Russian Federation, project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' FEUZ-2020-0016 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Generation Registration User initialization Distribution Operation Update Recovery Cancellation Storage & archiving ErasureB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Threats We’ll consider potential threats for each of the stages based on public reports about implemented attacks or attempted attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' We propose the following structure for describing threats: Name of the threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Description of the threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Mitigation measures to reduce the likelihood of the threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The main threats at the generation stage: – Threat of generating weak cryptographic key [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' This threat is a result of unsuccessful random number generator choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' It can produce a number which will result in generation of weak cryptographic key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat, strength of cryptographic keys must match the value of data it protects, and the period of time during which the data is protected by that key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Cryptographic key must be long enough for its purpose and generated using high quality random number generator, ideally the one that collects entropy from a suitable hardware source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The main threats at the distribution and operation stages: – Threat of insecure distribution of cryptographic keys [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Key distribution is one of the most vulnerable stages in the whole lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' If this process is not secure enough, keys can easily become compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat only secure protocols should be used for the distribution of cryptographic keys and certain rules of conduct should be followed by end users (do not transfer keys by open channels, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' – Threat of key misuse [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' If cryptographic keys are used for tasks that do not correspond with its generation purpose, that may lead to problems with maintenance of required protection level protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat each cryptographic key should be generated for one specific purpose only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' – Threat of key reuse [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Key being reused after its expiration can lead to compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat reuse of cryptographic keys should not be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' For a new operation new cryptographic key should be generated instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' – Threat of using a single cryptographic key to encrypt large amounts of data [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Using one single cryptographic key to encrypt large amounts of data (at the same time or at multiple instances) can make this key more vulnerable for attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat a single cryptographic key should not be used for high amount of data encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' If necessary, a new cryptographic key should be generated instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' – Internal threats (attacker can have access to cryptographic keys) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' If attacker has constant access to the cryptographic keys, that access can be used for malicious purposes or keys can be transferred and/or sold to a third party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat, it is necessary to employ user authentication, dual control and separation of roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The main threats at the storage stage: – Threat of improper storage [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Any bypass attack with filtration of protected data can compromise cryptographic keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat cryptographic keys must never be stored along with data they are protecting (for example, on a same server, in a same database, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' – Threat of weak protection [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The cryptographic keys, even if they are stored only in server memory, may be vulnerable to compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat: regardless of the storage, keys must be stored only in encrypted form (exception: if they are stored in an environment which is protected from unauthorized access).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' – Threat of inaccessibility [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' If cryptographic keys are not available when they are required or are lost due to any malfunction, and the backup is not available, data encrypted with these keys may also be inaccessible or lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat storage location of cryptographic keys must be reliably protected from failure and backups should be kept for the keys themselves if they need to be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' – Threat of logging absence [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' If cryptographic key lifecycle is not fully recorded or logged it will be more difficult to determine when exactly the security incident happened, which will affect the provision of security information for other cryptographic keys in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat, it is necessary to keep full action logs for cryptographic keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The journal should be protected from third-party access and ideally stored on a distributed ledger so it would be impossible to remove entries from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The main threats at the key erasure stage: – Threat of leaving cryptographic keys intact after their expiration [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Keys which are not properly destroyed can be compromised and used by attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat cryptographic keys must be destroyed (safely removed without leaving any traces) after their expiration, if they are not clearly required for later use (for example, to decrypt data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' – Threat of manual management [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Manual management of cryptographic keys using paper or unsuitable electronic tools, such as Excel spreadsheets, and accompanying manual input of cryptographic keys will lead to mistakes that can make keys extremely vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' To mitigate this threat management of cryptographic keys should be entrusted to an automated system, which is designated for key management and corresponds to modern information security requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Overall block diagram of cryptographic key management system threats with corresponding stages can be seen on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Block diagram of a blockchain cryptographic key lifecycle threats with corresponding stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' After analyzing potential threats to cryptographic keys management process, we will consider them specifically for the Azure Blockchain Workbench [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' AZURE BLOCKCHAIN WORKBENCH SECURITY SPECIFICATIONS The Azure Blockchain Workbench key management system is based on the Azure Key Vault [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Blockchain participants are using Azure Key Vault or third-party certificate authority in order to create identifiers and associated certificates which are required for all nodes to carry out transactions in blockchain network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Here are the rules and recommendations described in Azure Blockchain Workbench documentation [8,9] for cryptographic key management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Generation Stage: Generation of public and private keys occurs according to the EC-HSM (Elliptic Curve - Hardware Security Module) algorithm with P-256 elliptic curve key in base version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' If the private key is lost and cannot be restored or updated, then it will be necessary to generate a new private key and register a new identity at the certificate authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=" Storage stage: Keys are stored in the browser cache when they are initially added to the participant's blockchain wallet so that the console could use them to control blockchain components." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Customers are encouraged to export keys and import them into their own cryptographic key management system in case they clear browser cache or switch browsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Customers are responsible for storage, backup and recovery of all exported keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' For generating and storing a cryptographic key Azure Blockchain Workbench also offers the use of HSM (Hardware Security Module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' It complies with the FIPS 140-2 Level 3 standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The blockchain network node is responsible for the correct configuration and operation of the HSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Also in the HSM case the following cryptographic key management rules are used [6]: The private key for the node should not be stored in browser cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The private key must be accessible from the HSM via proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' When registering administrator IDs of another node or client application in certification authority using the console, their private keys should not be stored in the HSM because they need the private key to operate in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' RESULTS AND CONCLUSION Based on the analysis of Azure Blockchain Workbench, technical documentation for Azure Blockchain Workbench and Azure Key Vault and the compiled description we will analyze threats for Azure Blockchain Workbench key management system and assess the degree of protection from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' In our study we’ll use the following degrees of protection: Fully protected – security measures of Azure Blockchain Workbench completely eliminate the possibility of threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Partially protected – security measures of Azure Blockchain Workbench partially (or completely, but not in all cases) eliminate the possibility of a threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The likelihood of a threat is still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Not protected – security measures of Azure Blockchain Workbench for this threat do not exist or do not fulfill their functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' High likelihood of a threat present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' The results of our threat analysis can be seen in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='KEY MANAGEMENT THREAT TABLE FOR AZURE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='BLOCKCHAIN WORKBENCH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='№ Threat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Security element of Azure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Blockchain key management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='system for this threat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Degree of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of generating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='weak cryptographic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Generation of public and private ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='keys occurs according to EC-HSM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='algorithm with P-256 elliptic curve ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='key in base version ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Fully ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of key misuse Not found ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of key reuse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Key management rules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Fully ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of using a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='single cryptographic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='key to encrypt large ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='amounts of data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Not found ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of improper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='storage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Key management system or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='certified Hardware Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Fully ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of weak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Key management system or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='certified Hardware Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Fully ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of insecure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='distribution of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='cryptographic keys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Key management system or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='certified Hardware Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Fully ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Cryptographic key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='lifecycle threats ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Distribution & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Storage stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Erasure stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='operation stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Weak key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Insecure key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Improper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Leaving keys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='storage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='intact after ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='expiration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Key misuse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Weak protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Manual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Key reuse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='inaccessibility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Using single key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='to encrypt large ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Absence of logs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='amounts of data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Internal threats8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Threat of leaving ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='cryptographic keys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='intact after their ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='expiration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Not found ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Not ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='protected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Internal threats ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='(attacker can have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='access to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='cryptographic keys) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Certified Hardware Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='Module system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' No protection if Hardware Security Module is not used Partially protected 10 Threat of inaccessibility Clients are responsible for storage, backup and emergency recovery of all exported keys Partially protected 11 Threat of logging absence Certified Hardware Security Module system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' No protection if Hardware Security Module is not used Partially protected 12 Threat of manual management Key management system or certified Hardware Security Module Fully protected As a result of the threat analysis for the Azure Blockchain Workbench, we were able to assess the degree of protection from various attack vectors within its key management system and identify potential threats for the information security of Azure Blockchain Workbench key management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' We’d like to underline that for many attack vectors protection relies solely on certified Hardware Security Module, which might be unavailable for broader user base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Additionally, for a few potential vulnerabilities – using single cryptographic key to encrypt large amounts of data and leaving cryptographic key intact after its expiration there were no security measures at all, which means that end users must be extra careful to prevent those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Identified threats can be used by Azure Blockchain developers to further mitigate them or by end users in their risk minimization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' REFERENCES [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Gaudry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content=' Golovnev, “Breaking the encryption scheme of the Moscow Internet voting system”, Available at: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FJT4oBgHgl3EQfiSxv/content/2301.11569v1.pdf'} +page_content='org/abs/1908.' metadata={'source': 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Polarimetric Reverberation +Mapping in Medium-Band Filters. +Universe 2023, 1, 0. https://doi.org/ +Academic Editor: Francesco Shankar, +Ascension Del Olmo, Paola Marziani +Received: 09 December 2022 +Revised: 31 December 2022 +Accepted: 09 January 2023 +Published: +Copyright: +© 2023 by the authors. +Licensee MDPI, Basel, Switzerland. +This article is an open access article +distributed +under +the +terms +and +conditions of the Creative Commons +Attribution (CC BY) license (https:// +creativecommons.org/licenses/by/ +4.0/). +universe +Article +Polarimetric Reverberation Mapping in Medium-Band Filters +Elena Shablovinskaya 1,* +, Luka ˇC. Popovi´c 2,3 +, Roman Uklein 1 +, Eugene Malygin 1 +, Dragana Ili´c 3,4 +, Stefano +Ciroi 5 +, Dmitry Oparin 1 +, Luca Crepaldi 5 +, Lyuba Slavcheva-Mihova 6 +, Boyko Mihov 6 +and Yanko Nikolov 6 +1 +Special Astrophysical Observatory of RAS, 369167 Nizhny Arkhyz, Russia; uklein@sao.ru (R.U.); +male@sao.ru (E.M.); doparin2@gmail.com (D.O.) +2 +Astronomical Observatory, Volgina 7, 11000 Belgrade, Serbia; lpopovic@aob.rs +3 +Department of Astronomy, University of Belgrade - Faculty of Mathematics, Studentski trg 16, Belgrade, Serbia; +dilic@matf.bg.ac.rs +4 +Humboldt Research Fellow, Hamburger Sternwarte, Universitat Hamburg, Gojenbergsweg 112, D-21029 Hamburg, +Germany; +5 +Dipartimento di Fisica e Astronomia, Università di Padova, 35122 Padova, Italy; stefano.ciroi@unipd.it (S.C.); +luca.crepaldi.1@phd.unipd.it (L.C.) +6 +Institute of Astronomy and NAO, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee Blvd., +1784 Sofia, Bulgaria; lslavcheva@nao-rozhen.org (L.S.-M.); bmihov@astro.bas.bg (B.M.); +ynikolov@nao-rozhen.org (Y.N.) +* +Correspondence: e.shablie@yandex.com +Abstract: Earlier, we suggested the “reload” concept of the polarimetric reverberation mapping of +active galactic nuclei (AGN), proposed for the first time more than 10 years ago. We have successfully +tested this approach of reverberation mapping of the broad emission line on the galaxy Mrk 6. It was +shown that such an idea allows one to look at the AGN central parsec structure literally in a new light. +However, the method originally assumed the use of spectropolarimetric observations, expensive in terms +of telescope time, and implemented on rare large telescopes. Currently, we propose an adaptation of +the polarimetric reverberation mapping of broad lines in medium-band filters following the idea of the +photometric reverberation mapping, when filters are selected so that their bandwidth is oriented to the +broad line and the surrounding continuum near. In this paper, we present the progress status of such +monitoring conducted jointly at the Special astrophysical observatory and Asiago Cima Ekar observatory +(OAPd/INAF) with support from Rozhen National Astronomical Observatory (NAO), some first results +for the most frequently observed AGNs Mrk 335, Mrk 509, and Mrk 817, and the discussion of the future +perspectives of the campaign. +Keywords: polarization; active galactic nuclei; reverberation mapping +1. Introduction +According to the unified model of active galactic nuclei (AGN) [1,2], the central parts of the +central machine are surrounded by a gas–dust region, the so-called dusty torus. The presence +of a dusty region is key to explaining the observed dichotomy of type 1 and type 2 AGN +(see [3]). Characteristics of dust surrounding AGN, e.g. its location and chemical composition +determine the accretion properties of the AGN. Thanks to high-angular-resolution observations +of local active galaxies in the infrared (IR) (e.g., [4–8]) and molecular lines (e.g., [9,10]) now it +becomes possible to obtain direct images of the dusty region, while in the optical range, this +structure is still unresolvable. The development of observational capabilities made it possible +to determine the geometry of the dusty region, which turned out to be different from the +Universe 2023, 1, 0. https://doi.org/10.3390/universe1010000 +https://www.mdpi.com/journal/universe +arXiv:2301.05267v1 [astro-ph.GA] 12 Jan 2023 + +BYUniverse 2023, 1, 0 +2 of 24 +toroidal (see [11] for a review), and also to move from the simple models of a clumpy [12] and +smooth-distributed [13] dusty “torus” to more complex ones ([7,14,15] etc.). +The equatorial scattering observed in the optical range in many central regions of type 1 +AGN [16–18] is also associated with the presence of a dusty region. This process is responsible +for the specific polarization signatures along the emission line profiles: an S-shaped swing in +the polarization angle and a dip in the polarization degree along the emission line profile [19], +which cannot be explained by any other polarization mechanisms in AGN. Scattering by parti- +cles of the medium (mainly electrons) occurs in the plane of rotation of the AGN at a distance of +Rsc, where the optical depth becomes greater than 1 [16,18,20]. Based on physical assumptions, +Rsc is consistent with the dust sublimation radius. In particular, [21–23] use IR measurements +as Rsc for estimations of supermassive black hole (SMBH) masses by spectropolarimetric data +of angle swings in broad lines. However, there are no direct observations of the equatorial +scattering region, and the regions observed in IR may be located farther from the AGN center +at a greater optical depth than the optical radiation scattering region. +Since the scattering and emitting regions are spatially separated, polarimetric reverberation +mapping can be used to determine their sizes. The simulation done by Goosmann et al. [24] +showed that the equatorially scattered polarized emission of the AGN must lag behind the +continuum emission. However, the first observational test for NGC 4151 showed Rsc < +RBLR [25]. Due to the use of broad-band filters covering mostly continuum, the polarization +observed in NGC 4151 was contributed not only by equatorial scattering but also by sources +of the polarized continuum, such as the accretion disk or the base of the jet. Shablovinskaya +et al. [26] revised the approach and proposed the idea of AGN reverberation mapping in +polarized broad lines. When using a polarized flux in an emission line with a continuum flux +subtracted from it, the influence of other polarization mechanisms is minimized, which allows +us to measure the time delay that occurs precisely due to scattering by the equatorial region. +The new approach was applied to the analysis of data from spectropolarimetric monitoring +of the Seyfert galaxy Mrk 6. The detected delay between the polarized emission in the broad +Hα line and the continuum at a wavelength of 5100 Å was about 100 days, which is close +to the theoretical value (∼115 days, [27]), but about two times less than the expected delay +according to the spatial estimate of the size of the dust region, obtained by IR-interferometry +(∼214 days, [5]). This discrepancy requires a detailed analysis, but without static reinforcement +based on monitoring other galaxies, it cannot clarify the physics of AGNs. +Spectropolarimetric monitoring, which initially underlay the broad line polarimetric rever- +beration method, requires not only a large amount of time on a large telescope but also the use +of a device equipped with this mode, which is implemented only in a few observatories. Small +telescopes are best suited for monitoring a large sample of objects. To adapt the technique for +small instruments, it was necessary to switch from spectroscopy to direct images, as was done +in the case of photometric reverberation mapping [28]1. Similarly, AGN reverberation mapping +in polarized broad lines at small telescopes can be implemented using image-polarimetry in +mid-band filters oriented to the emission line and continuum nearby. +In this paper, we consider the adaptation of the method of polarimetric reverberation +mapping of broad lines to observations with small telescopes and some preliminary results. The +paper structure is as follows. Section 2 describes the observational technique used on 1- and 2-m +class telescopes and the sample of the AGNs chosen for the first stage of the monitoring project. +In Section 3, the first results for the three most frequently observed AGNs—Mrk 335, Mrk +509, and Mrk 817—are given, which are then discussed and compared with other estimations +in Section 4. The perspectives of the observational approach are described in Section 5. The +summary of the current project state is in Section 6. + +Universe 2023, 1, 0 +3 of 24 +2. Observational Technique and Sample +Since the beginning of 2020, we have been conducting polarimetric monitoring of a sample +of type 1 AGN with equatorial scattering at the 1-m telescope Zeiss-1000 [30] of the Special +Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS), at the Copernico +1.82-m telescope of the Asiago-Cima Ekar Observatory and at 2-m telescope of Rozhen National +Astronomical Observatory (NAO). +On the Zeiss-1000 telescope of the SAO RAS at different times, we used two instruments +“StoP” and “MAGIC” using medium-band 250 Å-wide filters from the SED-set2 and 100 Å-wide +filters called Sy671 and Sy685. In 2020, observations were made on the photometer-polarimeter +“StoP” [31]. Using a double Wollaston prism [32] as a polarization analyzer, the device made +it possible to simultaneously registered four images in the detector plane corresponding to +electric vector oscillations in the directions 0◦, 45◦, 90◦ and 135◦ and, consequently, three Stokes +parameters I, Q, and U within one exposure. This method of observation makes it possible to +minimize the effect of atmospheric depolarization and increase the accuracy of polarimetric +observations (for more details, see [33]). In the polarimetry mode of the “StoP” device with +a CCD system (2k × 2k px) Andor iKon-L 936 [a detailed study of the detector is described +in 34] in the 2 × 2 binning mode the scale is 0′′.42/pix with the field of view (FoV) for each +direction of polarization 0′.9 × 6′.1. +Since the end of 2020, we have switched to a new device—the “MAGIC” multimode focal +reducer [34,35]. Retaining all the advantages of the predecessor instrument in the technique of +polarimetric observations (the linear polarization measurement accuracy is up to 0.1% for stellar +sources up to 16 mag.), the new device has a large FoV: a Wollaston quadrupole prism [36], +used as polarization analyzer, projects onto the CCD-detector 4 images of the input mask, +corresponding to the directions of oscillation of the electric vector 0◦, 90◦, 45◦ and 135◦, each +with a size of 6′.5 × 6′.5. This allows using several local standard stars in the field of an object +in differential polarimetry. When using the focal reducer with the same Andor iKon-L 936 CCD +system in the 1 × 1 binning mode, the scale was 0′′.45/pix. +At Asiago Cima Ekar observatory (OAPd/INAF), data were obtained using Hα, 671 and +680 filters (70 Å-, 100 Å-, and 100 Å-widths, respectively, and centred at 656, 671, and 680 +nm, respectively) and a double Wollaston prism as a polarization analyzer, placed inside the +Asiago Faint Object Spectrographic Camera3 (AFOSC) of the 1.82-m Copernico telescope. The +same technique allows simultaneously obtaining four images of the input mask in different +directions of polarization in the FoV 0′.8 × 9′.4 with Andor iKon-L 936 CCD system with a +scale of 0′′.51/pix in 2 × 2 binning mode. +The data at Rozhen National Astronomical Observatory (NAO) were obtained at the +2-m Ritchey–Chrétien–Coudé (RCC) telescope using the two-channel Focal Reducer Rozhen +(FoReRo-2) [37], equipped with a double Wollaston prism and a 2k × 2k px Andor iKon-L CCD +camera. The images were taken through IF642 and IF672 narrow-band filters (with 26 Å and 33 +Å FWHMs, centered at 6416 Å and 6719 Å, respectively). The four images have FoV of 50′′.7 +× 50′′.7 each and a scale of 0′′.994/px in the 2 × 2 binning mode used. However, due to the +infrequent observations and the small FoV not allowing one to apply the reduction technique +shown below, we have not used further this data for the time series analysis. +During each observational night, we received calibration images (flat frames for each filter, +bias) to correct data for additive and multiplicative errors. For each object, the series of images +(at least 7 frames in each filter) were taken, the exposure times depend on the object brightness, +and weather conditions and are usually ranged from 2 to 5 min. To correct statistics each frame +is processed independently, and statistical evaluation is made by averaging the random value +by robust methods giving its unbiased estimate. In this case, the polarimetric errors are the +standard deviation of the robust distribution. + +Universe 2023, 1, 0 +4 of 24 +AGN observations were accompanied by observations of polarized standard stars and +stars with zero polarization. Introducing the instrumental parameters KQ and KU, which +characterize the transmission of polarization channels, determined from observations of un- +polarized standard stars, as well as I0, I45, I90, and I135 as the intensity at four polarization +directions, we can measure three Stokes parameters: +I = I0 + I90KQ + I45 + I135KU +(1) +Q = I0 − I90KQ +I0 + I90KQ +(2) +U = I45 − I135KU +I45 + I135KU +(3) +Here and below, we use Q and U to denote the normalized Stokes parameters. +Then, the degree of linear polarization P and the polarization angle ϕ as: +P = +� +Q2 + U2 +(4) +ϕ = 1 +2 arctan +� +U +Q +� +(5) +The observation technique and data reduction are described in more detail in [33]. Note +here that the interstellar medium (ISM) polarization is corrected using only one local standard +star in the field of the AGN, which may introduce a slight bias in measured polarization +parameters. Yet, this bias is about to be small and stable within the monitoring campaign. +When the signal-to-noise ratio of the measured polarization in AGNs was small (σP/P ≳ +0.7, where σP is the error of the polarization degree P measurement), the polarization degree +was corrected for the polarization bias [38]: +Punbiased = P · +� +1 − (1.41 · σP/P)2. +(6) +However, >95% of obtained data is of high signal-to-noise ratio (σP/P < 0.7). +Over the past two years, we have concentrated on observations of the 6 brightest (12–15 +mag) objects in the sample (see Table 1) with equatorial scattering, confirmed by spectropo- +larimetric observations at the 6-m BTA SAO RAS. All type 1 AGNs are observed sequentially +in the polarimetry mode in several mid-band filters, the passbands of which are spectrally +oriented toward the emission of the broad Hα line and the continuum near the line. Note that +in all cases, it is the broad Hα line that we observe since the equatorial scattering effect is most +detectable there. The selection of filters for three objects from the sample Mrk 335, Mrk 509, +and Mrk 817 is shown in Figures 1 and 2. Depending on the available filter sets one filter or the +combination of two filters was used for obtaining the emission line flux. Observations were +carried out approximately once a month, depending on the weather conditions and according +to the allocated telescope time for the implementation of programs. + +Universe 2023, 1, 0 +5 of 24 +Table 1. AGN sample and calculated expected values of the polarization parameters (Q, U, P and ϕ) in +the filters used in the observations based on the data from [22]. It is important to note that the earlier +published data for the objects Mrk 335 and Mrk 79 have been corrected after more thorough processing. +Object +Filters +Q, % +U, % +P, % +ϕ, ◦ +Mrk 335 +SED675 +0.28 +−0.16 +0.32 +165.1 +SED650 +0.55 +−0.51 +0.75 +158.6 +680 +0.41 +−0.14 +0.43 +170.6 +671 +0.12 +−0.13 +0.18 +156.4 +Hα +0.40 +−0.34 +0.52 +159.8 +Mrk 817 +SED625 +−0.69 +−0.43 +0.81 +106.0 +Sy685 +−0.01 +−0.62 +0.62 +134.5 +Sy671 +−0.82 +0.36 +0.89 +78.1 +Table 1. Cont. +Object +Filters +Q, % +U, % +P, % +ϕ, ◦ +Mrk 6 +SED675 +0.16 +−0.66 +0.68 +141.8 +SED650 +0.44 +−0.62 +0.76 +152.7 +SED625 +0.33 +−0.67 +0.75 +148.1 +Mrk 79 +SED675 +−0.42 +0.02 +0.42 +88.6 +SED650 +−0.40 +0.04 +0.40 +87.1 +NGC 4151 +SED650 +−0.13 +0.18 +0.22 +62.9 +SED600 +−0.18 +0.24 +0.30 +63.4 +Mrk 509 +SED675 +0.74 +−0.63 +0.97 +159.8 +SED650 +0.63 +−0.60 +0.87 +158.2 + +Universe 2023, 1, 0 +6 of 24 +0 +5.0•104 +1.0•105 +1.5•105 +2.0•105 +Mrk 335 − Zeiss−100/MAGIC +0 +5.0•104 +1.0•105 +1.5•105 +2.0•105 +F, ADU +(1) +−4 +−2 +0 +2 +4 +Q, % +(2) +−4 +−2 +0 +2 +4 +U, % +(3) +−4 +−2 +0 +2 +4 +P, % +(4) +6300 +6400 +6500 +6600 +6700 +6800 +6900 +Wavelength, Å +100 +150 +200 +ϕ, deg +(5) +0 +5.0•104 +1.0•105 +1.5•105 +2.0•105 +Mrk 335 − 1.82m Asiago/AFOSC +0 +5.0•104 +1.0•105 +1.5•105 +2.0•105 +F, ADU +(1) +−4 +−2 +0 +2 +4 +Q, % +(2) +−4 +−2 +0 +2 +4 +U, % +(3) +−4 +−2 +0 +2 +4 +P, % +(4) +6300 +6400 +6500 +6600 +6700 +6800 +6900 +Wavelength, Å +100 +150 +200 +ϕ, deg +(5) +Figure 1. The spectropolarimetric data for Mrk 335 from [22] taken at 09 November 2013 with the +overplotted selected filters. Left: SED675 (dark green) and SED650 (light green) transmission curves are +given. In panels 2–5, the data of image-polarimetry obtained with Zeiss-1000/MAGIC 01 November +2022 are overplotted with black dots. Right: Hα (yellow), 671 (red), and 680 (purple) transmission +curves are given. In panels 2–5, the data of image-polarimetry obtained with AFOSC 31 October 2022 are +overplotted with black dots. In both figures: flux in ADU (1), the Stokes parameters Q (2) and U (3) in %, +the polarization degree P in % (4), the polarization angle ϕ in degrees (5). + +Universe 2023, 1, 0 +7 of 24 +0 +2.0•104 +4.0•104 +6.0•104 +8.0•104 +1.0•105 +1.2•105 +1.4•105 +Mrk 509 − Zeiss−1000/MAGIC +0 +2.0•104 +4.0•104 +6.0•104 +8.0•104 +1.0•105 +1.2•105 +1.4•105 +F, ADU +(1) +−4 +−2 +0 +2 +4 +Q, % +(2) +−4 +−2 +0 +2 +4 +U, % +(3) +−4 +−2 +0 +2 +4 +P, % +(4) +6300 +6400 +6500 +6600 +6700 +6800 +6900 +Wavelength, Å +100 +150 +200 +ϕ, deg +(5) +0 +1•104 +2•104 +3•104 +4•104 +5•104 +6•104 +Mrk 817 − Zeiss−1000/MAGIC +0 +1•104 +2•104 +3•104 +4•104 +5•104 +6•104 +F, ADU +(1) +−4 +−2 +0 +2 +4 +Q, % +(2) +−4 +−2 +0 +2 +4 +U, % +(3) +−4 +−2 +0 +2 +4 +P, % +(4) +6200 +6400 +6600 +6800 +7000 +Wavelength, Å +50 +100 +150 +ϕ, deg +(5) +Figure 2. The spectropolarimetric data for Mrk 509 (left) and Mrk 817 (right) from [22] taken at 21 +October 2014 and 29 May 2014, respectively, with the overplotted selected filters. Left: SED675 (dark +green) and SED650 (light green) transmission curves are given for Mrk 509. In panels 2-5, the data of +image-polarimetry obtained with Zeiss-1000/MAGIC 29 August 2021 are overplotted with black dots. +Right: Sy685 (dark green), Sy671 (medium green) and SED625 (light green) transmission curves are given +for Mrk 817. In panels, 2-5 the data of image-polarimetry obtained with Zeiss-1000/MAGIC 28 August +2021 are overplotted with black dots. In both figures: flux in ADU (1), the Stokes parameters Q (2) and U +(3) in %, the polarization degree P in % (4), the polarization angle ϕ in degrees (5). +We estimated the expected values of the polarization effect due to equatorial scattering in +the observations of all studied AGNs in medium-band filters for a broad line and continuum +based on the previously obtained spectropolarimetric data [22]. Since the transmittance of +medium-band filters is measured in the laboratory, we denote it as a filter’s response function +f ilter(ν) and multiply it by the spectral distribution of the polarization parameters ξν [here we + +Universe 2023, 1, 0 +8 of 24 +used Q(ν) and U(ν) in per cent] over the frequencies of the investigated AGNs, to determine +its expected values X (in terms of Q and U) in specific filters:4 +X = +� +ξν · f ilter(ν) · dν +� +f ilter(ν) · dν +(7) +The estimated values of Q and U are given in Table 1. The values of P and ϕ are calculated +using Equations (4) and (5). It is interesting to note that in the case when the observations +are carried out in two mid-band filters, one of which is oriented to the continuum, and the +second is so that the flux from the broad emission Hα line falls into it, the difference between +the normalized Stokes parameters between the continuum and the line is small and does not +exceed ∼0.3%, which is comparable to the linear polarization measurement error for AGN +(0.1−0.2% in good weather conditions). The difference in the degree of polarization in the +two filters is ∼0.1-−0.4%, and the difference in the polarization angle is no more than 10 +degrees. Thus, the swing seen in the spectropolarimetric observations could not be resolved by +photometric polarimetry in filters, but this could indicate a difference between the emission +line and continuum polarization parameters. Here, the configuration for the Mrk 817 object +deserves special attention, when a broad emission line is observed in two filters oriented to the +“blue” and “red” wings of its profile. For Mrk 817 (the spectrum of the object with overplotted +transmission curves of the filters used is shown in Figure 2 on the right), the difference between +the normalized Stokes parameters for the continuum and the line wings reaches ∼0.7%, and +the deviation of the polarization angle from its value in the continuum is ± 28◦. It should +be noted here that the Sy685 filter is also oriented to the atmospheric absorption B-band λ = +6860–6917 Å (and Table 1 shows calculations without correction for this band). Nevertheless, +the Mrk 817 case most clearly shows that using medium-band filters oriented to different wings +of the Hα broad line profile, we can trace the characteristic changes in the polarization angle +profile, the wavelength dependence of which acquires a characteristic S-shaped profile during +equatorial scattering on a gas–dust torus. +3. First results +We performed polarimetric observations of the AGN sample on the 1-m and 1.82-m +telescopes in 2020–2022. The weather and the time allocated within the schedules did not allow +us to observe objects with a high cadence, and the total amount of data on the light curves +does not exceed 25 epochs, even in the case of the most regularly observed objects. Such a +meagre amount of data does not allow us to get a reliable result yet. In this section, we will +consider the current status of monitoring of three objects—Mrk 335, Mrk 509, and Mrk 817. The +monitoring period and the number of epochs are mean values of the nonpolarized continuum, +and the broad line flux and mean polarization degree of the broad line are given in Table 2. +There also, we provide the measure of variability calculated using Equation (3) from [39]. The +full observational data are given in Appendix A. + +Universe 2023, 1, 0 +9 of 24 +Table 2. Mean values of nonpolarized continuum and broad line flux and mean polarization degree of +the broad line obtained for Mrk 335, Mrk 509, and Mrk 817 during the observations: (1) object name, +(2) monitoring period (dd/mm/yyyy), (3) the number of epochs, (4) mean continuum flux in mJy, (5) +variability measure of the continuum flux, (6) mean broad Hα line flux with the continuum subtracted, in +mJy, (7) variability measure of the broad line flux, (8) mean polarization degree of the broad line in %. +For Mrk 817, the upper values of Iline, Fline +var and Pline are for the “blue” line profile wing and the bottom +values are for the “red” wing. +Period +N +Icont, mJy +Fcont +var +Iline, mJy +Fline +var +Pline, % +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +Mrk 335 +9 September 2020–1 November 2022 +23 +90.2 ± 7.9 +0.101 +140.1 ± 9.9 +0.085 +0.9 ± 0.3 +Mrk 509 +26 May 2020–29 August 2021 +11 +12.2 ± 0.5 +0.018 +21.0 ± 2.2 +0.092 +1.9 ± 0.8 +Mrk 817 +14 December 2020–28 August 2021 +8 +23.7 ± 1.1 +0.036 +36.9 ± 1.3 +0.044 +1.9 ± 0.4 +23.3 ± 1.2 +0.072 +2.1 ± 0.8 +3.1. Mrk 335 +Mrk 335 (z = 0.025, RA 00 06 19.5 Dec +20 12 10.6 J2000) is a well-known narrow-line +Sy 1 galaxy. The signs of the equatorial scattering in broad lines were observed in Mrk 335 +spectropolarimetric data for the first time in [40]. The violent polarization angle swing along +the Hα line profile was confirmed in [22]; here, we present the same data obtained at 6-m BTA +telescope of SAO RAS in Figure 1. As it could be seen the polarization angle variations are of +about ± 50◦, yet the polarization degree changes relative to the continuum are minor. In Figure +1, left, the data of image-polarimetry obtained with Zeiss-1000/MAGIC 01 November 2022 are +overplotted; in Figure 1, right, the data of image-polarimetry obtained with AFOSC 31 October +2022 at Asiago observatory are given. Note here that in all the cases, only slight differences of +the polarization parameters between continuum and broad line bands are detected. +In total, 10 epochs of Mrk 335 polarimetric data were obtained with MAGIC and 13 epochs +with AFOSC. Unfortunately, due to the different brightness of the field stars in the filters used +in MAGIC and AFOSC, we were unable to use the same local standard. For the MAGIC data +reduction, we used the reference star [GKG2008] 5 nearby at a distance of ∼1′.3 from the source, +and the star TYC 1184-771-1 at a distance of ∼2′.5 for AFOSC data. The light curves polarized +and broad integral line fluxes and continuum flux are shown in Figure 3. For AFOSC data, +the broad line flux is the sum of fluxes measured in two filters (671 and 680). In all cases, the +fluxes are given in mJy. One can see that despite the broad line curves the continuum flux light +curves seem to behave in a different way in the data sets obtained with MAGIC and AFOSC. +That was the reason not to merge the light curves not to introduce the systematical errors in the +correlation analysis. + +Universe 2023, 1, 0 +10 of 24 +0 +1 +2 +3 +Mrk 335 − Zeiss−1000/MAGIC + 1.82m Asiago/AFOSC +0 +1 +2 +3 +Ip +line +110 +120 +130 +140 +150 +160 +170 +Iline +9200 +9400 +9600 +9800 +JD − 2450000 +60 +70 +80 +90 +100 +110 +Icont +AFOSC +MAGIC +Figure 3. Mrk 335 light curves. From top to bottom: polarized broad line flux Ip +line, integral broad line flux +Iline with subtracted continuum and integral continuum flux Icont. Fluxes are given in mJy. Red circles are +used to denote the AFOSC data, and green squares are for the MAGIC data. For AFOSC, the broad line +flux is the sum of fluxes measured in two filters (671 and 680). +Additionally, we have estimated the polarization of Mrk 335 using observations at Rozhen +observatory taken at 15 August 21. For IF642 oriented to the continuum P = 1.25 ± 0.26%, +and ϕ = 78◦.5 ± 6◦.0; for IF672 oriented to the Hα emission line P = 1.13 ± 0.13%, and +ϕ = 89◦.0 ± 3◦.3. Here, one can detect the slight difference of the absolute polarization level, +particularly seen in the polarization angle. However, due to the lack of the local standard +stars in the FoV of the source we are not able to correctly take into account the atmospheric +depolarization, ISM effects, etc. Moreover, the flux calibration is also absent which makes it +complicated to exclude the polarized continuum flux from the polarized line flux in a proper +way. Thus, these data illustrate the possibilities of the medium-band polarimetry at 2-m +telescope, yet it is not used for the further analysis. +To determine the time delay between the light curves, we performed a cross-correlation +analysis using two approaches. As the main analysis tool, we used the JAVELIN code [41– +43], widely used in AGN reverberation mapping campaigns. In Figure 4 the results of the +JAVELIN analysis of the time delay of the polarized broad line emission Ip +line relative to the +variable continuum flux Icont for AFOSC (red histogram) and MAGIC (green histogram) data +are presented. Additionally, we conducted a joint analysis of both light curves, combining them +so that one of the light curves is shifted in time relative to the other by more than the duration +of the entire monitoring period. The results of the analysis of this synthetic curve are shown in +Figure 4 in black and provide only additional information. Similarly, in Figure 5 the time delay +of nonpolarized broad line emission Iline relative to the continuum Icont is analyzed. Also, to +estimate the delay between the light curves, we used the code ZDCF [44,45]. Separately for the +MAGIC and AFOSC light curves, cross-correlation analysis using ZDCF did not show results +due to large uncertainties caused by a small number of points. The results of estimating the +delay between the combined synthetic curves are given in Figures 4 and 5 in grey. + +Universe 2023, 1, 0 +11 of 24 +Mrk 335: continuum vs polarized line emission +0 +100 +200 +300 +400 +500 +Time-lag, days +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +Histogram Density +-1.0 +-0.5 +0.0 +0.5 +1.0 +DCF +Figure 4. Time delay analysis of the polarized broad line emission Ip +line relative to the variable continuum +flux Icont for Mrk 335. Histograms show the results of JAVELIN analysis based on AFOSC data (red +histogram), MAGIC data (green histogram), and combined time-shifted data (black histogram). On the +left y-axis the frequency of occurrence of parameter values sets during MCMC sampling is shown. 5000 +sets of parameter values were used in the simulation. The grey curve shows the results of the ZDCF +analysis of the combined time-shifted data (values are given on the y-axis on the right). +Despite the number of epochs comparable to what we previously obtained for Mrk 6 in +spectropolarimetric mode [26], the analysis of the delay between Ip +line and Icont does not show +an unambiguous peak for Mrk 335. In Figure 4 it can be seen that the histogram of estimates of +time delays for AFOSC and MAGIC data is close, about 180 and 150 days, respectively, but +the peak of the time-lag distribution has an error of 25–40%. Synthetic data show two peaks +at 224 ± 24 days and 157 ± 18 days, where the given errors are formally calculated as the +standard deviations of the given Gaussian-like peaks. Here, the larger peak is definitely an +artifact, since it is repeated in the analysis of photometric data (Figure 5). The second peak +is ∼4 times weaker than the first one, but its position roughly coincides with other estimates. +Thus, we see the tendency of the Ip +line light curves to show a delay of about 150–180 days. +However, such a time lag is close to the half of year, which characterize the typical length of +the observational periods of the source, and is shorter than the gaps between these periods. +This might indicate the measured value as the analysis artefact. Additionally, we performed +data analysis of the time delay of Iline relatively to Icont to estimate RBLR if possible. JAVELIN +histograms for AFOSC and MAGIC data, as well as analysis of synthetic light curves by the +ZDCF method, indicate an estimated delay between 73 ± 18 and 87 ± 17 days. The AFOSC +data separately demonstrate a peak at the value of 27 ± 17 days, which is close to the cadence +of observations (about 1 time per month) and can be an artifact of analysis. + +Universe 2023, 1, 0 +12 of 24 +Mrk 335: continuum vs line emission +0 +100 +200 +300 +400 +500 +Time-lag, days +0 +500 +1000 +1500 +2000 +2500 +Histogram Density +-1.0 +-0.5 +0.0 +0.5 +1.0 +DCF +Figure 5. Time delay analysis of the broad line emission Iline relative to the variable continuum flux Icont +for Mrk 335. The coloured histograms and labels are the same as in Figure 4. +3.2. Mrk 509 +As well as in the case of Mrk 335, Mrk 509 (z = 0.035, RA 20 44 09.8 Dec −10 43 24.7 +J2000) was observed in spectropolarimetric mode firstly in [40] and later in [22]. The latter +data were used in Figure 2 (left). As can be seen in the figure, we selected for observations +two medium-band (FWHM = 250 Å) filters oriented to a broad line and a continuum near. The +overplotted image-polarimetry data was obtained with Zeiss-1000/MAGIC 29 August 2021. +As in the case of Mrk 335, Mrk 509 shows only a slight difference in the polarization parameters +between continuum and broad line bands, more detectable in the polarization angle variations. +As far as the object can be observed for only four months a year, in 2020–2021, we gained +only 11 epochs using Zeiss-1000/MAGIC. To reduce the data, we used the reference star TYC +5760-1396-1 nearby at a distance of ∼1′.5 from the source. The light curves are shown in Figure +6. For all measured fluxes Ip +line, Iline, and Icont the variability is observed, and the pattern +of Ip +line variations differs from other light curves. The curves show a large gap between the +observational epochs associated with the inability to observe the object evenly throughout the +year. +To estimate the time-delay in a broad polarized line, we applied the JAVELIN code to the +received data. It turned out that despite a small number of epochs, the analysis revealed an +unambiguous peak at 114+12.7 +−8.8 days (Figure 7). We have also applied the JAVELIN analysis to +the data taken only in 2020 excluding the epochs from 2021, and we have obtained the same +time-delay. This estimate corresponds to the size of the dusty region ∼0.1 pc. Note, however, +that the ZDCF analysis did not show a significant correlation. Additionally, we performed 2020 +data analysis of the time delay of Iline relatively to Icont to estimate RBLR following the Mrk +335 case. JAVELIN histogram is shown in Figure 8 demonstrating two clear peaks at 39 ± 5 +days and at 85 ± 11 days (which is approximately 39 ± 5 days × 2). Taking into account that +the median cadence of observations in 2020 is ∼16 days, we could not unambiguously make a +conclusion about the origin of the double-peaked correlation histogram. + +Universe 2023, 1, 0 +13 of 24 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Mrk 509 − Zeiss−1000/MAGIC +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Ip +line +14 +16 +18 +20 +22 +24 +26 +Iline +9000 +9100 +9200 +9300 +9400 +JD − 2450000 +8 +10 +12 +14 +16 +Icont +continuum +line +Figure 6. Mrk 509 light curves obtained with MAGIC. From top to bottom: polarized broad line flux Ip +line +and integral broad line flux Iline with subtracted continuum and integral continuum flux Icont. Fluxes are +given in mJy. +Mrk 509: continuum vs polarized line emission +0 +100 +200 +300 +400 +500 +Time−lag, days +0 +1000 +2000 +3000 +4000 +5000 +6000 +Histogram Density +Figure 7. Time delay analysis of the polarized broad line emission Ip +line relative to the variable continuum +flux Icont for Mrk 509. Histogram show the results of JAVELIN analysis based on MAGIC data. On y-axis +the frequency of occurrence of parameter values sets of during MCMC sampling is shown. 10000 sets of +parameter values were used in the simulation. The time-delay estimation equal to 114+12.7 +−8.8 days is shown +with the yellow vertical line. + +Universe 2023, 1, 0 +14 of 24 +Mrk 509: continuum vs line emission +0 +100 +200 +300 +400 +500 +Time−lag, days +0 +500 +1000 +1500 +2000 +2500 +3000 +Histogram Density +Figure 8. Time delay analysis of the broad line emission Iline relative to the variable continuum flux Icont +for Mrk 509. Histogram show the results of JAVELIN analysis based on MAGIC data. On y-axis the +frequency of occurrence of parameter values sets of during MCMC sampling is shown. 10,000 sets of +parameter values were used in the simulation. +3.3. Mrk 817 +Mrk 817 (z = 0.031, RA 14 36 22.1 +58 47 39.4 J2000) is a Sy 1.2 AGN, where the equatorial +scattering was discovered in [22]. The data published in that work is presented in Figure 2, +right, where the transmission curves of the two filters 100 Å-width oriented to “red” and “blue” +broad line wings and selected for monitoring are also shown. A broader (FWHM = 250 Å) +filter was chosen for continuum polarimetry. The spectropolarimetric data demonstrate small +changes in the polarization degree P and a violent switch of the polarization angle ϕ along the +line profile. In Figure 2, right, the data of Mrk 817 image-polarimetry obtained on the MAGIC +28 August 2021 device is also plotted. Comparing the values obtained in two filters oriented to +different wings of the lines, one can see a significant difference in the polarization parameters. +This indicates that the use of a similar filter configuration in image-polarimetry mode may be an +alternative approach for identifying signs of equatorial scattering using small-class telescopes +or large instruments for observations of faint AGNs where the spectropolarimetric data show +too low a signal-to-noise ratio. Note here that in Figure 2 for the Mrk 817 data, there are visible +differences in the polarization parameters between the observations of 2014 and 2021, especially +in Sy685 band. In this case, it is important to obtain newer spectropolarimetric data in order +to confirm whether such a difference is the result of the influence of external factors (e.g., the +variability of atmospheric B-band 6860–6917 Å) or internal changes in the spectrum of Mrk 817 +in polarized light. +During the Mrk 817 monitoring, 8 epochs of observations were obtained using the MAGIC +device in the period from December 2020 to August 2021. To reduce the data, we used a +reference star of comparable brightness in the field of the object (RA 14 36 06.7 +58 50 38.4 +J2000) at a distance of ∼3′.6 from the source. The data were obtained relatively evenly, once or +twice every two months. Unfortunately, during the monitoring period, Mrk 817 did not show + +Universe 2023, 1, 0 +15 of 24 +significant variability either in the continuum or in the broad line. The light curves of Mrk 817 +are given in Figure 9. It can be seen that Iline does not show differences between “red” and +“blue” wings in integral light (see the middle panel in Figure 9). However, in polarized light, +for several epochs of observations, both the difference in Ip +line between filters is visible (e.g., +in the epoch of 18 December 2020), and a violent change of Ip +line between epochs (e.g., 02 July +2021 and 07 July 2021, see the upper panel in Figure 9). The meagre amount of data with no +significant variability did not allow us to obtain any result in cross-correlation analysis. +0.4 +0.8 +1.2 +Mrk 817 - Zeiss-1000/MAGIC +0.4 +0.8 +1.2 +Ip +line +10 +20 +30 +40 +Iline +9200 +9250 +9300 +9350 +9400 +9450 +JD - 2450000 +16 +18 +20 +22 +24 +26 +28 +Icont +continuum +blue wing +red wing +Figure 9. Mrk 817 light curves obtained with MAGIC. From top to bottom: polarized broad line flux Ip +line +and integral broad line flux Iline with subtracted continuum and integral continuum flux Icont. Red and +blue dots in Ip +line and Iline light curves denote the fluxes of the “red” and “blue” broad line profile wings, +respectively. Fluxes are given in mJy. +4. Discussion +Since 2020 due to the lack of stable weather meeting our requirements in two observatories +involved in the project (SAO RAS and Asiago) we have not reached the desired cadence when +observing a sample of objects, and the total number of epochs obtained has reached 23 for only +one object. Despite this, we managed to obtain some first results for three sample objects Mrk +335, Mrk 509, and Mrk 817, presented in this paper. As these AGNs are studied in deep detail +in various multi-wavelength campaigns, here we discuss our results in comparison with the +measurements given in the literature to investigate if the provided estimations are reliable. +Mrk 335 was intensively studied in numerous reverberation mapping campaigns. The +BLR size RBLR was measured as 16.4+5.2 +−3.2 [46,47], 17.3+4.9 +−4.3 [48], 15.7+3.4 +−4.0 [49], 14.3 ± 0.7 [50,51], +10.6+1.7 +−2.9 [52], 17.0+2.5 +−3.2 lt days [53] in Hβ broad line and 20.5+2.0 +−2.8 lt days [28] in Hα broad line. +Given estimations of the BLR size are ∼10 times larger than the accretion disk size of ∼1 lt +day [54]. According to the scale relation from [22], Rsc ≃ 5.1RBLR, so Rsc for Mrk 335 could be +estimated as ∼70 lt days. IR reverberation mapping in K band provided RIR ≈ 166 lt days [55] +which is two times greater than a value obtained using the scale relation. Lyu et al. [56] found +the size of the dusty region in WISE W1 band RW1 ≈ 1300 lt days. According to the relation of +the dusty region sizes in different bands RK:RW1 = 0.6:1 given in the same paper, RK ≈ 770 lt +days which is much larger than other estimations and seems not to be reliable. Thus, the value +of the polarized emission line time lag is predicted to be in ∼70–170 days range. Throughout + +Universe 2023, 1, 0 +16 of 24 +our polarimetric reverberation mapping monitoring the polarized Hα emission showed a delay +of about 150–180 days, which is in good agreement with the predictions of the size of the dusty +region for Mrk 335. However, we prefer to refrain from the statement of such a result, primarily +due to the fact that such an estimation may be caused by a correlation artefact since it is close +to the value of 1/2 year. Moreover, as it was mentioned above, this estimate is longer than the +length observation periods of the source, but shorter than the gaps between them. In addition, +the maxima of cross-correlation functions have large uncertainties of 25–40%, which makes +them unconfident. Moreover, the results of the analysis of the corresponding photometric data +do not coincide with the known estimates of RBLR for Mrk 335. These facts give reasons to +doubt the sustainability of the results we have obtained so far. +Sy1 galaxy Mrk 509 was also studied in multiple campaigns covering the entire electromag- +netic spectrum (see, e.g., [57] for a review). The BLR size RBLR was measured as 76.7+6.3 +−6.0 [46,47] +and 79.6+6.1 +−5.4 lt days [49] in Hβ broad line. From our estimations, relying on the maximum value +from the double-peaked histogram in Figure 8 RBLR =85 ± 11 lt days which coincides with +the measurements given in the literature. However, we cannot still explain the existence of the +second estimation of the time delay being two times shorter. The estimations of RBLR predict a +very extended BLR region, which is much larger (∼40 times) than the accretion disk size of +∼2 lt day [58,59], on the one hand, and only ∼2 times less then IR reverberation mapping in +K band estimations RIRRM ≈ 131 lt days [55]. GRAVITY Collaboration et al. [60] resolved the +hot gas structure in Mrk 509 with VLTI/GRAVITY near-infrared interferometry and measured +the size of the dusty region RIRIF ≈ 296 ± 30 lt days. Using given RBLR measurements and the +scale relation from [22], Rsc ≈ 408 lt days. While RIRIF > RIRRM is usually predicted (see [26] +for references), Rsc should be less then dusty structures in AGN. Apparently, such controversial +measurements rise issues of the dusty region size. During our monitoring, we estimated Rsc ≈ +114+12.7 +−8.8 lt days, or ∼0.1 pc. A comparison with near-IR torus interferometric data [60] shows +that the equatorial scattering region is 2 times smaller than the radius of the dusty structure +in the IR band, which is similar to what we previously obtained for Mrk 6 [26]. However, our +estimate of Rsc, although consistent with the estimates of the size of the dusty region obtained +by two independent methods, is only ∼1.3–1.6RBLR. In general, all other estimates of the size of +structures inside the central Mrk 509 parsec obtained independently indicate that RBLR is most +likely overestimated, for example, due to the presence of outflows driven by AGN observed +for Mrk 509 (e.g., [61]). This reveals the necessity of more intensive homogeneous monitoring +in polarized and integral light. +Mrk 817 is in the focus of several vast monitoring campaigns, e.g., AGN STORM 2 [62]. +The BLR size RBLR was measured as 15.0+4.2 +−3.4 [46,47], 21.8+2.4 +−3.0 [49], 14.0+3.4 +−3.5 lt days[63] in Hβ +broad line, and 28.3+2.1 +−1.8, 26.8+2.8 +−2.5, 51.7+14.9 +−1.3 lt days simultaneously in Hβ, Hγ and FeII lines, +respectively [64]. Given estimations of the BLR size are ∼3-6 times larger than the accretion +disk size of ∼4.5 lt day [54]. Mrk 817 was observed within IR RM monitoring and the dusty +region size was estimated as RIRRM = 89 ± 9 lt days [55]. This coincides with the expected +estimates of Rsc using the scale relation Rsc ≈ 95 ± 15 lt days, which is predicted for our +measurements. Due to the insignificant variability of the polarized and non-polarized fluxes, +despite the monitoring period being two times longer than the expected time lag no result was +obtained for Mrk 817. However, the variability of blue and red wings of polarized broad Hα +line is intriguing enough to go on with the observations more intensively. +5. Future perspectives +The new approach of polarimetric reverberation mapping in broad lines looks promising +since it can provide additional information about the size of structures in AGN, and therefore, +better understand the nature of processes associated with accretion onto SMBH. As we have +shown in the given paper, the technique in medium-band filters together with one-shot differen- + +Universe 2023, 1, 0 +17 of 24 +tial polarimetry is suitable for small telescopes, yet needs a careful adaptation. It is important to +note that the polarimetry of faint polarized sources, in contrast to, e.g., differential photometry, +put high restrictions on permissible weather conditions. Even weak cirruses or a haze can +significantly depolarize the radiation of observed objects, and the variability of atmospheric +transparency between exposures significantly degrades the quality of data. Here we consider +several issues related to the adaptation of observations in the framework of monitoring. +1. Filter selection. At the beginning of the monitoring observations, we selected filters +from our existing sets (see Section 2), focusing on our experience of observations in the frame- +work of photometric reverberation mapping of AGN [65,66]. Because of this, we mainly aimed +to use pairs of 250 Å-width filters oriented to a broad line and a continuum near. However, +as shown in Table 1 via the convolution of spectropolarimetric data with filter transmission +and in Figures 1 and 2 using the example of image-polarimetric data for Mrk 335 and Mrk +509, this strategy does not always seem optimal. This is due to that since the variations of the +polarization parameters along the wavelength during equatorial scattering are small, and ϕ has +an S-shaped profile, even the 250 Å-width filter may be too broad, and the average value of the +line polarization in the filter, summing up by wavelengths, will not differ markedly from the +continuum. However, it is important to note that the differences are small when we consider +polarization normalized by intensity. When the polarized flux in the line is considered with the +subtracted polarized flux of the continuum, the behaviour of the variability in the broad line +begins to differ significantly in polarized and nonpolarized light. It corresponds to the fact that +we see this radiation coming from different regions of the AGN. This is what we observe in the +case of objects Mrk 335 and Mrk 509. Thus, it can be argued that even if a medium-band filter +covering the whole line profile is selected, the variable flux Ip +line is detectable. +In cases of bright AGNs, a more optimal strategy may be to choose narrower filters +oriented on different sides from the centre of a broad emission line. In our case, we were +able to implement this by using 100 Å-width filters for Mrk 817 monitoring. The light curves +we obtained are not yet sufficient to reveal the approach efficiency within the monitoring +framework. However, as was shown above, such a strategy is more suitable to check AGNs for +signs of equatorial scattering in polarized light efficiently using telescope time. +Another problem for us was the combination of observational data obtained using a +different set of filters. One can see this in the example of Mrk 335 observations, which were +carried out at the 1-m telescope of the SAO RAS and 1.82-m in Asiago. Having the same trend, +the variability of radiation, especially in the continuum, differs significantly between the data +obtained in 250 Å-width filter SED650 and in 70 Å-width Hα. It is unlikely that the reason for +this difference was the AGN emission lines being on the transmission edge of the SED650 filter, +e.g., the [FeX] 6374 Å line. The more likely reason may be that different reference stars were +used when processing the data sets obtained from MAGIC and AFOSC. In any case, the data +obtained require additional analysis. +2. Aperture selection and host-galaxy subtraction. Two objects presented in this article, +Mrk 335 and Mrk 509, are almost star-shaped sources. Their host-galaxies, which fitting can be +found in [49], have a small contribution to the optical band. In our observations, taking into +account the image quality, the profile of objects was indistinguishable from the profiles of stars +in the field. Thus, we were able to choose the size of the aperture for photometry so that the +signal-to-noise ratio was maximum. However, when the host galaxy is extended, the choice +of aperture is complicated, as the larger the aperture size, the more galactic flux is recorded, +lowering the contrast of the polarized radiation of the nucleus. For Mrk 817 polarimetry, a fixed +aperture size of ∼4” was chosen so that when processing data with different image quality +(data with a seeing better than 3” was used), the same contribution of the host-galaxy would +be inside the aperture. However, greater accuracy will be achieved if, when processing AGN +images with extended galaxies, a galaxy model is subtracted from the frames. If this is not so + +Universe 2023, 1, 0 +18 of 24 +critical in the case of Mrk 817, then for, e.g., NGC 4151, where the host-galaxy has a size >3′ and +the contrast of the nucleus is relatively small, subtraction of the galaxy fitting may be necessary +to construct the light curves. This should be the subject of a separate detailed check. +3. Cadence of observations. Currently, based on simulated AGN light curves, attempts +are being made to determine the optimal cadence for reverberation mapping observations. +Improving cadence leads to fewer artifacts in cross-correlation analysis, but requires a large +amount of telescope time. The upper limit for time resolution is the expected time delay since +when observations are made with a lower frequency, the observed variability will not be related. +For example, Kovaˇcevi´c et al. [67] offer ∼5-days cadence for estimates of the accretion disk +size of typically 1–10 lt days size using LSST. Woo et al. [68] suggested having a factor of 5 +or better time resolution for a given time lag. Due to the time allocation on the telescopes, +the typical cadence of our observations was planned to be about 1 month. Such a cadence +is close to optimal with the expected sizes of Rsc for Mrk 509; for Mrk 335 and Mrk 817, a +cadence of ∼20 days would be more effective. However, due to weather conditions, it turned +out to be impossible to conduct observations every month, so the real-time resolution is worse. +Moreover, our estimates show that it is important to simultaneously measure Rsc and RBLR +to improve the scale relation (which may, generally speaking, have a different appearance +for different objects). In this case, observations should be carried out at least 1 (preferably 2) +times a week. Such a cadence can be achieved using a telescope, observations on which are +fully oriented only for such a task. According to the adaptation of the method of reverberation +mapping of the polarized lines to observations on a 1-m-class telescope, such a project has +prospects for development. +6. Conclusions +We presented the first results of reverberation mapping in polarized broad lines conducted +at the 1-m telescope of SAO RAS and at 1.82-m at Astronomical observatory Asiago. Since 2020, +we obtained the first results for the three most frequently observed objects from our sample of +type 1 AGNs with equatorial scattering, namely, Mrk 335, Mrk 509, and Mrk 817. +• +For Mrk 335, the measured dusty region size is Rsc ∼ 150–180 lt days. This result coincides +with the values predicted concerning the several estimations of the dusty structure in +the IR band and measurements of RBLR via optical reverberation mapping campaigns. +However, due to the irregular observations, the monitoring is going on to check whether +our result is a cross-correlation artefact. +• +For Mrk 509, we obtained Rsc ≈ 114+12.7 +−8.8 lt days, or ∼0.1 pc. This is 2 times smaller than +the radius of the dusty structure in the IR band. +• +For Mrk 817, no result is obtained due to the low variability of the object during the +monitoring period. However, observations of the polarized flux in the two line profile +wings demonstrate a sharp variability between epochs as well as a significant difference in +the polarized flux in the two wings during one epoch. This shows the potential possibility +of recording the delay of a polarized signal of a broad line in different parts of its profile. +Author Contributions: Conceptualization, Elena Shablovinskaya and Luka ˇC. Popovi´c; Methodology, +Elena Shablovinskaya and Luka ˇC. Popovi´c; Software, Elena Shablovinskaya and Roman Uklein; Valida- +tion, Dragana Ili´c; Formal analysis, Luka ˇC. Popovi´c, Roman Uklein and Eugene Malygin; Investigation, +Elena Shablovinskaya and Eugene Malygin; Data curation, Elena Shablovinskaya, Dragana Ili´c, Ste- +fano Ciroi, Dmitry Oparin, Luca Crepaldi, Lyuba Slavcheva-Mihova, Boyko Mihov and Yanko Nikolov; +Writing—original draft, Elena Shablovinskaya; Writing—review & editing, Luka ˇC. Popovi´c, Eugene +Malygin and Dragana Ili´c; Visualization, Elena Shablovinskaya and Eugene Malygin; Supervision, Luka +ˇC. Popovi´c; Project administration, Elena Shablovinskaya. + +Universe 2023, 1, 0 +19 of 24 +Funding: E.S., E.M. and R.U. were supported by RFBR grant, project number 20-02-00048 while conduct- +ing observations on 1-m telescope of SAO RAS, reducing and analyzing the polarimetric data. L. ˇC.P., +and D.I. acknowledge funding provided by Astronomical Observatory (the contract 451-03-68/2022-14/ +200002) and by University of Belgrade-Faculty of Mathematics (the contract 451-03-68/2022-14/200104), +through the grants by the Ministry of Education, Science, and Technological Development of the Republic +of Serbia. L.S.M. and B.M. acknowledge the project “Reverberation mapping of quasars in polarized light” +within the agreement between Bulgarian Academy of Sciences and Serbian Academy of Sciences and +Arts, 2020-2022. D.I. acknowledges the support of the Alexander von Humboldt Foundation. +Institutional Review Board Statement: Not applicable. +Informed Consent Statement: Not applicable. +Data Availability Statement: The observational data underlying this article is available on request 1 yr +after the publication of this paper. +Acknowledgments: Observations with the SAO RAS telescopes are supported by the Ministry of Science +and Higher Education of the Russian Federation. The renovation of telescope equipment is currently +provided within the national project “Science and Universities”. +Conflicts of Interest: The authors declare no conflict of interest. + +Universe 2023, 1, 0 +20 of 24 +Appendix A +Table A1. Observed values of nonpolarized continuum and nonpolarized and polarized broad line flux +for Mrk 335, Mrk 509 and Mrk 817 : (1) date of observations (dd/mm/yyyy), (2) the same in Julian +form JD-2450000, (3) continuum flux in mJy, (4) the Stokes Q parameters of the continuum flux in %, (5) +the Stokes U parameters of the continuum flux in %, (6) broad Hα line flux with subtracted continuum, +in mJy, (7) the Stokes Q parameters of the line flux with subtracted continuum, in %, (8) the Stokes +U parameters of the line flux with subtracted continuum, in %, (9) polarized broad Hα line flux with +subtracted continuum, in mJy. For Mrk 817, the upper values of Iline, Qline, Uline and Ip +line are for the +“blue” line profile wing and the bottom values are for the “red” wing. The values are calculated by robust +average, and the errors are the robust standard deviation (see [69] for more details). +Date +JD +Icont +Qcont +Ucont +Iline +Qline +Uline +Ip +line +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +Mrk 335 (MAGIC) +20 September 20 +9112 +92.6 ± 0.1 +0.0 ± 0.1 +−0.4 ± 0.4 +157.1 ± 0.2 +0.8 ± 0.2 +−0.7 ± 0.1 +2.0 ± 0.2 +21 October 20 +9143 +88.1 ± 0.1 +−0.4 ± 0.1 +0.5 ± 0.2 +147.9 ± 0.1 +−0.8 ± 0.1 +0.1 ± 0.1 +1.5 ± 0.1 +25 October 20 +9147 +84.6 ± 0.1 +−0.5 ± 0.3 +0.9 ± 0.4 +151.8 ± 0.3 +−0.3 ± 0.1 +−0.2 ± 0.1 +0.4 ± 0.1 +19 November 20 +9172 +88.6 ± 0.1 +−0.5 ± 0.1 +−0.5 ± 0.2 +149.8 ± 0.1 +0.1 ± 0.1 +−0.1 ± 0.1 +0.7 ± 0.2 +14 December 20 +9197 +96.2 ± 0.2 +0.1 ± 0.2 +−0.5 ± 0.4 +150.1 ± 0.6 +1.3 ± 0.2 +−0.1 ± 0.4 +2.3 ± 0.1 +18 December 20 +9201 +97.4 ± 0.1 +−0.1 ± 0.1 +−0.8 ± 0.6 +149.0 ± 0.1 +0.2 ± 0.3 +0.1 ± 0.3 +1.4 ± 0.4 +29 August 21 +9455 +79.3 ± 0.1 +0.7 ± 0.1 +0.6 ± 0.2 +146.8 ± 0.1 +0.8 ± 0.1 +1.1 ± 0.1 +2.3 ± 0.1 +07 September 21 +9464 +77.7 ± 1.1 +0.2 ± 0.2 +0.3 ± 2.1 +145.4 ± 0.1 +0.4 ± 0.5 +−0.4 ± 0.1 +0.9 ± 0.1 +29 October 22 +9881 +75.8 ± 0.1 +0.8 ± 0.1 +−0.2 ± 0.6 +117.8 ± 0.6 +0.4 ± 0.1 +−0.7 ± 0.2 +1.5 ± 0.2 +01 November 22 +9884 +77.9 ± 0.1 +0.7 ± 0.1 +−1.1 ± 0.4 +118.0 ± 0.6 +1.0 ± 0.1 +−0.6 ± 0.1 +1.4 ± 0.1 +Mrk 335 (AFOSC) +09 September 20 +9101 +102.3 ± 0.1 +0.7 ± 0.1 +−0.4 ± 0.2 +153.7 ± 0.2 +0.0 ± 0.1 +0.5 ± 0.1 +0.8 ± 0.1 +07 October 20 +9129 +104.6 ± 0.1 +−0.2 ± 0.2 +0.0 ± 0.5 +148.7 ± 0.1 +0.6 ± 0.3 +−0.5 ± 0.4 +1.0 ± 0.1 +24 November 20 +9177 +102.8 ± 0.2 +0.2 ± 0.3 +0.0 ± 0.6 +143.7 ± 0.1 +0.6 ± 0.2 +0.2 ± 0.1 +0.9 ± 0.1 +25 November 20 +9178 +104.1 ± 0.1 +−0.2 ± 0.8 +−0.3 ± 0.8 +145.4 ± 0.2 +0.8 ± 0.1 +0.3 ± 0.4 +1.6 ± 0.1 +26 November 20 +9179 +103.8 ± 0.1 +0.1 ± 0.4 +−0.2 ± 0.7 +144.6 ± 0.2 +0.6 ± 0.1 +0.2 ± 0.2 +2.8 ± 0.1 +28 September 21 +9485 +93.9 ± 0.1 +0.6 ± 0.4 +−0.1 ± 0.5 +140.9 ± 0.1 +0.7 ± 0.2 +0.2 ± 0.3 +1.2 ± 0.2 +12 October 21 +9499 +92.2 ± 0.1 +0.3 ± 0.1 +−0.3 ± 0.5 +137.2 ± 0.1 +−0.3 ± 0.1 +0.6 ± 0.1 +0.8 ± 0.1 +13 October 21 +9500 +91.7 ± 0.4 +0.5 ± 0.1 +0.2 ± 0.3 +138.1 ± 0.1 +0.4 ± 0.1 +0.7 ± 0.1 +1.2 ± 0.1 +14 October 21 +9501 +92.5 ± 0.1 +0.4 ± 0.1 +0.1 ± 0.6 +137.0 ± 0.1 +0.3 ± 0.1 +0.3 ± 0.4 +0.9 ± 0.1 +13 December 21 +9561 +87.5 ± 0.1 +0.4 ± 0.1 +0.2 ± 0.9 +134.8 ± 0.3 +0.4 ± 0.2 +0.3 ± 0.4 +1.2 ± 0.3 +11 January 22 +9590 +85.2 ± 0.1 +0.5 ± 0.5 +−0.3 ± 0.9 +128.5 ± 0.1 +0.0 ± 0.1 +0.7 ± 0.1 +1.4 ± 0.1 +20 August 22 +9811 +77.6 ± 0.1 +0.9 ± 0.4 +0.2 ± 0.6 +119.3 ± 0.2 +0.4 ± 0.1 +0.3 ± 0.2 +0.8 ± 0.1 +31 October 22 +9883 +79.0 ± 0.1 +0.4 ± 0.2 +−0.3 ± 0.8 +116.3 ± 0.1 +0.9 ± 0.1 +0.3 ± 0.4 +1.1 ± 0.1 +Mrk 509 +26 May 20 +8995 +11.1 ± 0.1 +0.2 ± 0.1 +−0.4 ± 0.1 +18.9 ± 0.1 +2.3 ± 0.1 +0.4 ± 0.1 +0.4 ± 0.1 +21 June 20 +9021 +12.0 ± 0.1 +0.2 ± 0.1 +−0.1 ± 0.1 +18.5 ± 0.1 +3.1 ± 0.1 +1.3 ± 0.1 +0.6 ± 0.1 +01 July 20 +9031 +12.2 ± 0.1 +0.4 ± 0.1 +0.1 ± 0.3 +18.1 ± 0.1 +2.6 ± 0.1 +1.0 ± 0.1 +0.5 ± 0.1 +23 July 20 +9053 +10.7 ± 0.9 +0.4 ± 0.1 +0.6 ± 0.2 +19.8 ± 0.8 +2.6 ± 0.1 +0.4 ± 0.1 +0.5 ± 0.1 +23 August 20 +9084 +12.3 ± 0.1 +0.9 ± 0.1 +−0.6 ± 0.2 +19.1 ± 0.1 +2.9 ± 0.2 +0.0 ± 0.1 +0.5 ± 0.1 +27 August 20 +9088 +12.2 ± 0.1 +0.5 ± 0.2 +−0.3 ± 0.1 +19.2 ± 0.1 +1.9 ± 0.2 +−0.7 ± 0.1 +0.4 ± 0.1 +20 September 20 +9112 +13.0 ± 0.1 +−0.7 ± 0.1 +−0.4 ± 0.2 +23.6 ± 0.1 +0.1 ± 0.1 +−0.5 ± 0.1 +0.1 ± 0.1 +22 September 20 +9114 +12.3 ± 0.1 +−0.3 ± 0.1 +−1.4 ± 0.1 +23.0 ± 0.1 +−0.5 ± 0.1 +−1.0 ± 0.1 +0.2 ± 0.1 +21 October 20 +9143 +13.3 ± 0.1 +1.7 ± 0.1 +−0.3 ± 0.2 +22.4 ± 0.1 +1.8 ± 0.1 +−0.9 ± 0.1 +0.4 ± 0.1 +04 August 21 +9430 +12.4 ± 0.1 +−0.1 ± 0.1 +−1.0 ± 0.2 +23.7 ± 0.1 +0.0 ± 0.1 +−1.4 ± 0.1 +0.3 ± 0.1 +29 August 21 +9455 +12.6 ± 0.1 +1.0 ± 0.1 +−0.9 ± 0.2 +24.6 ± 0.1 +0.8 ± 0.1 +−0.7 ± 0.1 +0.2 ± 0.1 + +Universe 2023, 1, 0 +21 of 24 +Table A1. Cont. +Date +JD +Icont +Qcont +Ucont +Iline (blue) +Qline (blue) +Uline (blue) +Ip +line (blue) +Iline (red) +Qline (red) +Uline (red) +Ip +line (red) +Mrk 817 +14 December 20 +9197 +24.0 ± 0.1 +0.0 ± 0.1 +−0.6 ± 0.2 +34.7 ± 0.1 +−0.1 ± 0.1 +−1.5 ± 0.4 +0.7 ± 0.2 +24.2 ± 0.1 +−1.0 ± 0.5 +−0.7 ± 0.7 +0.4 ± 0.1 +18 December 20 +9201 +25.0 ± 0.1 +1.2 ± 0.1 +0.2 ± 0.1 +36.5 ± 0.1 +−1.1 ± 0.3 +−0.9 ± 0.5 +0.7 ± 0.1 +24.9 ± 0.1 +−2.1 ± 0.1 +−0.2 ± 0.1 +0.6 ± 0.1 +07 March 21 +9280 +25.3 ± 0.1 +−1.7 ± 0.1 +−1.0 ± 0.1 +38.9 ± 0.2 +−2.1 ± 0.3 +−0.1 ± 0.2 +0.9 ± 0.2 +25.5 ± 0.1 +−2.9 ± 0.1 +−0.6 ± 0.1 +0.8 ± 0.1 +08 March 21 +9281 +24.9 ± 0.1 +−1.9 ± 0.2 +−1.1 ± 0.1 +38.7 ± 0.1 +−1.9 ± 0.1 +−0.5 ± 0.1 +0.9 ± 0.1 +22.4 ± 2.5 +−1.7 ± 0.2 +−0.6 ± 0.6 +0.4 ± 0.1 +05 May 21 +9339 +23.6 ± 0.1 +−2.6 ± 0.2 +−0.8 ± 0.1 +38.2 ± 0.1 +−1.1 ± 0.1 +−0.5 ± 0.3 +0.4 ± 0.1 +23.5 ± 0.1 +−1.6 ± 0.6 +−2.1 ± 0.1 +0.3 ± 0.1 +02 July 21 +9397 +22.2 ± 0.1 +−2.4 ± 0.4 +1.4 ± 0.1 +36.7 ± 0.8 +−1.0 ± 0.3 +1.7 ± 0.4 +0.8 ± 0.1 +23.3 ± 0.2 +−2.8 ± 1.6 +2.7 ± 0.1 +0.7 ± 0.1 +07 July 21 +9402 +21.6 ± 0.1 +−1.4 ± 0.1 +−0.7 ± 0.1 +36.2 ± 1.3 +−1.0 ± 0.1 +−0.3 ± 0.1 +0.5 ± 0.1 +20.3 ± 2.0 +−1.6 ± 0.1 +−1.1 ± 0.1 +0.3 ± 0.1 +28 August 21 +9454 +23.2 ± 0.1 +−1.6 ± 0.8 +−0.6 ± 0.1 +35.4 ± 0.1 +0.1 ± 0.1 +−0.2 ± 0.3 +0.6 ± 0.1 +22.6 ± 0.1 +−2.0 ± 0.1 +−3.9 ± 0.1 +0.8 ± 0.2 +Notes +1 +It is worth remembering that it was the approach using narrow spectral bands in the photometric observations of Seyfert galaxies that +was used in the pioneering work Cherepashchuk and Lyutyi [29]. +2 +Details about the characteristics of medium-band filters can be found on the https://www.sao.ru/hq/lsfvo/devices/scorpio-2/ +filters_eng.html (accessed on 31 December 2022) +3 +Details describing the instrument can be found on the https://www.oapd.inaf.it/sede-di-asiago/telescopes-and-instrumentations/ +copernico-182cm-telescope/afosc (accessed on 31 December 2022). +4 +The multiplication of SED by the filter response function is usually called “convolution” in literature yet is not equal to the real +mathematical convolution. +References +1. +Antonucci, R. +Unified models for active galactic nuclei and quasars. +Annu. +Rev. +Astron. +Astrophys. +1993, 31, 473–521. +https://doi.org/10.1146/annurev.aa.31.090193.002353. +2. +Urry, C.M.; Padovani, P. +Unified Schemes for Radio-Loud Active Galactic Nuclei. +Publ. 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MDPI and/or the editor(s) disclaim responsibility for any injury to people or +property resulting from any ideas, methods, instructions or products referred to in the content. + diff --git a/z9E4T4oBgHgl3EQfyg29/content/tmp_files/load_file.txt b/z9E4T4oBgHgl3EQfyg29/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f617c818158a8e25b34981a30f10391e0f31070 --- /dev/null +++ b/z9E4T4oBgHgl3EQfyg29/content/tmp_files/load_file.txt @@ -0,0 +1,3044 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf,len=3043 +page_content='���������� ������� Citation: Shablovinskaya, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Popovi´c, L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Slavcheva-Mihova, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mihov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Polarimetric Reverberation Mapping in Medium-Band Filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='org/ Academic Editor: Francesco Shankar, Ascension Del Olmo, Paola Marziani Received: 09 December 2022 Revised: 31 December 2022 Accepted: 09 January 2023 Published: Copyright: © 2023 by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Licensee MDPI, Basel, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='org/licenses/by/ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' universe Article Polarimetric Reverberation Mapping in Medium-Band Filters Elena Shablovinskaya 1,* , Luka ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Popovi´c 2,3 , Roman Uklein 1 , Eugene Malygin 1 , Dragana Ili´c 3,4 , Stefano Ciroi 5 , Dmitry Oparin 1 , Luca Crepaldi 5 , Lyuba Slavcheva-Mihova 6 , Boyko Mihov 6 and Yanko Nikolov 6 1 Special Astrophysical Observatory of RAS, 369167 Nizhny Arkhyz, Russia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' uklein@sao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='ru (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' male@sao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='ru (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' doparin2@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='com (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=') 2 Astronomical Observatory, Volgina 7, 11000 Belgrade, Serbia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' lpopovic@aob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='rs 3 Department of Astronomy, University of Belgrade - Faculty of Mathematics, Studentski trg 16, Belgrade, Serbia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' dilic@matf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='bg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='rs 4 Humboldt Research Fellow, Hamburger Sternwarte, Universitat Hamburg, Gojenbergsweg 112, D-21029 Hamburg, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 5 Dipartimento di Fisica e Astronomia, Università di Padova, 35122 Padova, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' stefano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='ciroi@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='it (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='crepaldi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='it (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=') 6 Institute of Astronomy and NAO, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', 1784 Sofia, Bulgaria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' lslavcheva@nao-rozhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='org (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' bmihov@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='bas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='bg (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ynikolov@nao-rozhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='org (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=') Correspondence: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='shablie@yandex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='com Abstract: Earlier, we suggested the “reload” concept of the polarimetric reverberation mapping of active galactic nuclei (AGN), proposed for the first time more than 10 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' We have successfully tested this approach of reverberation mapping of the broad emission line on the galaxy Mrk 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It was shown that such an idea allows one to look at the AGN central parsec structure literally in a new light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, the method originally assumed the use of spectropolarimetric observations, expensive in terms of telescope time, and implemented on rare large telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Currently, we propose an adaptation of the polarimetric reverberation mapping of broad lines in medium-band filters following the idea of the photometric reverberation mapping, when filters are selected so that their bandwidth is oriented to the broad line and the surrounding continuum near.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In this paper, we present the progress status of such monitoring conducted jointly at the Special astrophysical observatory and Asiago Cima Ekar observatory (OAPd/INAF) with support from Rozhen National Astronomical Observatory (NAO), some first results for the most frequently observed AGNs Mrk 335, Mrk 509, and Mrk 817, and the discussion of the future perspectives of the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Keywords: polarization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' active galactic nuclei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' reverberation mapping 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Introduction According to the unified model of active galactic nuclei (AGN) [1,2], the central parts of the central machine are surrounded by a gas–dust region, the so-called dusty torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The presence of a dusty region is key to explaining the observed dichotomy of type 1 and type 2 AGN (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Characteristics of dust surrounding AGN, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' its location and chemical composition determine the accretion properties of the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Thanks to high-angular-resolution observations of local active galaxies in the infrared (IR) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', [4–8]) and molecular lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', [9,10]) now it becomes possible to obtain direct images of the dusty region, while in the optical range, this structure is still unresolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The development of observational capabilities made it possible to determine the geometry of the dusty region, which turned out to be different from the Universe 2023, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3390/universe1010000 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='mdpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='com/journal/universe arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='05267v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='GA] 12 Jan 2023 BYUniverse 2023, 1, 0 2 of 24 toroidal (see [11] for a review), and also to move from the simple models of a clumpy [12] and smooth-distributed [13] dusty “torus” to more complex ones ([7,14,15] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The equatorial scattering observed in the optical range in many central regions of type 1 AGN [16–18] is also associated with the presence of a dusty region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This process is responsible for the specific polarization signatures along the emission line profiles: an S-shaped swing in the polarization angle and a dip in the polarization degree along the emission line profile [19], which cannot be explained by any other polarization mechanisms in AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Scattering by parti- cles of the medium (mainly electrons) occurs in the plane of rotation of the AGN at a distance of Rsc, where the optical depth becomes greater than 1 [16,18,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Based on physical assumptions, Rsc is consistent with the dust sublimation radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In particular, [21–23] use IR measurements as Rsc for estimations of supermassive black hole (SMBH) masses by spectropolarimetric data of angle swings in broad lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, there are no direct observations of the equatorial scattering region, and the regions observed in IR may be located farther from the AGN center at a greater optical depth than the optical radiation scattering region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Since the scattering and emitting regions are spatially separated, polarimetric reverberation mapping can be used to determine their sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The simulation done by Goosmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' [24] showed that the equatorially scattered polarized emission of the AGN must lag behind the continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, the first observational test for NGC 4151 showed Rsc < RBLR [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Due to the use of broad-band filters covering mostly continuum, the polarization observed in NGC 4151 was contributed not only by equatorial scattering but also by sources of the polarized continuum, such as the accretion disk or the base of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Shablovinskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' [26] revised the approach and proposed the idea of AGN reverberation mapping in polarized broad lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' When using a polarized flux in an emission line with a continuum flux subtracted from it, the influence of other polarization mechanisms is minimized, which allows us to measure the time delay that occurs precisely due to scattering by the equatorial region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The new approach was applied to the analysis of data from spectropolarimetric monitoring of the Seyfert galaxy Mrk 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The detected delay between the polarized emission in the broad Hα line and the continuum at a wavelength of 5100 Å was about 100 days, which is close to the theoretical value (∼115 days, [27]), but about two times less than the expected delay according to the spatial estimate of the size of the dust region, obtained by IR-interferometry (∼214 days, [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This discrepancy requires a detailed analysis, but without static reinforcement based on monitoring other galaxies, it cannot clarify the physics of AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Spectropolarimetric monitoring, which initially underlay the broad line polarimetric rever- beration method, requires not only a large amount of time on a large telescope but also the use of a device equipped with this mode, which is implemented only in a few observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Small telescopes are best suited for monitoring a large sample of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' To adapt the technique for small instruments, it was necessary to switch from spectroscopy to direct images, as was done in the case of photometric reverberation mapping [28]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Similarly, AGN reverberation mapping in polarized broad lines at small telescopes can be implemented using image-polarimetry in mid-band filters oriented to the emission line and continuum nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In this paper, we consider the adaptation of the method of polarimetric reverberation mapping of broad lines to observations with small telescopes and some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The paper structure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Section 2 describes the observational technique used on 1- and 2-m class telescopes and the sample of the AGNs chosen for the first stage of the monitoring project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In Section 3, the first results for the three most frequently observed AGNs—Mrk 335, Mrk 509, and Mrk 817—are given, which are then discussed and compared with other estimations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The perspectives of the observational approach are described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The summary of the current project state is in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 3 of 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Observational Technique and Sample Since the beginning of 2020, we have been conducting polarimetric monitoring of a sample of type 1 AGN with equatorial scattering at the 1-m telescope Zeiss-1000 [30] of the Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS), at the Copernico 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='82-m telescope of the Asiago-Cima Ekar Observatory and at 2-m telescope of Rozhen National Astronomical Observatory (NAO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' On the Zeiss-1000 telescope of the SAO RAS at different times, we used two instruments “StoP” and “MAGIC” using medium-band 250 Å-wide filters from the SED-set2 and 100 Å-wide filters called Sy671 and Sy685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In 2020, observations were made on the photometer-polarimeter “StoP” [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Using a double Wollaston prism [32] as a polarization analyzer, the device made it possible to simultaneously registered four images in the detector plane corresponding to electric vector oscillations in the directions 0◦, 45◦, 90◦ and 135◦ and, consequently, three Stokes parameters I, Q, and U within one exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This method of observation makes it possible to minimize the effect of atmospheric depolarization and increase the accuracy of polarimetric observations (for more details, see [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In the polarimetry mode of the “StoP” device with a CCD system (2k × 2k px) Andor iKon-L 936 [a detailed study of the detector is described in 34] in the 2 × 2 binning mode the scale is 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='42/pix with the field of view (FoV) for each direction of polarization 0′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 × 6′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Since the end of 2020, we have switched to a new device—the “MAGIC” multimode focal reducer [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Retaining all the advantages of the predecessor instrument in the technique of polarimetric observations (the linear polarization measurement accuracy is up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1% for stellar sources up to 16 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ), the new device has a large FoV: a Wollaston quadrupole prism [36], used as polarization analyzer, projects onto the CCD-detector 4 images of the input mask, corresponding to the directions of oscillation of the electric vector 0◦, 90◦, 45◦ and 135◦, each with a size of 6′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 × 6′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This allows using several local standard stars in the field of an object in differential polarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' When using the focal reducer with the same Andor iKon-L 936 CCD system in the 1 × 1 binning mode, the scale was 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='45/pix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' At Asiago Cima Ekar observatory (OAPd/INAF), data were obtained using Hα, 671 and 680 filters (70 Å-, 100 Å-, and 100 Å-widths, respectively, and centred at 656, 671, and 680 nm, respectively) and a double Wollaston prism as a polarization analyzer, placed inside the Asiago Faint Object Spectrographic Camera3 (AFOSC) of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='82-m Copernico telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The same technique allows simultaneously obtaining four images of the input mask in different directions of polarization in the FoV 0′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 × 9′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 with Andor iKon-L 936 CCD system with a scale of 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='51/pix in 2 × 2 binning mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The data at Rozhen National Astronomical Observatory (NAO) were obtained at the 2-m Ritchey–Chrétien–Coudé (RCC) telescope using the two-channel Focal Reducer Rozhen (FoReRo-2) [37], equipped with a double Wollaston prism and a 2k × 2k px Andor iKon-L CCD camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The images were taken through IF642 and IF672 narrow-band filters (with 26 Å and 33 Å FWHMs, centered at 6416 Å and 6719 Å, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The four images have FoV of 50′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 × 50′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 each and a scale of 0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='994/px in the 2 × 2 binning mode used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, due to the infrequent observations and the small FoV not allowing one to apply the reduction technique shown below, we have not used further this data for the time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' During each observational night, we received calibration images (flat frames for each filter, bias) to correct data for additive and multiplicative errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For each object, the series of images (at least 7 frames in each filter) were taken, the exposure times depend on the object brightness, and weather conditions and are usually ranged from 2 to 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' To correct statistics each frame is processed independently, and statistical evaluation is made by averaging the random value by robust methods giving its unbiased estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In this case, the polarimetric errors are the standard deviation of the robust distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 4 of 24 AGN observations were accompanied by observations of polarized standard stars and stars with zero polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Introducing the instrumental parameters KQ and KU, which characterize the transmission of polarization channels, determined from observations of un- polarized standard stars, as well as I0, I45, I90, and I135 as the intensity at four polarization directions, we can measure three Stokes parameters: I = I0 + I90KQ + I45 + I135KU (1) Q = I0 − I90KQ I0 + I90KQ (2) U = I45 − I135KU I45 + I135KU (3) Here and below, we use Q and U to denote the normalized Stokes parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Then, the degree of linear polarization P and the polarization angle ϕ as: P = � Q2 + U2 (4) ϕ = 1 2 arctan � U Q � (5) The observation technique and data reduction are described in more detail in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Note here that the interstellar medium (ISM) polarization is corrected using only one local standard star in the field of the AGN, which may introduce a slight bias in measured polarization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Yet, this bias is about to be small and stable within the monitoring campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' When the signal-to-noise ratio of the measured polarization in AGNs was small (σP/P ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7, where σP is the error of the polarization degree P measurement), the polarization degree was corrected for the polarization bias [38]: Punbiased = P · � 1 − (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='41 · σP/P)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (6) However, >95% of obtained data is of high signal-to-noise ratio (σP/P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Over the past two years, we have concentrated on observations of the 6 brightest (12–15 mag) objects in the sample (see Table 1) with equatorial scattering, confirmed by spectropo- larimetric observations at the 6-m BTA SAO RAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' All type 1 AGNs are observed sequentially in the polarimetry mode in several mid-band filters, the passbands of which are spectrally oriented toward the emission of the broad Hα line and the continuum near the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Note that in all cases, it is the broad Hα line that we observe since the equatorial scattering effect is most detectable there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The selection of filters for three objects from the sample Mrk 335, Mrk 509, and Mrk 817 is shown in Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Depending on the available filter sets one filter or the combination of two filters was used for obtaining the emission line flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Observations were carried out approximately once a month, depending on the weather conditions and according to the allocated telescope time for the implementation of programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 5 of 24 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' AGN sample and calculated expected values of the polarization parameters (Q, U, P and ϕ) in the filters used in the observations based on the data from [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It is important to note that the earlier published data for the objects Mrk 335 and Mrk 79 have been corrected after more thorough processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Object Filters Q, % U, % P, % ϕ, ◦ Mrk 335 SED675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='32 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 SED650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='75 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='43 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='18 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 Hα 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='52 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 Mrk 817 SED625 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='81 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 Sy685 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='62 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 Sy671 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='89 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Object Filters Q, % U, % P, % ϕ, ◦ Mrk 6 SED675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='68 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 SED650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='76 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 SED625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='75 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 Mrk 79 SED675 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='42 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 SED650 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='40 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 NGC 4151 SED650 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='22 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 SED600 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='30 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 Mrk 509 SED675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='97 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 SED650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='87 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 Universe 2023, 1, 0 6 of 24 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5•105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 Mrk 335 − Zeiss−100/MAGIC 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5•105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 F, ADU (1) −4 −2 0 2 4 Q, % (2) −4 −2 0 2 4 U, % (3) −4 −2 0 2 4 P, % (4) 6300 6400 6500 6600 6700 6800 6900 Wavelength, Å 100 150 200 ϕ, deg (5) 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5•105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 Mrk 335 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='82m Asiago/AFOSC 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5•105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 F, ADU (1) −4 −2 0 2 4 Q, % (2) −4 −2 0 2 4 U, % (3) −4 −2 0 2 4 P, % (4) 6300 6400 6500 6600 6700 6800 6900 Wavelength, Å 100 150 200 ϕ, deg (5) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The spectropolarimetric data for Mrk 335 from [22] taken at 09 November 2013 with the overplotted selected filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Left: SED675 (dark green) and SED650 (light green) transmission curves are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In panels 2–5, the data of image-polarimetry obtained with Zeiss-1000/MAGIC 01 November 2022 are overplotted with black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Right: Hα (yellow), 671 (red), and 680 (purple) transmission curves are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In panels 2–5, the data of image-polarimetry obtained with AFOSC 31 October 2022 are overplotted with black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In both figures: flux in ADU (1), the Stokes parameters Q (2) and U (3) in %, the polarization degree P in % (4), the polarization angle ϕ in degrees (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 7 of 24 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2•105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4•105 Mrk 509 − Zeiss−1000/MAGIC 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0•105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2•105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4•105 F, ADU (1) −4 −2 0 2 4 Q, % (2) −4 −2 0 2 4 U, % (3) −4 −2 0 2 4 P, % (4) 6300 6400 6500 6600 6700 6800 6900 Wavelength, Å 100 150 200 ϕ, deg (5) 0 1•104 2•104 3•104 4•104 5•104 6•104 Mrk 817 − Zeiss−1000/MAGIC 0 1•104 2•104 3•104 4•104 5•104 6•104 F, ADU (1) −4 −2 0 2 4 Q, % (2) −4 −2 0 2 4 U, % (3) −4 −2 0 2 4 P, % (4) 6200 6400 6600 6800 7000 Wavelength, Å 50 100 150 ϕ, deg (5) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The spectropolarimetric data for Mrk 509 (left) and Mrk 817 (right) from [22] taken at 21 October 2014 and 29 May 2014, respectively, with the overplotted selected filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Left: SED675 (dark green) and SED650 (light green) transmission curves are given for Mrk 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In panels 2-5, the data of image-polarimetry obtained with Zeiss-1000/MAGIC 29 August 2021 are overplotted with black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Right: Sy685 (dark green), Sy671 (medium green) and SED625 (light green) transmission curves are given for Mrk 817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In panels, 2-5 the data of image-polarimetry obtained with Zeiss-1000/MAGIC 28 August 2021 are overplotted with black dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In both figures: flux in ADU (1), the Stokes parameters Q (2) and U (3) in %, the polarization degree P in % (4), the polarization angle ϕ in degrees (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' We estimated the expected values of the polarization effect due to equatorial scattering in the observations of all studied AGNs in medium-band filters for a broad line and continuum based on the previously obtained spectropolarimetric data [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Since the transmittance of medium-band filters is measured in the laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' we denote it as a filter’s response function f ilter(ν) and multiply it by the spectral distribution of the polarization parameters ξν [here we Universe 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 0 8 of 24 used Q(ν) and U(ν) in per cent] over the frequencies of the investigated AGNs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' to determine its expected values X (in terms of Q and U) in specific filters:4 X = � ξν · f ilter(ν) · dν � f ilter(ν) · dν (7) The estimated values of Q and U are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The values of P and ϕ are calculated using Equations (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It is interesting to note that in the case when the observations are carried out in two mid-band filters, one of which is oriented to the continuum, and the second is so that the flux from the broad emission Hα line falls into it, the difference between the normalized Stokes parameters between the continuum and the line is small and does not exceed ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3%, which is comparable to the linear polarization measurement error for AGN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2% in good weather conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The difference in the degree of polarization in the two filters is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1-−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4%, and the difference in the polarization angle is no more than 10 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Thus, the swing seen in the spectropolarimetric observations could not be resolved by photometric polarimetry in filters, but this could indicate a difference between the emission line and continuum polarization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Here, the configuration for the Mrk 817 object deserves special attention, when a broad emission line is observed in two filters oriented to the “blue” and “red” wings of its profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For Mrk 817 (the spectrum of the object with overplotted transmission curves of the filters used is shown in Figure 2 on the right), the difference between the normalized Stokes parameters for the continuum and the line wings reaches ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7%, and the deviation of the polarization angle from its value in the continuum is ± 28◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It should be noted here that the Sy685 filter is also oriented to the atmospheric absorption B-band λ = 6860–6917 Å (and Table 1 shows calculations without correction for this band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Nevertheless, the Mrk 817 case most clearly shows that using medium-band filters oriented to different wings of the Hα broad line profile, we can trace the characteristic changes in the polarization angle profile, the wavelength dependence of which acquires a characteristic S-shaped profile during equatorial scattering on a gas–dust torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' First results We performed polarimetric observations of the AGN sample on the 1-m and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='82-m telescopes in 2020–2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The weather and the time allocated within the schedules did not allow us to observe objects with a high cadence, and the total amount of data on the light curves does not exceed 25 epochs, even in the case of the most regularly observed objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Such a meagre amount of data does not allow us to get a reliable result yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In this section, we will consider the current status of monitoring of three objects—Mrk 335, Mrk 509, and Mrk 817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The monitoring period and the number of epochs are mean values of the nonpolarized continuum, and the broad line flux and mean polarization degree of the broad line are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' There also, we provide the measure of variability calculated using Equation (3) from [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The full observational data are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 9 of 24 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mean values of nonpolarized continuum and broad line flux and mean polarization degree of the broad line obtained for Mrk 335, Mrk 509, and Mrk 817 during the observations: (1) object name, (2) monitoring period (dd/mm/yyyy), (3) the number of epochs, (4) mean continuum flux in mJy, (5) variability measure of the continuum flux, (6) mean broad Hα line flux with the continuum subtracted, in mJy, (7) variability measure of the broad line flux, (8) mean polarization degree of the broad line in %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For Mrk 817, the upper values of Iline, Fline var and Pline are for the “blue” line profile wing and the bottom values are for the “red” wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Period N Icont, mJy Fcont var Iline, mJy Fline var Pline, % (1) (2) (3) (4) (5) (6) (7) (8) Mrk 335 9 September 2020–1 November 2022 23 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='101 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 Mrk 509 26 May 2020–29 August 2021 11 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='018 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='092 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 Mrk 817 14 December 2020–28 August 2021 8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='036 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='044 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='072 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 335 Mrk 335 (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='025, RA 00 06 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 Dec +20 12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 J2000) is a well-known narrow-line Sy 1 galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The signs of the equatorial scattering in broad lines were observed in Mrk 335 spectropolarimetric data for the first time in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The violent polarization angle swing along the Hα line profile was confirmed in [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' here, we present the same data obtained at 6-m BTA telescope of SAO RAS in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' As it could be seen the polarization angle variations are of about ± 50◦, yet the polarization degree changes relative to the continuum are minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In Figure 1, left, the data of image-polarimetry obtained with Zeiss-1000/MAGIC 01 November 2022 are overplotted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' in Figure 1, right, the data of image-polarimetry obtained with AFOSC 31 October 2022 at Asiago observatory are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Note here that in all the cases, only slight differences of the polarization parameters between continuum and broad line bands are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In total, 10 epochs of Mrk 335 polarimetric data were obtained with MAGIC and 13 epochs with AFOSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Unfortunately, due to the different brightness of the field stars in the filters used in MAGIC and AFOSC, we were unable to use the same local standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For the MAGIC data reduction, we used the reference star [GKG2008] 5 nearby at a distance of ∼1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 from the source, and the star TYC 1184-771-1 at a distance of ∼2′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 for AFOSC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The light curves polarized and broad integral line fluxes and continuum flux are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For AFOSC data, the broad line flux is the sum of fluxes measured in two filters (671 and 680).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In all cases, the fluxes are given in mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' One can see that despite the broad line curves the continuum flux light curves seem to behave in a different way in the data sets obtained with MAGIC and AFOSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' That was the reason not to merge the light curves not to introduce the systematical errors in the correlation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 10 of 24 0 1 2 3 Mrk 335 − Zeiss−1000/MAGIC + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='82m Asiago/AFOSC 0 1 2 3 Ip line 110 120 130 140 150 160 170 Iline 9200 9400 9600 9800 JD − 2450000 60 70 80 90 100 110 Icont AFOSC MAGIC Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 335 light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' From top to bottom: polarized broad line flux Ip line, integral broad line flux Iline with subtracted continuum and integral continuum flux Icont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Fluxes are given in mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Red circles are used to denote the AFOSC data, and green squares are for the MAGIC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For AFOSC, the broad line flux is the sum of fluxes measured in two filters (671 and 680).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Additionally, we have estimated the polarization of Mrk 335 using observations at Rozhen observatory taken at 15 August 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For IF642 oriented to the continuum P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='26%, and ϕ = 78◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 ± 6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' for IF672 oriented to the Hα emission line P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='13%, and ϕ = 89◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 ± 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Here, one can detect the slight difference of the absolute polarization level, particularly seen in the polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, due to the lack of the local standard stars in the FoV of the source we are not able to correctly take into account the atmospheric depolarization, ISM effects, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Moreover, the flux calibration is also absent which makes it complicated to exclude the polarized continuum flux from the polarized line flux in a proper way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Thus, these data illustrate the possibilities of the medium-band polarimetry at 2-m telescope, yet it is not used for the further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' To determine the time delay between the light curves, we performed a cross-correlation analysis using two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' As the main analysis tool, we used the JAVELIN code [41– 43], widely used in AGN reverberation mapping campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In Figure 4 the results of the JAVELIN analysis of the time delay of the polarized broad line emission Ip line relative to the variable continuum flux Icont for AFOSC (red histogram) and MAGIC (green histogram) data are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Additionally, we conducted a joint analysis of both light curves, combining them so that one of the light curves is shifted in time relative to the other by more than the duration of the entire monitoring period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The results of the analysis of this synthetic curve are shown in Figure 4 in black and provide only additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Similarly, in Figure 5 the time delay of nonpolarized broad line emission Iline relative to the continuum Icont is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Also, to estimate the delay between the light curves, we used the code ZDCF [44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Separately for the MAGIC and AFOSC light curves, cross-correlation analysis using ZDCF did not show results due to large uncertainties caused by a small number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The results of estimating the delay between the combined synthetic curves are given in Figures 4 and 5 in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 11 of 24 Mrk 335: continuum vs polarized line emission 0 100 200 300 400 500 Time-lag, days 0 500 1000 1500 2000 2500 3000 3500 Histogram Density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 DCF Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Time delay analysis of the polarized broad line emission Ip line relative to the variable continuum flux Icont for Mrk 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Histograms show the results of JAVELIN analysis based on AFOSC data (red histogram), MAGIC data (green histogram), and combined time-shifted data (black histogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' On the left y-axis the frequency of occurrence of parameter values sets during MCMC sampling is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 5000 sets of parameter values were used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The grey curve shows the results of the ZDCF analysis of the combined time-shifted data (values are given on the y-axis on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Despite the number of epochs comparable to what we previously obtained for Mrk 6 in spectropolarimetric mode [26], the analysis of the delay between Ip line and Icont does not show an unambiguous peak for Mrk 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In Figure 4 it can be seen that the histogram of estimates of time delays for AFOSC and MAGIC data is close, about 180 and 150 days, respectively, but the peak of the time-lag distribution has an error of 25–40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Synthetic data show two peaks at 224 ± 24 days and 157 ± 18 days, where the given errors are formally calculated as the standard deviations of the given Gaussian-like peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Here, the larger peak is definitely an artifact, since it is repeated in the analysis of photometric data (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The second peak is ∼4 times weaker than the first one, but its position roughly coincides with other estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Thus, we see the tendency of the Ip line light curves to show a delay of about 150–180 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, such a time lag is close to the half of year, which characterize the typical length of the observational periods of the source, and is shorter than the gaps between these periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This might indicate the measured value as the analysis artefact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Additionally, we performed data analysis of the time delay of Iline relatively to Icont to estimate RBLR if possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' JAVELIN histograms for AFOSC and MAGIC data, as well as analysis of synthetic light curves by the ZDCF method, indicate an estimated delay between 73 ± 18 and 87 ± 17 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The AFOSC data separately demonstrate a peak at the value of 27 ± 17 days, which is close to the cadence of observations (about 1 time per month) and can be an artifact of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 12 of 24 Mrk 335: continuum vs line emission 0 100 200 300 400 500 Time-lag, days 0 500 1000 1500 2000 2500 Histogram Density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 DCF Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Time delay analysis of the broad line emission Iline relative to the variable continuum flux Icont for Mrk 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The coloured histograms and labels are the same as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 509 As well as in the case of Mrk 335, Mrk 509 (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='035, RA 20 44 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 Dec −10 43 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 J2000) was observed in spectropolarimetric mode firstly in [40] and later in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The latter data were used in Figure 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' As can be seen in the figure, we selected for observations two medium-band (FWHM = 250 Å) filters oriented to a broad line and a continuum near.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The overplotted image-polarimetry data was obtained with Zeiss-1000/MAGIC 29 August 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' As in the case of Mrk 335, Mrk 509 shows only a slight difference in the polarization parameters between continuum and broad line bands, more detectable in the polarization angle variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' As far as the object can be observed for only four months a year, in 2020–2021, we gained only 11 epochs using Zeiss-1000/MAGIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' To reduce the data, we used the reference star TYC 5760-1396-1 nearby at a distance of ∼1′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The light curves are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For all measured fluxes Ip line, Iline, and Icont the variability is observed, and the pattern of Ip line variations differs from other light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The curves show a large gap between the observational epochs associated with the inability to observe the object evenly throughout the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' To estimate the time-delay in a broad polarized line, we applied the JAVELIN code to the received data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It turned out that despite a small number of epochs, the analysis revealed an unambiguous peak at 114+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 days (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' We have also applied the JAVELIN analysis to the data taken only in 2020 excluding the epochs from 2021, and we have obtained the same time-delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This estimate corresponds to the size of the dusty region ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Note, however, that the ZDCF analysis did not show a significant correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Additionally, we performed 2020 data analysis of the time delay of Iline relatively to Icont to estimate RBLR following the Mrk 335 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' JAVELIN histogram is shown in Figure 8 demonstrating two clear peaks at 39 ± 5 days and at 85 ± 11 days (which is approximately 39 ± 5 days × 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Taking into account that the median cadence of observations in 2020 is ∼16 days, we could not unambiguously make a conclusion about the origin of the double-peaked correlation histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 13 of 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 Mrk 509 − Zeiss−1000/MAGIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 Ip line 14 16 18 20 22 24 26 Iline 9000 9100 9200 9300 9400 JD − 2450000 8 10 12 14 16 Icont continuum line Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 509 light curves obtained with MAGIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' From top to bottom: polarized broad line flux Ip line and integral broad line flux Iline with subtracted continuum and integral continuum flux Icont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Fluxes are given in mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 509: continuum vs polarized line emission 0 100 200 300 400 500 Time−lag, days 0 1000 2000 3000 4000 5000 6000 Histogram Density Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Time delay analysis of the polarized broad line emission Ip line relative to the variable continuum flux Icont for Mrk 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Histogram show the results of JAVELIN analysis based on MAGIC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' On y-axis the frequency of occurrence of parameter values sets of during MCMC sampling is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 10000 sets of parameter values were used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The time-delay estimation equal to 114+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 days is shown with the yellow vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 14 of 24 Mrk 509: continuum vs line emission 0 100 200 300 400 500 Time−lag, days 0 500 1000 1500 2000 2500 3000 Histogram Density Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Time delay analysis of the broad line emission Iline relative to the variable continuum flux Icont for Mrk 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Histogram show the results of JAVELIN analysis based on MAGIC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' On y-axis the frequency of occurrence of parameter values sets of during MCMC sampling is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 10,000 sets of parameter values were used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 817 Mrk 817 (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='031, RA 14 36 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 +58 47 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 J2000) is a Sy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 AGN, where the equatorial scattering was discovered in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The data published in that work is presented in Figure 2, right, where the transmission curves of the two filters 100 Å-width oriented to “red” and “blue” broad line wings and selected for monitoring are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' A broader (FWHM = 250 Å) filter was chosen for continuum polarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The spectropolarimetric data demonstrate small changes in the polarization degree P and a violent switch of the polarization angle ϕ along the line profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In Figure 2, right, the data of Mrk 817 image-polarimetry obtained on the MAGIC 28 August 2021 device is also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Comparing the values obtained in two filters oriented to different wings of the lines, one can see a significant difference in the polarization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This indicates that the use of a similar filter configuration in image-polarimetry mode may be an alternative approach for identifying signs of equatorial scattering using small-class telescopes or large instruments for observations of faint AGNs where the spectropolarimetric data show too low a signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Note here that in Figure 2 for the Mrk 817 data, there are visible differences in the polarization parameters between the observations of 2014 and 2021, especially in Sy685 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In this case, it is important to obtain newer spectropolarimetric data in order to confirm whether such a difference is the result of the influence of external factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', the variability of atmospheric B-band 6860–6917 Å) or internal changes in the spectrum of Mrk 817 in polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' During the Mrk 817 monitoring, 8 epochs of observations were obtained using the MAGIC device in the period from December 2020 to August 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' To reduce the data, we used a reference star of comparable brightness in the field of the object (RA 14 36 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 +58 50 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 J2000) at a distance of ∼3′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The data were obtained relatively evenly, once or twice every two months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Unfortunately, during the monitoring period, Mrk 817 did not show Universe 2023, 1, 0 15 of 24 significant variability either in the continuum or in the broad line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The light curves of Mrk 817 are given in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It can be seen that Iline does not show differences between “red” and “blue” wings in integral light (see the middle panel in Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, in polarized light, for several epochs of observations, both the difference in Ip line between filters is visible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', in the epoch of 18 December 2020), and a violent change of Ip line between epochs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', 02 July 2021 and 07 July 2021, see the upper panel in Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The meagre amount of data with no significant variability did not allow us to obtain any result in cross-correlation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 Mrk 817 - Zeiss-1000/MAGIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 Ip line 10 20 30 40 Iline 9200 9250 9300 9350 9400 9450 JD - 2450000 16 18 20 22 24 26 28 Icont continuum blue wing red wing Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 817 light curves obtained with MAGIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' From top to bottom: polarized broad line flux Ip line and integral broad line flux Iline with subtracted continuum and integral continuum flux Icont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Red and blue dots in Ip line and Iline light curves denote the fluxes of the “red” and “blue” broad line profile wings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Fluxes are given in mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Discussion Since 2020 due to the lack of stable weather meeting our requirements in two observatories involved in the project (SAO RAS and Asiago) we have not reached the desired cadence when observing a sample of objects, and the total number of epochs obtained has reached 23 for only one object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Despite this, we managed to obtain some first results for three sample objects Mrk 335, Mrk 509, and Mrk 817, presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' As these AGNs are studied in deep detail in various multi-wavelength campaigns, here we discuss our results in comparison with the measurements given in the literature to investigate if the provided estimations are reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 335 was intensively studied in numerous reverberation mapping campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The BLR size RBLR was measured as 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 [46,47], 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 [48], 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 [49], 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 [50,51], 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 [52], 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 lt days [53] in Hβ broad line and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 lt days [28] in Hα broad line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Given estimations of the BLR size are ∼10 times larger than the accretion disk size of ∼1 lt day [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' According to the scale relation from [22], Rsc ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1RBLR, so Rsc for Mrk 335 could be estimated as ∼70 lt days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' IR reverberation mapping in K band provided RIR ≈ 166 lt days [55] which is two times greater than a value obtained using the scale relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' [56] found the size of the dusty region in WISE W1 band RW1 ≈ 1300 lt days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' According to the relation of the dusty region sizes in different bands RK:RW1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6:1 given in the same paper, RK ≈ 770 lt days which is much larger than other estimations and seems not to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Thus, the value of the polarized emission line time lag is predicted to be in ∼70–170 days range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Throughout Universe 2023, 1, 0 16 of 24 our polarimetric reverberation mapping monitoring the polarized Hα emission showed a delay of about 150–180 days, which is in good agreement with the predictions of the size of the dusty region for Mrk 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, we prefer to refrain from the statement of such a result, primarily due to the fact that such an estimation may be caused by a correlation artefact since it is close to the value of 1/2 year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Moreover, as it was mentioned above, this estimate is longer than the length observation periods of the source, but shorter than the gaps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In addition, the maxima of cross-correlation functions have large uncertainties of 25–40%, which makes them unconfident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Moreover, the results of the analysis of the corresponding photometric data do not coincide with the known estimates of RBLR for Mrk 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' These facts give reasons to doubt the sustainability of the results we have obtained so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Sy1 galaxy Mrk 509 was also studied in multiple campaigns covering the entire electromag- netic spectrum (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', [57] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The BLR size RBLR was measured as 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 [46,47] and 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 lt days [49] in Hβ broad line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' From our estimations, relying on the maximum value from the double-peaked histogram in Figure 8 RBLR =85 ± 11 lt days which coincides with the measurements given in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, we cannot still explain the existence of the second estimation of the time delay being two times shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The estimations of RBLR predict a very extended BLR region, which is much larger (∼40 times) than the accretion disk size of ∼2 lt day [58,59], on the one hand, and only ∼2 times less then IR reverberation mapping in K band estimations RIRRM ≈ 131 lt days [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' GRAVITY Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' [60] resolved the hot gas structure in Mrk 509 with VLTI/GRAVITY near-infrared interferometry and measured the size of the dusty region RIRIF ≈ 296 ± 30 lt days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Using given RBLR measurements and the scale relation from [22], Rsc ≈ 408 lt days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' While RIRIF > RIRRM is usually predicted (see [26] for references), Rsc should be less then dusty structures in AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Apparently, such controversial measurements rise issues of the dusty region size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' During our monitoring, we estimated Rsc ≈ 114+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 lt days, or ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' A comparison with near-IR torus interferometric data [60] shows that the equatorial scattering region is 2 times smaller than the radius of the dusty structure in the IR band, which is similar to what we previously obtained for Mrk 6 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, our estimate of Rsc, although consistent with the estimates of the size of the dusty region obtained by two independent methods, is only ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6RBLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In general, all other estimates of the size of structures inside the central Mrk 509 parsec obtained independently indicate that RBLR is most likely overestimated, for example, due to the presence of outflows driven by AGN observed for Mrk 509 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', [61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This reveals the necessity of more intensive homogeneous monitoring in polarized and integral light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 817 is in the focus of several vast monitoring campaigns, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', AGN STORM 2 [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The BLR size RBLR was measured as 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 [46,47], 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 [49], 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 lt days[63] in Hβ broad line, and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7+14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 lt days simultaneously in Hβ, Hγ and FeII lines, respectively [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Given estimations of the BLR size are ∼3-6 times larger than the accretion disk size of ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 lt day [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 817 was observed within IR RM monitoring and the dusty region size was estimated as RIRRM = 89 ± 9 lt days [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This coincides with the expected estimates of Rsc using the scale relation Rsc ≈ 95 ± 15 lt days, which is predicted for our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Due to the insignificant variability of the polarized and non-polarized fluxes, despite the monitoring period being two times longer than the expected time lag no result was obtained for Mrk 817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, the variability of blue and red wings of polarized broad Hα line is intriguing enough to go on with the observations more intensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Future perspectives The new approach of polarimetric reverberation mapping in broad lines looks promising since it can provide additional information about the size of structures in AGN, and therefore, better understand the nature of processes associated with accretion onto SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' As we have shown in the given paper, the technique in medium-band filters together with one-shot differen- Universe 2023, 1, 0 17 of 24 tial polarimetry is suitable for small telescopes, yet needs a careful adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It is important to note that the polarimetry of faint polarized sources, in contrast to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', differential photometry, put high restrictions on permissible weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Even weak cirruses or a haze can significantly depolarize the radiation of observed objects, and the variability of atmospheric transparency between exposures significantly degrades the quality of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Here we consider several issues related to the adaptation of observations in the framework of monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Filter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' At the beginning of the monitoring observations, we selected filters from our existing sets (see Section 2), focusing on our experience of observations in the frame- work of photometric reverberation mapping of AGN [65,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Because of this, we mainly aimed to use pairs of 250 Å-width filters oriented to a broad line and a continuum near.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, as shown in Table 1 via the convolution of spectropolarimetric data with filter transmission and in Figures 1 and 2 using the example of image-polarimetric data for Mrk 335 and Mrk 509, this strategy does not always seem optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This is due to that since the variations of the polarization parameters along the wavelength during equatorial scattering are small, and ϕ has an S-shaped profile, even the 250 Å-width filter may be too broad, and the average value of the line polarization in the filter, summing up by wavelengths, will not differ markedly from the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, it is important to note that the differences are small when we consider polarization normalized by intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' When the polarized flux in the line is considered with the subtracted polarized flux of the continuum, the behaviour of the variability in the broad line begins to differ significantly in polarized and nonpolarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It corresponds to the fact that we see this radiation coming from different regions of the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This is what we observe in the case of objects Mrk 335 and Mrk 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Thus, it can be argued that even if a medium-band filter covering the whole line profile is selected, the variable flux Ip line is detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In cases of bright AGNs, a more optimal strategy may be to choose narrower filters oriented on different sides from the centre of a broad emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In our case, we were able to implement this by using 100 Å-width filters for Mrk 817 monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The light curves we obtained are not yet sufficient to reveal the approach efficiency within the monitoring framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, as was shown above, such a strategy is more suitable to check AGNs for signs of equatorial scattering in polarized light efficiently using telescope time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Another problem for us was the combination of observational data obtained using a different set of filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' One can see this in the example of Mrk 335 observations, which were carried out at the 1-m telescope of the SAO RAS and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='82-m in Asiago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Having the same trend, the variability of radiation, especially in the continuum, differs significantly between the data obtained in 250 Å-width filter SED650 and in 70 Å-width Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' It is unlikely that the reason for this difference was the AGN emission lines being on the transmission edge of the SED650 filter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', the [FeX] 6374 Å line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The more likely reason may be that different reference stars were used when processing the data sets obtained from MAGIC and AFOSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In any case, the data obtained require additional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Aperture selection and host-galaxy subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Two objects presented in this article, Mrk 335 and Mrk 509, are almost star-shaped sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Their host-galaxies, which fitting can be found in [49], have a small contribution to the optical band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In our observations, taking into account the image quality, the profile of objects was indistinguishable from the profiles of stars in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Thus, we were able to choose the size of the aperture for photometry so that the signal-to-noise ratio was maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, when the host galaxy is extended, the choice of aperture is complicated, as the larger the aperture size, the more galactic flux is recorded, lowering the contrast of the polarized radiation of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For Mrk 817 polarimetry, a fixed aperture size of ∼4” was chosen so that when processing data with different image quality (data with a seeing better than 3” was used), the same contribution of the host-galaxy would be inside the aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, greater accuracy will be achieved if, when processing AGN images with extended galaxies, a galaxy model is subtracted from the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' If this is not so Universe 2023, 1, 0 18 of 24 critical in the case of Mrk 817, then for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', NGC 4151, where the host-galaxy has a size >3′ and the contrast of the nucleus is relatively small, subtraction of the galaxy fitting may be necessary to construct the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This should be the subject of a separate detailed check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Cadence of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Currently, based on simulated AGN light curves, attempts are being made to determine the optimal cadence for reverberation mapping observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Improving cadence leads to fewer artifacts in cross-correlation analysis, but requires a large amount of telescope time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The upper limit for time resolution is the expected time delay since when observations are made with a lower frequency, the observed variability will not be related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For example, Kovaˇcevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' [67] offer ∼5-days cadence for estimates of the accretion disk size of typically 1–10 lt days size using LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' [68] suggested having a factor of 5 or better time resolution for a given time lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Due to the time allocation on the telescopes, the typical cadence of our observations was planned to be about 1 month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Such a cadence is close to optimal with the expected sizes of Rsc for Mrk 509;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' for Mrk 335 and Mrk 817, a cadence of ∼20 days would be more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, due to weather conditions, it turned out to be impossible to conduct observations every month, so the real-time resolution is worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Moreover, our estimates show that it is important to simultaneously measure Rsc and RBLR to improve the scale relation (which may, generally speaking, have a different appearance for different objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' In this case, observations should be carried out at least 1 (preferably 2) times a week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Such a cadence can be achieved using a telescope, observations on which are fully oriented only for such a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' According to the adaptation of the method of reverberation mapping of the polarized lines to observations on a 1-m-class telescope, such a project has prospects for development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Conclusions We presented the first results of reverberation mapping in polarized broad lines conducted at the 1-m telescope of SAO RAS and at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='82-m at Astronomical observatory Asiago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Since 2020, we obtained the first results for the three most frequently observed objects from our sample of type 1 AGNs with equatorial scattering, namely, Mrk 335, Mrk 509, and Mrk 817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For Mrk 335, the measured dusty region size is Rsc ∼ 150–180 lt days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This result coincides with the values predicted concerning the several estimations of the dusty structure in the IR band and measurements of RBLR via optical reverberation mapping campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, due to the irregular observations, the monitoring is going on to check whether our result is a cross-correlation artefact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For Mrk 509, we obtained Rsc ≈ 114+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 lt days, or ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This is 2 times smaller than the radius of the dusty structure in the IR band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For Mrk 817, no result is obtained due to the low variability of the object during the monitoring period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' However, observations of the polarized flux in the two line profile wings demonstrate a sharp variability between epochs as well as a significant difference in the polarized flux in the two wings during one epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' This shows the potential possibility of recording the delay of a polarized signal of a broad line in different parts of its profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Author Contributions: Conceptualization, Elena Shablovinskaya and Luka ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Popovi´c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Methodology, Elena Shablovinskaya and Luka ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Popovi´c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Software, Elena Shablovinskaya and Roman Uklein;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Valida- tion, Dragana Ili´c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Formal analysis, Luka ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Popovi´c, Roman Uklein and Eugene Malygin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Investigation, Elena Shablovinskaya and Eugene Malygin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Data curation, Elena Shablovinskaya, Dragana Ili´c, Ste- fano Ciroi, Dmitry Oparin, Luca Crepaldi, Lyuba Slavcheva-Mihova, Boyko Mihov and Yanko Nikolov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Writing—original draft, Elena Shablovinskaya;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Writing—review & editing, Luka ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Popovi´c, Eugene Malygin and Dragana Ili´c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Visualization, Elena Shablovinskaya and Eugene Malygin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Supervision, Luka ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Popovi´c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Project administration, Elena Shablovinskaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 19 of 24 Funding: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' were supported by RFBR grant, project number 20-02-00048 while conduct- ing observations on 1-m telescope of SAO RAS, reducing and analyzing the polarimetric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' acknowledge funding provided by Astronomical Observatory (the contract 451-03-68/2022-14/ 200002) and by University of Belgrade-Faculty of Mathematics (the contract 451-03-68/2022-14/200104), through the grants by the Ministry of Education, Science, and Technological Development of the Republic of Serbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' acknowledge the project “Reverberation mapping of quasars in polarized light” within the agreement between Bulgarian Academy of Sciences and Serbian Academy of Sciences and Arts, 2020-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' acknowledges the support of the Alexander von Humboldt Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Institutional Review Board Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Informed Consent Statement: Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Data Availability Statement: The observational data underlying this article is available on request 1 yr after the publication of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Acknowledgments: Observations with the SAO RAS telescopes are supported by the Ministry of Science and Higher Education of the Russian Federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The renovation of telescope equipment is currently provided within the national project “Science and Universities”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Conflicts of Interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Universe 2023, 1, 0 20 of 24 Appendix A Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Observed values of nonpolarized continuum and nonpolarized and polarized broad line flux for Mrk 335,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Mrk 509 and Mrk 817 : (1) date of observations (dd/mm/yyyy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (2) the same in Julian form JD-2450000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (3) continuum flux in mJy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (4) the Stokes Q parameters of the continuum flux in %,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (5) the Stokes U parameters of the continuum flux in %,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (6) broad Hα line flux with subtracted continuum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' in mJy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (7) the Stokes Q parameters of the line flux with subtracted continuum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' in %,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (8) the Stokes U parameters of the line flux with subtracted continuum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' in %,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' (9) polarized broad Hα line flux with subtracted continuum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' in mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' For Mrk 817, the upper values of Iline, Qline, Uline and Ip line are for the “blue” line profile wing and the bottom values are for the “red” wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' The values are calculated by robust average, and the errors are the robust standard deviation (see [69] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Date JD Icont Qcont Ucont Iline Qline Uline Ip line (1) (2) (3) (4) (5) (6) (7) (8) (9) Mrk 335 (MAGIC) 20 September 20 9112 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 21 October 20 9143 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 25 October 20 9147 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 14 December 20 9197 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='3 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2 Notes 1 It is worth remembering that it was the approach using narrow spectral bands in the photometric observations of Seyfert galaxies that was used in the pioneering work Cherepashchuk and Lyutyi [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 2 Details about the characteristics of medium-band filters can be found on the https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='sao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='ru/hq/lsfvo/devices/scorpio-2/ filters_eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='html (accessed on 31 December 2022) 3 Details describing the instrument can be found on the https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='oapd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='it/sede-di-asiago/telescopes-and-instrumentations/ copernico-182cm-telescope/afosc (accessed on 31 December 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 4 The multiplication of SED by the filter response function is usually called “convolution” in literature yet is not equal to the real mathematical convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Antonucci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Unified models for active galactic nuclei and quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 1993, 31, 473–521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='aa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='090193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='002353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Marconi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Millour, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Petrov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='1365- 2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='07610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='E.' metadata={'source': 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not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 2022, in print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Afanasiev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ;' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Minezaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Yoshii, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Kobayashi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Sakata, Y.' 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Gelbord, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Cackett, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Peterson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Horne, K.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Kovaˇcevi´c, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Radovi´c, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Ili´c, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Popovi´c, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Assef, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'} +page_content=' Sánchez-Sáez, P.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E4T4oBgHgl3EQfyg29/content/2301.05267v1.pdf'}